text: 200 30+ 20 15 150 25 10 100 5 20 50 0 0 50 100 150 200 250 300 350 0 15 50 Average Error 40 0 95th Percentile Error 50 10 30 100 20 5 150 10 0 200 0 0 50 100 150 200 250 300 350 200 150 100 50 0 50 100 150 200 Number of Detected (Non-zero) Entries X (m) Fig. 6. Prediction error and sample count per number of detected non-zero Fig. 7. Average error per considered position, assuming binary data samples entries in a sample, considering binary data samples and a noise σ of 6 dB. As and a noise σ of 6 dB. Given that the transmitter is at the center of the image observed, when few entries are detected, the prediction error soars, explaining (red triangle), it is possible to verify that being in a NLOS position is not a the least accurate predictions shown in Fig. 5. constraint for the proposed system. information improve the system performance, the presence of [3] J. A. del Peral-Rosado, R. Raulefs, J. A. Lo´pez-Salcedo, and G. Seco- non-zero entries originating from other BSs should increase Granados, “Survey of cellular mobile radio localization methods: from the prediction accuracy, making the proposed system ﬁt for 1G to 5G,” IEEE Communications Surveys Tutorials, 2017. dense BS deployment schemes. [4] K. Witrisal, P. Meissner, E. Leitinger, Y. Shen, C. Gustafson, F. Tufves- Finally, in Fig. 7 the average prediction error per position son, K. Haneda, D. Dardari, A. F. Molisch, A. Conti, and M. Z. Win, “High-accuracy localization for assisted living: 5G systems will is shown, considering once again a binary dataset with a turn multipath channels from foe to friend,” IEEE Signal Processing noise σ of 6 dB. Comparing these results to Fig. 3, it is Magazine, vol. 33, no. 2, March 2016. possible to conclude that the system is always able to return a [5] M. Koivisto, A. Hakkarainen, M. Costa, P. Kela, K. Leppanen, and positioning prediction, as long as the mobile user is covered by M. Valkama, “High-efﬁciency device positioning and location-aware communications in dense 5G networks,” IEEE Communications Maga- the mmWave network. Moreover, it is interesting to notice that zine, vol. 55, no. 8, 2017. the prediction quality is not dictated by the received power: [6] X. Han, J. Wang, W. Shi, Q. Niu, and L. Xu, “Indoor precise positioning for instance, inside the building block at (75, 50), the mobile algorithm using 60GHz pulse based on compressive sensing,” Journal receiver is expected to have a weak signal and a high-quality of Mathematics and Computer Science, 2016. [7] Z. Wei, Y. Zhao, X. Liu, and Z. Feng, “DoA-LF: A location ﬁngerprint position estimate. This can be easily explained due to a unique positioning algorithm with millimeter-wave,” IEEE Access, vol. 5, 2017. data pattern, which is identiﬁed by the CNN. As a ﬁnal key [8] A. Shahmansoori, G. E. Garc´ıa, G. Destino, G. Seco-Granados, and remark, it is clearly observable that being in a NLOS position H. Wymeersch, “Position and orientation estimation through millimeter is not a constraint to obtain an accurate positioning prediction. wave MIMO in 5G systems,” CoRR, vol. abs/1702.01605, 2017. [Online]. Available: http://arxiv.org/abs/1702.01605 VI. CONCLUSIONS [9] V. Savic and E. G. Larsson, “Fingerprinting-based positioning in Throughout this paper, a system that is able to predict a de- distributed massive MIMO systems,” in 2015 IEEE 82nd Vehicular vice geolocation through mmWave transmissions and CNNs is Technology Conference (VTC2015-Fall), Sept 2015. proposed. This system will be able to function whenever there [10] J. Gante, G. Falca˜o, and L. Sousa, “Data-aided fast beamforming selection for 5G,” IEEE International Conference on Acoustics, Speech, is 5G coverage, and should require no additional hardware in and Signal Processing (ICASSP), 2018. devices that have mmWave reception capabilities. [11] Y. Azar, G. N. Wong, K. Wang, R. Mayzus, J. K. Schulz, H. Zhao, In the preliminary simulations performed, the average esti- F. Gutierrez, D. Hwang, and T. S. Rappaport, “28 GHz propagation measurements for outdoor cellular communications using steerable beam mation error can drop below 10 meters, in a scenario con- antennas in New York city,” in 2013 IEEE International Conference on taining mostly NLOS positions, outperforming the existing Communications (ICC), June 2013, pp. 5143–5147. algorithms for such positions. Therefore, by providing accurate [12] Y. Bengio, Y. LeCun, and G. Hinton, “Deep learning,” Nature, vol. 521, estimates for NLOS positions, where other sub-meter-accuracy pp. 436–444, 2015. mmWave positioning algorithms struggle, the proposed system [13] “Wireless InSite web-page,” https://www.remcom.com/ wireless-insite-em-propagation-software/, accessed: 9th of April, can be seen as a enabling component of the mmWave position- 2018. ing techniques ecosystem, enhancing the end-user experience. [14] “Open source new york 3D building model,” http://www1.nyc.gov/site/ [2] T. S. Rappaport, R. W. Heath, R. C. Daniels, and J. N. Murdock, doitt/initiatives/3d-building.page, accessed: 9th of April, 2018. Millimeter Wave Wireless Communications. Prentice Hall, 2014. [15] T. Obara et al., “Experiment of 28 ghz band 5g super wideband REFERENCES transmission using beamforming and beam tracking in high mobility environment,” in 2016 IEEE 27th Annual International Symposium on [1] P. Pirinen, “A brief overview of 5G research activities,” in 5G for Personal, Indoor, and Mobile Radio Communications (PIMRC), Sept Ubiquitous Connectivity (5GU), 2014 1st International Conference on, 2016. Nov 2014, pp. 17–22. )sdnasuohT( tnuoC elpmaS )m( rorrE )m( Y Average Error (m) ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

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text: error

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text: dB.

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text: that

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text: the

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text: the

text: image

text: observed,

text: when

text: few

text: entries

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text: detected,

text: the

text: prediction

text: error

text: soars,

text: explaining

text: (red

text: triangle),

text: it

text: is

text: possible

text: to

text: verify

text: that

text: being

text: in

text: a

text: NLOS

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text: not

text: a

text: the

text: least

text: accurate

text: predictions

text: shown

text: in

text: Fig.

text: constraint

text: for

text: the

text: proposed

text: system.

text: information

text: improve

text: the

text: system

text: performance,

text: the

text: presence

text: of

text: J.

text: A.

text: del

text: Peral-Rosado,

text: R.

text: Raulefs,

text: J.

text: A.

text: Lo´pez-Salcedo,

text: and

text: G.

text: Seco-

text: non-zero

text: entries

text: originating

text: from

text: other

text: BSs

text: should

text: increase

text: Granados,

text: “Survey

text: of

text: cellular

text: mobile

text: radio

text: localization

text: methods:

text: from

text: the

text: prediction

text: accuracy,

text: making

text: the

text: proposed

text: system

text: ﬁt

text: for

text: to

text: IEEE

text: Communications

text: Surveys

text: Tutorials,

text: dense

text: BS

text: deployment

text: schemes.

text: K.

text: Witrisal,

text: P.

text: Meissner,

text: E.

text: Leitinger,

text: Y.

text: Shen,

text: C.

text: Gustafson,

text: F.

text: Tufves-

text: Finally,

text: in

text: Fig.

text: the

text: average

text: prediction

text: error

text: per

text: position

text: son,

text: K.

text: Haneda,

text: D.

text: Dardari,

text: A.

text: F.

text: Molisch,

text: A.

text: Conti,

text: and

text: M.

text: Z.

text: Win,

text: “High-accuracy

text: localization

text: for

text: assisted

text: living:

text: systems

text: will

text: is

text: shown,

text: considering

text: once

text: again

text: a

text: binary

text: dataset

text: with

text: a

text: turn

text: multipath

text: channels

text: from

text: foe

text: to

text: friend,”

text: IEEE

text: Signal

text: Processing

text: noise

text: σ

text: of

text: dB.

text: Comparing

text: these

text: results

text: to

text: Fig.

text: it

text: is

text: Magazine,

text: vol.

text: no.

text: March

text: possible

text: to

text: conclude

text: that

text: the

text: system

text: is

text: always

text: able

text: to

text: return

text: a

text: M.

text: Koivisto,

text: A.

text: Hakkarainen,

text: M.

text: Costa,

text: P.

text: Kela,

text: K.

text: Leppanen,

text: and

text: positioning

text: prediction,

text: as

text: long

text: as

text: the

text: mobile

text: user

text: is

text: covered

text: by

text: M.

text: Valkama,

text: “High-efﬁciency

text: device

text: positioning

text: and

text: location-aware

text: communications

text: in

text: dense

text: networks,”

text: IEEE

text: Communications

text: Maga-

text: the

text: mmWave

text: network.

text: Moreover,

text: it

text: is

text: interesting

text: to

text: notice

text: that

text: zine,

text: vol.

text: no.

text: the

text: prediction

text: quality

text: is

text: not

text: dictated

text: by

text: the

text: received

text: power:

text: X.

text: Han,

text: J.

text: Wang,

text: W.

text: Shi,

text: Q.

text: Niu,

text: and

text: L.

text: Xu,

text: “Indoor

text: precise

text: positioning

text: for

text: instance,

text: inside

text: the

text: building

text: block

text: at

text: the

text: mobile

text: algorithm

text: using

text: pulse

text: based

text: on

text: compressive

text: sensing,”

text: Journal

text: receiver

text: is

text: expected

text: to

text: have

text: a

text: weak

text: signal

text: and

text: a

text: high-quality

text: of

text: Mathematics

text: and

text: Computer

text: Science,

text: Z.

text: Wei,

text: Y.

text: Zhao,

text: X.

text: Liu,

text: and

text: Z.

text: Feng,

text: “DoA-LF:

text: A

text: location

text: ﬁngerprint

text: position

text: estimate.

text: This

text: can

text: be

text: easily

text: explained

text: due

text: to

text: a

text: unique

text: positioning

text: algorithm

text: with

text: millimeter-wave,”

text: IEEE

text: Access,

text: vol.

text: data

text: pattern,

text: which

text: is

text: identiﬁed

text: by

text: the

text: CNN.

text: As

text: a

text: ﬁnal

text: key

text: A.

text: Shahmansoori,

text: G.

text: E.

text: Garc´ıa,

text: G.

text: Destino,

text: G.

text: Seco-Granados,

text: and

text: remark,

text: it

text: is

text: clearly

text: observable

text: that

text: being

text: in

text: a

text: NLOS

text: position

text: H.

text: Wymeersch,

text: “Position

text: and

text: orientation

text: estimation

text: through

text: millimeter

text: is

text: not

text: a

text: constraint

text: to

text: obtain

text: an

text: accurate

text: positioning

text: prediction.

text: wave

text: MIMO

text: in

text: systems,”

text: CoRR,

text: vol.

text: [Online].

text: Available:

text: VI.

text: CONCLUSIONS

text: V.

text: Savic

text: and

text: E.

text: G.

text: Larsson,

text: “Fingerprinting-based

text: positioning

text: in

text: Throughout

text: this

text: paper,

text: a

text: system

text: that

text: is

text: able

text: to

text: predict

text: a

text: de-

text: distributed

text: massive

text: MIMO

text: systems,”

text: in

text: IEEE

text: Vehicular

text: vice

text: geolocation

text: through

text: mmWave

text: transmissions

text: and

text: CNNs

text: is

text: Technology

text: Conference

text: Sept

text: proposed.

text: This

text: system

text: will

text: be

text: able

text: to

text: function

text: whenever

text: there

text: J.

text: Gante,

text: G.

text: Falca˜o,

text: and

text: L.

text: Sousa,

text: “Data-aided

text: fast

text: beamforming

text: selection

text: for

text: IEEE

text: International

text: Conference

text: on

text: Acoustics,

text: Speech,

text: is

text: coverage,

text: and

text: should

text: require

text: no

text: additional

text: hardware

text: in

text: and

text: Signal

text: Processing

text: (ICASSP),

text: devices

text: that

text: have

text: mmWave

text: reception

text: capabilities.

text: Y.

text: Azar,

text: G.

text: N.

text: Wong,

text: K.

text: Wang,

text: R.

text: Mayzus,

text: J.

text: K.

text: Schulz,

text: H.

text: Zhao,

text: In

text: the

text: preliminary

text: simulations

text: performed,

text: the

text: average

text: esti-

text: F.

text: Gutierrez,

text: D.

text: Hwang,

text: and

text: T.

text: S.

text: Rappaport,

text: GHz

text: propagation

text: measurements

text: for

text: outdoor

text: cellular

text: communications

text: using

text: steerable

text: beam

text: mation

text: error

text: can

text: drop

text: below

text: meters,

text: in

text: a

text: scenario

text: con-

text: antennas

text: in

text: New

text: York

text: city,”

text: in

text: IEEE

text: International

text: Conference

text: on

text: taining

text: mostly

text: NLOS

text: positions,

text: outperforming

text: the

text: existing

text: Communications

text: (ICC),

text: June

text: pp.

text: algorithms

text: for

text: such

text: positions.

text: Therefore,

text: by

text: providing

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text: Y.

text: LeCun,

text: and

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text: Hinton,

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text: positions,

text: where

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text: mmWave

text: positioning

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text: the

text: proposed

text: system

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text: accessed:

text: of

text: April,

text: can

text: be

text: seen

text: as

text: a

text: enabling

text: component

text: of

text: the

text: mmWave

text: position-

text: ing

text: techniques

text: ecosystem,

text: enhancing

text: the

text: end-user

text: experience.

text: “Open

text: source

text: new

text: york

text: building

text: model,”

text: T.

text: S.

text: Rappaport,

text: R.

text: W.

text: Heath,

text: R.

text: C.

text: Daniels,

text: and

text: J.

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text: of

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text: Wave

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text: Communications.

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text: Ubiquitous

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text: Conference

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text: Nov

text: pp.

text: )sdnasuohT(

text: tnuoC

text: elpmaS

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text: MCF WITH SURGERY FOR LOW ENTROPY HYPERSURFACES 12 (cid:90) λ(Σ) = sup F (Σ) = sup F (Σ) ∈R x0,r ∈ ×R x0,r x0 n+1,r>0 x0 B1(0,1) n,r<r1 = sup 1 e−|x−x0|2 dµ (3.1) ∈ ×R (4πr)n 4r x x0 B1(0,1) n,r<r1 Σ 2 namely the entropy will only be approximated by F-functionals on a bounded range of scales. ∈ × R Proof. That the sup only needs to be taken with x B1(0, 1) n follows 0 from lemma 2.5 above and that the surface is supported in this solid round cylinder. By the entropy bound, we can get a uniform Euclidean volume bound on the ∩ ≤ (cid:90) surface Σ. Vol(Σ Bn+1(p, r)) Crn for any p, r and C is a universal constant. The lemma follows if we can show lim sup 1 e−|x−x0|2 dµ = 0 r→∞ Rn+1 Σ (4πr)n2 4r x By breaking the integral up into integration on concentric annuli, it can be esti- (cid:90) mated by: (cid:90) 1 e−|x−x0|2 dµ (4πr)n 4r x Σ 2 = (cid:88) (cid:90) 1 e−|x−x0|2 dµ (4π)n 4 x Σ 2 ∞r = (cid:88) (cid:18)Σ∩[(cid:19)Bn+1(0,k)\Bn+1(0,k−1)] (4π1)n2 e−|x−4x0|2 dµx (3.2) k∞=1 r ≤ C n · − − (cid:101) e (k 1)2/4 r k=0 C = rn → 0 → ∞ ∩ \ − ≤ as r . The volume bound Vol(Σ [Bn+1(0, k) Bn+1(0, k 1)]) C is r r (cid:3) because after rescaling Σ is contained in a round solid cylinder of radius 1. r r ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: MCF

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text: λ(Σ)

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text: R

text: Proof.

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text: cylinder.

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text: the

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text: surface

text: Σ.

text: Vol(Σ

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text: Crn

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text: a

text: universal

text: constant.

text: The

text: lemma

text: follows

text: if

text: we

text: can

text: show

text: lim

text: sup

text: dµ

text: r→∞

text: Σ

text: x

text: By

text: breaking

text: the

text: integral

text: up

text: into

text: integration

text: on

text: concentric

text: annuli,

text: it

text: can

text: be

text: esti-

text: mated

text: by:

text: dµ

text: x

text: Σ

text: dµ

text: x

text: Σ

text: ∞r

text: dµx

text: r

text: C

text: n

text: e

text: (k

text: r

text: C

text: rn

text: as

text: r

text: The

text: volume

text: bound

text: Vol(Σ

text: k)

text: k

text: C

text: is

text: r

text: r

text: because

text: after

text: rescaling

text: Σ

text: is

text: contained

text: in

text: a

text: round

text: solid

text: cylinder

text: of

text: radius

text: r

text: r

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: MCF WITH SURGERY FOR LOW ENTROPY HYPERSURFACES 18 So, taking s = · 1 << s , depending on H , and relabeling s = s we have ∩ (cid:15)1 Hcan((cid:15)1) 2 can 2 M B(x, s ) is (cid:15) close to a round cylinder after rescaling (with our previous choice 2 1 of s , instead this was true in every neighborhood of every point) when H = H 2 can somewhere within the ball. Of course, we continue for the time being to assume our stipulations (1) - (3) listed above. (cid:82) By properties of Gaussian distribution,√there exists a universal constant N >> 1 independent of scale or basepoint so that· if r < 4Ns2 , then M∩Bc(x0,r) (4π1r)n2 e−|x−4xr0|2 < (cid:15) . By making (cid:15) small, we can force s2 Hcan = 1· >> 100. 3 3 N N (cid:15)3 √ Now we estimate F (M) for some point x on the ﬂow with H(x) > H , and x0,r ≤ can we want to show that if r c = s2 , then the F functionals with centers at x 0 N 0 must be small in the sense they can be bounded above by Λ − + (cid:15) + (cid:15) . We note n 1 1 2 the following consequnece of the continunity of the ﬂow and that on the initial time slice, H < H ((cid:15) ) (for (cid:15) small enough): can 1 1 (cid:83) Lemma 3.7. Those points with mean curvature H(x, t) > H must be covered can by union of balls [B(x˜, s )]. 2 ≤ x˜,H(x˜,t˜)=Hcanfor some t˜ t ≥ ≤ We have for any point x with H(x , t) H , and r c2 0 0 can 0 (cid:90) F (M) x0,r = (cid:90) 1 e−|x−x0|2 4r (2πr)n (3.6) M 2 1 −|x|2 = − (2πrH2 )n e4rHc2an Hcan(M x0) can 2 By lemma 3.7 above, there are some(cid:101)points in the ball B(x , s ) of some previous 0 2 time slice of the unscaled ﬂow that wit(cid:101)h mean curvature exactly equal to H . can After rescaling, there must some point x with mean curvature exactly 1 in the ball B(0, s H2 ) of some previous time t < T of the rescaled ﬂow. 2 can 1 By the standard comparison argument with the(cid:101)self shrinking torus (threading the cylinder through the shrinking torus), the rescaled ﬂow will develop a singularity − by at least t + 100, or in other words so that δ = T t < 100. Thus we have the 1 1 following: ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: MCF

text: WITH

text: SURGERY

text: FOR

text: LOW

text: ENTROPY

text: HYPERSURFACES

text: So,

text: taking

text: s

text: s

text: depending

text: on

text: H

text: and

text: relabeling

text: s

text: s

text: we

text: have

text: can

text: M

text: B(x,

text: s

text: is

text: close

text: to

text: a

text: round

text: cylinder

text: after

text: rescaling

text: (with

text: our

text: previous

text: choice

text: of

text: s

text: instead

text: this

text: was

text: true

text: in

text: every

text: neighborhood

text: of

text: every

text: point)

text: when

text: H

text: H

text: can

text: somewhere

text: within

text: the

text: ball.

text: Of

text: course,

text: we

text: continue

text: for

text: the

text: time

text: being

text: to

text: assume

text: our

text: stipulations

text: listed

text: above.

text: By

text: properties

text: of

text: Gaussian

text: distribution,√there

text: exists

text: a

text: universal

text: constant

text: N

text: independent

text: of

text: scale

text: or

text: basepoint

text: so

text: that·

text: if

text: r

text: then

text: By

text: making

text: small,

text: we

text: can

text: force

text: Hcan

text: 1·

text: N

text: N

text: Now

text: we

text: estimate

text: F

text: (M)

text: for

text: some

text: point

text: x

text: on

text: the

text: ﬂow

text: with

text: H(x)

text: H

text: and

text: can

text: we

text: want

text: to

text: show

text: that

text: if

text: r

text: c

text: then

text: the

text: F

text: functionals

text: with

text: centers

text: at

text: x

text: N

text: must

text: be

text: small

text: in

text: the

text: sense

text: they

text: can

text: be

text: bounded

text: above

text: by

text: Λ

text: We

text: note

text: n

text: the

text: following

text: consequnece

text: of

text: the

text: continunity

text: of

text: the

text: ﬂow

text: and

text: that

text: on

text: the

text: initial

text: time

text: slice,

text: H

text: H

text: (for

text: small

text: enough):

text: can

text: Lemma

text: Those

text: points

text: with

text: mean

text: curvature

text: H(x,

text: t)

text: H

text: must

text: be

text: covered

text: can

text: by

text: union

text: of

text: balls

text: [B(x˜,

text: s

text: x˜,H(x˜,t˜)=Hcanfor

text: some

text: t˜

text: t

text: We

text: have

text: for

text: any

text: point

text: x

text: with

text: H(x

text: t)

text: H

text: and

text: r

text: can

text: F

text: (M)

text: M

text: )n

text: Hcan(M

text: can

text: By

text: lemma

text: above,

text: there

text: are

text: in

text: the

text: ball

text: B(x

text: s

text: of

text: some

text: previous

text: time

text: slice

text: of

text: the

text: unscaled

text: ﬂow

text: that

text: mean

text: curvature

text: exactly

text: equal

text: to

text: H

text: can

text: After

text: rescaling,

text: there

text: must

text: some

text: point

text: x

text: with

text: mean

text: curvature

text: exactly

text: in

text: the

text: ball

text: s

text: of

text: some

text: previous

text: time

text: t

text: T

text: of

text: the

text: rescaled

text: ﬂow.

text: can

text: By

text: the

text: standard

text: comparison

text: argument

text: with

text: shrinking

text: torus

text: (threading

text: the

text: cylinder

text: through

text: the

text: shrinking

text: torus),

text: the

text: rescaled

text: ﬂow

text: will

text: develop

text: a

text: singularity

text: by

text: at

text: least

text: t

text: or

text: in

text: other

text: words

text: so

text: that

text: δ

text: T

text: t

text: Thus

text: we

text: have

text: the

text: following:

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: MCF WITH SURGERY FOR LOW ENTROPY HYPERSURFACES 21 For example, in the diagram above (although in practice the circle might be quite a bit bigger relative to the scale of surgery), the circled tip of a high curature region is essentially modeled on a self translator and is not a neck point, but these are in the region of M# that will be thrown away under surgery by the time is allowed to continue again. Note that the choice of c only relies α, Π, and γ (initial curvature bound on the surface when start MCF with surgery), since we use the canonical neighborhood theorem, and that H ((cid:15) ) < H < 1H ((cid:15) ). Namely, we may take can 1 th 2 can 0 H freely in the next part of the argument. neck ≤ The next case is when c r. In this case we will see the F functionals are 0 in fact nonincreasing if H is taken large enough. To do this we will show if the neck surgery region U is suﬃciently small the Gaussian weights of F functionals with a e lower bound on r are nearly constant within it in a sense made precise below. Then to conclude we use the following observation corresponding to the cap having less volume than the cylinder: Lemma 3.8. For R > R chosen above, there is an 0 < η(R) < 1 depending only (cid:102) 0 (cid:102) on the surgery cap and R such that (cid:102) ∩ ∩ V ol(M+ B(0, 3R)) < ηV ol(M# B(0, 3R)) T T − where M = M q (cid:101) (cid:101) (cid:102) T H neck − − − Where above U = H (U q) , U = H (U q), and M = H (M e neck e f neck f T neck T q) denote the rescaled versions of U , U , and M where the surgery center q is e f T translated to the origin. To begin, note that the graduent of the Gaussian weight e−|x−x0|2 of a F 4r x0,r functional is given by: ##LTFigure## ##LTLine## ##LTLine## ##LTLine##

text: MCF

text: WITH

text: SURGERY

text: FOR

text: LOW

text: ENTROPY

text: HYPERSURFACES

text: For

text: example,

text: in

text: the

text: diagram

text: above

text: (although

text: in

text: practice

text: the

text: circle

text: might

text: be

text: quite

text: a

text: bit

text: bigger

text: relative

text: to

text: the

text: scale

text: of

text: surgery),

text: the

text: circled

text: tip

text: of

text: a

text: high

text: curature

text: region

text: is

text: essentially

text: modeled

text: on

text: a

text: self

text: translator

text: and

text: is

text: not

text: a

text: neck

text: point,

text: but

text: these

text: are

text: in

text: the

text: region

text: of

text: M#

text: that

text: will

text: be

text: thrown

text: away

text: under

text: surgery

text: by

text: the

text: time

text: is

text: allowed

text: to

text: continue

text: again.

text: Note

text: that

text: the

text: choice

text: of

text: c

text: only

text: relies

text: α,

text: Π,

text: and

text: γ

text: (initial

text: curvature

text: bound

text: on

text: the

text: surface

text: when

text: start

text: MCF

text: with

text: surgery),

text: since

text: we

text: use

text: the

text: canonical

text: neighborhood

text: theorem,

text: and

text: that

text: H

text: H

text: Namely,

text: we

text: may

text: take

text: can

text: th

text: can

text: H

text: freely

text: in

text: the

text: next

text: part

text: of

text: the

text: argument.

text: neck

text: The

text: next

text: case

text: is

text: when

text: c

text: r.

text: In

text: this

text: case

text: we

text: will

text: see

text: the

text: F

text: functionals

text: are

text: in

text: fact

text: nonincreasing

text: if

text: H

text: is

text: taken

text: large

text: enough.

text: To

text: do

text: this

text: we

text: will

text: show

text: if

text: the

text: neck

text: surgery

text: region

text: U

text: is

text: suﬃciently

text: small

text: the

text: Gaussian

text: weights

text: of

text: F

text: functionals

text: with

text: a

text: e

text: lower

text: bound

text: on

text: r

text: are

text: nearly

text: constant

text: within

text: it

text: in

text: a

text: sense

text: made

text: precise

text: below.

text: Then

text: to

text: conclude

text: we

text: use

text: the

text: following

text: observation

text: corresponding

text: to

text: the

text: cap

text: having

text: less

text: volume

text: than

text: the

text: cylinder:

text: Lemma

text: For

text: R

text: R

text: chosen

text: above,

text: there

text: is

text: an

text: η(R)

text: depending

text: only

text: on

text: the

text: surgery

text: cap

text: and

text: R

text: such

text: that

text: V

text: ol(M+

text: ηV

text: ol(M#

text: T

text: T

text: where

text: M

text: M

text: q

text: T

text: H

text: neck

text: Where

text: above

text: U

text: H

text: (U

text: q)

text: U

text: H

text: (U

text: q),

text: and

text: M

text: H

text: (M

text: e

text: neck

text: e

text: f

text: neck

text: f

text: T

text: neck

text: T

text: q)

text: denote

text: the

text: rescaled

text: versions

text: of

text: U

text: U

text: and

text: M

text: where

text: the

text: surgery

text: center

text: q

text: is

text: e

text: f

text: T

text: translated

text: to

text: the

text: origin.

text: To

text: begin,

text: note

text: that

text: the

text: graduent

text: of

text: the

text: Gaussian

text: weight

text: of

text: a

text: F

text: functional

text: is

text: given

text: by:

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: MCF WITH SURGERY FOR LOW ENTROPY HYPERSURFACES 22 − − ∇e−|x−x0|2 = 2(x x0)e−|x−x0|2 (3.12) 4r 4r 4r Since by lemma 2.5 the entropy for a compact hypersurface will be attained by an F functional centered in its convex hull, without loss of generality x is in the | − | ≤ ∞ 0 convex hull of M#. Since for such x we have x x D < 2, we see for a lower 0 0 bound c on r we have ∇e−|x−x0|2 | ≤ D < ∞ for any choice of x ∈ M . Denote this 4r 2c T upper bound by ρ. We also note similarly for r > c that the Gaussian weight of a F functional x0,r (with x in the convex hull of M) is bounded below by e−D2 > 0; denote this lower 0 4c bound by σ. Also denote by m and M the minimum and maximum respectively x0,r x0,r of the Gaussian weight of F in U . Then the following is true: x0,r e ≥ m ≥ m ≥ σ − r ρ 1 x0,r x0,r = 1 e (3.13) M m + r ρ σ + r ρ σ + r ρ x0,r x0,r 3 e e ∞ Since σ > 0 and ρ < we can make this quotient as close to one as we like by making r suﬃciently small; in other words we can make the ratio of the minimum e to the maximum of the weight in these F (cid:101)functionals as close to 1 as we want in U by increasing H . Switching to the translated and rescaled picture (the ratio e neck ∈ persists under rescaling), we have for x U and for r > c the following: (cid:102) 0 f 1 (cid:90) F (M+) x0,r = (cid:90)(cid:102) 1 e−|x−x0|2 (cid:90) 4r (4πr)n M+ 2 ≤ (cid:102) (cid:101) 1 e−|x−x0|2 + (cid:102) (cid:101) (cid:101) 1 e−|x−x0|2 4r 4r (cid:90)M+\Ue (4πr)n2 (cid:90)M+∩Ue (4πr)n2 (3.14) (because surgery only happens in U ) e = (cid:90)(cid:102) (cid:101) 1 e−|x−x0|2 + (cid:90)(cid:102) (cid:101) 1 e−|x−x0|2 4r 4r \ (4πr)n ∩ (4πr)n M# Ue 2 M+ Ue 2 ≤ (cid:102) (cid:101) 1 e−|x−x0|2 + (cid:102) (cid:101) 1 M \ (4πr)n 4r ∩ (4πr)n x0,r M# Ue 2 M+ Ue 2 2of course, the diameter is decreasing under the ﬂow so is uniformly bounded by the diameter of the initial time slice ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: MCF

text: WITH

text: SURGERY

text: FOR

text: LOW

text: ENTROPY

text: HYPERSURFACES

text: Since

text: by

text: lemma

text: the

text: entropy

text: for

text: a

text: compact

text: hypersurface

text: will

text: be

text: attained

text: by

text: an

text: F

text: functional

text: centered

text: in

text: its

text: convex

text: hull,

text: without

text: loss

text: of

text: generality

text: x

text: is

text: in

text: the

text: convex

text: hull

text: of

text: M#.

text: Since

text: for

text: such

text: x

text: we

text: have

text: x

text: x

text: D

text: we

text: see

text: for

text: a

text: lower

text: bound

text: c

text: on

text: r

text: we

text: have

text: D

text: for

text: any

text: choice

text: of

text: x

text: M

text: Denote

text: this

text: T

text: upper

text: bound

text: by

text: ρ.

text: We

text: also

text: note

text: similarly

text: for

text: r

text: c

text: that

text: the

text: Gaussian

text: weight

text: of

text: a

text: F

text: functional

text: (with

text: x

text: in

text: the

text: convex

text: hull

text: of

text: M)

text: is

text: bounded

text: below

text: by

text: denote

text: this

text: lower

text: bound

text: by

text: σ.

text: Also

text: denote

text: by

text: m

text: and

text: M

text: the

text: minimum

text: and

text: maximum

text: respectively

text: of

text: the

text: Gaussian

text: weight

text: of

text: F

text: in

text: U

text: Then

text: the

text: following

text: is

text: true:

text: e

text: m

text: m

text: σ

text: r

text: ρ

text: e

text: M

text: m

text: r

text: ρ

text: σ

text: r

text: ρ

text: σ

text: r

text: ρ

text: e

text: e

text: Since

text: σ

text: and

text: ρ

text: we

text: can

text: make

text: this

text: quotient

text: as

text: close

text: to

text: one

text: as

text: we

text: like

text: by

text: making

text: r

text: suﬃciently

text: small;

text: in

text: other

text: words

text: we

text: can

text: make

text: the

text: ratio

text: of

text: the

text: minimum

text: e

text: to

text: the

text: maximum

text: of

text: the

text: weight

text: in

text: these

text: F

text: as

text: close

text: to

text: as

text: we

text: want

text: in

text: U

text: by

text: increasing

text: H

text: Switching

text: to

text: the

text: translated

text: and

text: rescaled

text: picture

text: (the

text: ratio

text: e

text: neck

text: persists

text: under

text: rescaling),

text: we

text: have

text: for

text: x

text: U

text: and

text: for

text: r

text: c

text: the

text: following:

text: f

text: F

text: (M+)

text: M+

text: (because

text: surgery

text: only

text: happens

text: in

text: U

text: e

text: M#

text: Ue

text: M+

text: Ue

text: M

text: M#

text: Ue

text: M+

text: Ue

text: course,

text: the

text: diameter

text: is

text: decreasing

text: under

text: the

text: ﬂow

text: so

text: is

text: uniformly

text: bounded

text: by

text: the

text: diameter

text: of

text: the

text: initial

text: time

text: slice

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: MCF WITH SURGERY FOR LOW ENTROPY HYPERSURFACES 26 [7] Brakke, Kenneth. The Motion of a Surface by its Mean Curvature. Princeton University Press, 1978. [8] Brendle, Simon. A sharp bound for the inscribed radius under the mean curvature ﬂow. Inven- tiones Mathematicae 202, 217-237. [9] Brendle, Simon; Huisken, Gerhard. Mean curvature ﬂow with surgery of mean convex surfaces R in 3. Inventiones Mathematicae 203, 615-654 [10] Buzano, Reto; Haslhofer, Robert; Hershkovits, Or. The Moduli space of two-convex embedded spheres. Preprint, arXiv:1607.05604 [math.DG]. [11] Buzano, Reto; Haslhofer, Robert; Hershkovits, Or. The moduli space of two-convex embedded tori. Preprint, arXiv:1703.01758 [math.DG]. [12] Chen, Bing-Long; Yin, Le. Uniqueness and pseudolocality theorems of the mean curvature ﬂow. Communications in Analysis and Geometry. Volume 15, Number 3, 435-490, 2007. [13] Colding, Tobias; Minicozzi II, William. Generic mean curvature ﬂow I; generic singularities. Annals of Mathematics. (2) 175 (2012), 755-833. [14] Colding, Tobias; Minicozzi II, William. A course in minimal surfaces. Graduate studies in mathematics; v. 121. [15] Evans, L. C.; Spruck, J. Motion of level sets by mean curvature. I. J. Diﬀerential Geom. 33 (1991), no. 3, 635–681 [16] Chen, Yun Gang; Giga, Yoshikazu; Goto, Shun’ichi. Uniqueness and existence of viscosity solutions of generalized mean curvature ﬂow equations. J. Diﬀerential Geom. 33 (1991), no. 3, 749–786. [17] Hamilton, Richard. Four-manifolds with positive curvature operator. J. Diﬀerential Geom. 24 (1986), no. 2, 153–179 [18] Haslhofer, Robert; Kleiner, Bruce. Mean curvature ﬂow with surgery. Duke Math. J. Advance Publication, 23 February 2017. [19] Haslhofer, Robert; Kleiner, Bruce. Mean curvature ﬂow of mean convex hypersurfaces. arXiv. [20] Haslhofer, Robert; Kleiner, Bruce. On Brendle’s estimate for the inscribed radius under the mean curvature ﬂow. Int Math Res Notices 2015; 2015 (15): 6558-6561. [21] Head, John. The Surgery and Level-Set Approaches to Mean Curvature Flow. Thesis. [22] Huisken, Gerhard. Asymptotic behavior for singularities of the mean curvature ﬂow. J. Diﬀer- ential Geom. 31 (1990), no. 1, 285–299. [23] Huisken; Gerhard, Sinestrari, Carlo. Mean curvature ﬂow with surgeries of two-convex hyper- surfaces. Inventiones mathematicae 175, 137-221. [24] Illmanen, Tom. Elliptic regularization and partial regularity for motion by mean curvature, Mem. Amer. Math. Soc.108, (1994), no. 520, x+90. [25] Lauer, Joseph. Convergence of mean curvature ﬂows with surgery. Communications in Analysis and Geometry. Volume 21, Number 2, 355-363, 2013. [26] Mantegazza, Carlos. Lecture Notes on Mean Curvature Flow. Springer Basel, 2013. Print. [27] Mramor, Alexander. A ﬁniteness theorem via the mean curvature ﬂow with surgery. Journal of Geometric Analysis (2017) [28] Mramor, Alexander. Entropy and generic mean curvature ﬂow in curved ambient spaces. To appear in the Proceedings of the AMS. [29] Stone, Andrew. A density function and the structure of singularities of the mean curvature ﬂow. Calc. Var (1994) 2: 443. [30] White, Brian. The nature of singularities in mean curvature ﬂow of mean convex sets. Journal of the AMS, vol 16 no 1 pp 123-138. [31] Zhu, Jonathan. On the entropy of closed hypersurfaces and singular self-shrinkers. Preprint, arXiv:1607.07760[math.DG]

text: MCF

text: WITH

text: SURGERY

text: FOR

text: LOW

text: ENTROPY

text: HYPERSURFACES

label: reference

text: Brakke,

text: Kenneth.

text: The

text: Motion

text: of

text: a

text: Surface

text: by

text: its

text: Mean

text: Curvature.

text: Princeton

text: University

text: Press,

text: Brendle,

text: Simon.

text: A

text: sharp

text: bound

text: for

text: the

text: inscribed

text: radius

text: under

text: the

text: mean

text: curvature

text: ﬂow.

text: Inven-

text: tiones

text: Mathematicae

text: Brendle,

text: Simon;

text: Huisken,

text: Gerhard.

text: Mean

text: curvature

text: ﬂow

text: with

text: surgery

text: of

text: mean

text: convex

text: surfaces

text: R

text: in

text: Inventiones

text: Mathematicae

text: Buzano,

text: Reto;

text: Haslhofer,

text: Robert;

text: Hershkovits,

text: Or.

text: The

text: Moduli

text: space

text: of

text: two-convex

text: embedded

text: spheres.

text: Preprint,

text: [math.DG].

text: Buzano,

text: Reto;

text: Haslhofer,

text: Robert;

text: Hershkovits,

text: Or.

text: The

text: moduli

text: space

text: of

text: two-convex

text: embedded

text: tori.

text: Preprint,

text: [math.DG].

text: Chen,

text: Bing-Long;

text: Yin,

text: Le.

text: Uniqueness

text: and

text: pseudolocality

text: theorems

text: of

text: the

text: mean

text: curvature

text: ﬂow.

text: Communications

text: in

text: Analysis

text: and

text: Geometry.

text: Volume

text: Number

text: Colding,

text: Tobias;

text: Minicozzi

text: II,

text: William.

text: Generic

text: mean

text: curvature

text: ﬂow

text: I;

text: generic

text: singularities.

text: Annals

text: of

text: Mathematics.

text: Colding,

text: Tobias;

text: Minicozzi

text: II,

text: William.

text: A

text: course

text: in

text: minimal

text: surfaces.

text: Graduate

text: studies

text: in

text: mathematics;

text: v.

text: Evans,

text: L.

text: C.;

text: Spruck,

text: J.

text: Motion

text: of

text: level

text: sets

text: by

text: mean

text: curvature.

text: I.

text: J.

text: Diﬀerential

text: Geom.

text: no.

text: Chen,

text: Yun

text: Gang;

text: Giga,

text: Yoshikazu;

text: Goto,

text: Shun’ichi.

text: Uniqueness

text: and

text: existence

text: of

text: viscosity

text: solutions

text: of

text: generalized

text: mean

text: curvature

text: ﬂow

text: equations.

text: J.

text: Diﬀerential

text: Geom.

text: no.

text: Hamilton,

text: Richard.

text: Four-manifolds

text: with

text: positive

text: curvature

text: operator.

text: J.

text: Diﬀerential

text: Geom.

text: no.

text: Haslhofer,

text: Robert;

text: Kleiner,

text: Bruce.

text: Mean

text: curvature

text: ﬂow

text: with

text: surgery.

text: Duke

text: Math.

text: J.

text: Advance

text: Publication,

text: February

text: Haslhofer,

text: Robert;

text: Kleiner,

text: Bruce.

text: Mean

text: curvature

text: ﬂow

text: of

text: mean

text: convex

text: hypersurfaces.

text: arXiv.

text: Haslhofer,

text: Robert;

text: Kleiner,

text: Bruce.

text: On

text: Brendle’s

text: estimate

text: for

text: the

text: inscribed

text: radius

text: under

text: the

text: mean

text: curvature

text: ﬂow.

text: Int

text: Math

text: Res

text: Notices

text: Head,

text: John.

text: The

text: Surgery

text: and

text: Level-Set

text: Approaches

text: to

text: Mean

text: Curvature

text: Flow.

text: Thesis.

text: Huisken,

text: Gerhard.

text: Asymptotic

text: behavior

text: for

text: singularities

text: of

text: the

text: mean

text: curvature

text: ﬂow.

text: J.

text: Diﬀer-

text: ential

text: Geom.

text: no.

text: Huisken;

text: Gerhard,

text: Sinestrari,

text: Carlo.

text: Mean

text: curvature

text: ﬂow

text: with

text: surgeries

text: of

text: two-convex

text: hyper-

text: surfaces.

text: Inventiones

text: mathematicae

text: Illmanen,

text: Tom.

text: Elliptic

text: regularization

text: and

text: partial

text: regularity

text: for

text: motion

text: by

text: mean

text: curvature,

text: Mem.

text: Amer.

text: Math.

text: no.

text: Lauer,

text: Joseph.

text: Convergence

text: of

text: mean

text: curvature

text: ﬂows

text: with

text: surgery.

text: Communications

text: in

text: Analysis

text: and

text: Geometry.

text: Volume

text: Number

text: Mantegazza,

text: Carlos.

text: Lecture

text: Notes

text: on

text: Mean

text: Curvature

text: Flow.

text: Springer

text: Basel,

text: Print.

text: Mramor,

text: Alexander.

text: A

text: ﬁniteness

text: theorem

text: via

text: the

text: mean

text: curvature

text: ﬂow

text: with

text: surgery.

text: Journal

text: of

text: Geometric

text: Analysis

text: Mramor,

text: Alexander.

text: Entropy

text: and

text: generic

text: mean

text: curvature

text: ﬂow

text: in

text: curved

text: ambient

text: spaces.

text: To

text: appear

text: in

text: the

text: Proceedings

text: of

text: the

text: AMS.

text: Stone,

text: Andrew.

text: A

text: density

text: function

text: and

text: the

text: structure

text: of

text: singularities

text: of

text: the

text: mean

text: curvature

text: ﬂow.

text: Calc.

text: Var

text: White,

text: Brian.

text: The

text: nature

text: of

text: singularities

text: in

text: mean

text: curvature

text: ﬂow

text: of

text: mean

text: convex

text: sets.

text: Journal

text: of

text: the

text: AMS,

text: vol

text: no

text: pp

text: Zhu,

text: Jonathan.

text: On

text: the

text: entropy

text: of

text: closed

text: hypersurfaces

text: and

text: singular

text: self-shrinkers.

text: Preprint,


text: MCF WITH SURGERY FOR LOW ENTROPY HYPERSURFACES 27 Department of Mathematics, University of California Irvine, Irvine, CA 92617 Department of Mathematics, Johns Hopkins University, Baltimore, MD 21218 Email address: mramora@uci.edu,swang@math.jhu.edu

text: MCF

text: WITH

text: SURGERY

text: FOR

text: LOW

text: ENTROPY

text: HYPERSURFACES

text: Department

label: author

text: of

text: Mathematics,

text: University

text: of

text: California

text: Irvine,

text: Irvine,

text: CA

text: Department

text: of

text: Mathematics,

text: Johns

text: Hopkins

text: University,

text: Baltimore,

text: MD

text: Email

text: address:

text: mramora@uci.edu,swang@math.jhu.edu

label: footer


text: MCF WITH SURGERY FOR LOW ENTROPY HYPERSURFACES 8 strong neck points in the sense above on which they can do surgery; see claim 4.7 in [18]. If the ancient ﬂow found in the canonical neighborhood theorem is compact, it will be diﬀeomorphic to a sphere, see the discussion after claim 4.8 in [18]. Furthermore as long as H is taken large enough (roughly large enough to employ the canonical th neighborhood theorem for appropriately small (cid:15) as we do in section 3.3 below) all points in the intermediate region between H and H can be forced to be neck th neck points; we will also refer to this region as the neck region below. This is essentially also contained in the argument in the proof of corollary 1.25 in [18] following claim 4.8 therein. For readers perhaps more familiar with the approach to surgery of Huisken and Sinestrari in [23], this is essentially the content of their neck continuation theorem (more precisely, theorem 8.1 in [23]); one starts by ﬁnding a neck point, and the statement is essentially that one may continue the neck as long as H is large (in our context, H > H ), λ /H, and there are no previous surgeries in the way. If the th 1 second or third conditions are violated the case then is that the neck is ended by a convex cap. In [13] Colding and Minicozzi discovered a useful new quantity called the entropy ⊂ R to study the m∈eaRn curvature ﬂow. To elabor(cid:90)ate, consider a hypersurface Σk (cid:96); then given x (cid:96) and r > 0 deﬁne the functional F by 0 x0,r F (Σ) = 1 e−|x−x0|2 dµ (2.5) x0,r (4πt )k/2 4r 0 Σ Colding and Minicozzi then deﬁne the entropy λ(Σ) of a submanifold to be the supremum over all F functionals: x0,r λ(Σ) = sup F (Σ) (2.6) x0,r x0,r Important for below is to note that equivalently λ(Σ) is the supremum of F 0,1 when we vary over rescalings (changing r) and translations (choice of x ). For hy- 0 persurfaces with polynomial growth this supremum is attained and, for self shrinkers Σ, λ(Σ) = F (Σ). In fact, self shrinkers are critical points for the entropy so it is 0,1 natural next to ask what the stable ones are. If Σ is a normal variation of Σ and 2 x , t are variations with x = 0, r = 1, s s 0 | | | ∂ Σ = fν, ∂ x = y, and ∂ t = h (2.7) s s=0 s s s=0 s s s=0 s The second variation formula one ﬁnd is: ##LTLine## ##LTLine##

text: MCF

text: WITH

text: SURGERY

text: FOR

text: LOW

text: ENTROPY

text: HYPERSURFACES

text: strong

text: neck

text: points

text: in

text: the

text: sense

text: above

text: on

text: which

text: they

text: can

text: do

text: surgery;

text: see

text: claim

text: in

text: If

text: the

text: ancient

text: ﬂow

text: found

text: in

text: the

text: canonical

text: neighborhood

text: theorem

text: is

text: compact,

text: it

text: will

text: be

text: diﬀeomorphic

text: to

text: a

text: sphere,

text: see

text: the

text: discussion

text: after

text: claim

text: in

text: Furthermore

text: as

text: long

text: as

text: H

text: is

text: taken

text: large

text: enough

text: (roughly

text: large

text: enough

text: to

text: employ

text: the

text: canonical

text: th

text: neighborhood

text: theorem

text: for

text: appropriately

text: small

text: as

text: we

text: do

text: in

text: section

text: below)

text: all

text: points

text: in

text: the

text: intermediate

text: region

text: between

text: H

text: and

text: H

text: can

text: be

text: forced

text: to

text: be

text: neck

text: th

text: neck

text: points;

text: we

text: will

text: also

text: refer

text: to

text: this

text: region

text: as

text: the

text: neck

text: region

text: below.

text: This

text: is

text: essentially

text: also

text: contained

text: in

text: the

text: argument

text: in

text: the

text: proof

text: of

text: corollary

text: in

text: following

text: claim

text: therein.

text: For

text: readers

text: perhaps

text: more

text: familiar

text: with

text: the

text: approach

text: to

text: surgery

text: of

text: Huisken

text: and

text: Sinestrari

text: in

text: this

text: is

text: essentially

text: the

text: content

text: of

text: their

text: neck

text: continuation

text: theorem

text: (more

text: precisely,

text: theorem

text: in

text: one

text: starts

text: by

text: ﬁnding

text: a

text: neck

text: point,

text: and

text: the

text: statement

text: is

text: essentially

text: that

text: one

text: may

text: continue

text: the

text: neck

text: as

text: long

text: as

text: H

text: is

text: large

text: (in

text: our

text: context,

text: H

text: H

text: λ

text: /H,

text: and

text: there

text: are

text: no

text: previous

text: surgeries

text: in

text: the

text: way.

text: If

text: the

text: th

text: second

text: or

text: third

text: conditions

text: are

text: violated

text: the

text: case

text: then

text: is

text: that

text: the

text: neck

text: is

text: ended

text: by

text: a

text: convex

text: cap.

text: In

text: Colding

text: and

text: Minicozzi

text: discovered

text: a

text: useful

text: new

text: quantity

text: called

text: the

text: entropy

text: R

text: to

text: study

text: the

text: m∈eaRn

text: curvature

text: ﬂow.

text: To

text: consider

text: a

text: hypersurface

text: Σk

text: then

text: given

text: x

text: and

text: r

text: deﬁne

text: the

text: functional

text: F

text: by

text: F

text: (Σ)

text: dµ

text: Σ

text: Colding

text: and

text: Minicozzi

text: then

text: deﬁne

text: the

text: entropy

text: λ(Σ)

text: of

text: a

text: submanifold

text: to

text: be

text: the

text: supremum

text: over

text: all

text: F

text: functionals:

text: λ(Σ)

text: sup

text: F

text: (Σ)

text: Important

text: for

text: below

text: is

text: to

text: note

text: that

text: equivalently

text: λ(Σ)

text: is

text: the

text: supremum

text: of

text: F

text: when

text: we

text: vary

text: over

text: rescalings

text: (changing

text: r)

text: and

text: translations

text: (choice

text: of

text: x

text: For

text: hy-

text: persurfaces

text: with

text: polynomial

text: growth

text: this

text: supremum

text: is

text: attained

text: and,

text: for

text: self

text: shrinkers

text: Σ,

text: λ(Σ)

text: F

text: (Σ).

text: In

text: fact,

text: self

text: shrinkers

text: are

text: critical

text: points

text: for

text: the

text: entropy

text: so

text: it

text: is

text: natural

text: next

text: to

text: ask

text: what

text: the

text: stable

text: ones

text: are.

text: If

text: Σ

text: is

text: a

text: normal

text: variation

text: of

text: Σ

text: and

text: x

text: t

text: are

text: variations

text: with

text: x

text: r

text: s

text: s

text: Σ

text: fν,

text: x

text: y,

text: and

text: t

text: h

text: s

text: s

text: s

text: s

text: s

text: s

text: The

text: second

text: variation

text: formula

text: one

text: ﬁnd

text: is:

text: ##LTLine##

text: ##LTLine##


text: The diﬀusion of noble and other gases through polymers has been studied in various contexts, because its of relevance for industries including the oil and gas industry and food packaging. References such as [31], [32] and [33] report on the seepage of gases through very thin polymer membranes. The eﬀect on mechanical properties of the membrane, such as changes in shape or mass, are not reported. Because they are heavier and thus move more slowly in thermal equilibrium, a naive expectation might be that more massive noble gases would diﬀuse through materials more slowly than less massive ones. This is observed in only some conditions. For example, in [34], Xenon permeability ∼ ∼ through Kapton ﬁlms is shown to be a factor 10 times lower than krypton and 100 times lower than Argon at the same temperature. However, careful analysis of the time proﬁles and pressure dependencies of noble gas permeation as reported in Ref [32] illuminates more nuanced behaviour. There, two distinct absorption mechanisms are inferred in oriented polypropylene (OPP) from the two time constants in absorption and desorption curves. These are named the “normal pathway”, or standard diﬀusion through the solid polymer matrix, and “cavity condensation”, which occurs only for the heaviest gases and only at the highest pressures. Under this mechanism, small numbers of atoms may enter material voids, where interactions with the walls create potential wells that increase the local eﬀective pressure. This causes condensation of the gas in these voids, enhancing the absorption rate. Related phenomena have been observed in porous carbon media, where enhanced production of Xenon dimers is observed and eﬀective pore pressures in excess of 1000 bar can be generated [35]. This process is postulated to also be active in the amorphous regions of polymer matrices, with measurements from [32] suggesting that it accounts for 8% of diﬀusion of Xenon at 1 bar in OPP. The characteristic two-exponential behaviour is reported for Xe and Kr but not Ar, He or Ne, which have much lower boiling points. At the pressures considered in this work, the cavity condensation mechanism is expected to be further enhanced relative to that reported at 1 bar in [32]. This may provide an explanation for the striking qualitative diﬀerence between Xenon and Argon absorption in bulk HDPE, UHMW and PTFE. The diﬀerence between these materials and POM and PEEK would then be understood to lie in the diﬀerent density of microscopic voids in these materials that may enable the condensation mechanism. Fig.12 (right) displays mass versus length for HDPE, UHMW, and PTFE soaked in Xenon both with diﬀerent pressures and diﬀerent soaking times. There is a clear correlation between mass and length change ratio for each of the materials, with PTFE appearing to display a distinct slope compared to the other two. The dielectric strength of these materials show similar behaviours between high pressure Argon and Xenon gases. More scatter observed between points at diﬀerent pressures in Xenon. This may be because dielectric strength of Xenon is simply higher, and thus the relevant surfaces carry more charge which may ﬂuctuate to initiate a breakdown. This scatter is not observed for HDPE, which maintains a reliable and predictable breakdown strength even to the highest pressures. In both cases, POM is the weakest material, apparently carrying surface breakdowns at low voltages in both cases. We noted that under higher breakdown ﬁelds, some pitting occurred in the electrodes, and a dark residue was observed to accumulate on the HV cable insulation and posts, which we suspect may be the ejected material from the spark. This may potentially explain the more scattered behaviour in xenon gas, as this ejected material could introduce new but sporadic conductive paths. – 13 –

text: The

font: WMRDXZ+TeXGyreTermes-Regular

text: diﬀusion

text: of

text: noble

text: and

text: other

text: gases

text: through

text: polymers

text: has

text: been

text: studied

text: in

text: various

text: contexts,

text: because

text: its

text: of

text: relevance

text: for

text: industries

text: including

text: the

text: oil

text: and

text: gas

text: industry

text: and

text: food

text: packaging.

text: References

text: such

text: as

text: and

text: report

text: on

text: the

text: seepage

text: of

text: gases

text: through

text: very

text: thin

text: polymer

text: membranes.

text: The

text: eﬀect

text: on

text: mechanical

text: properties

text: of

text: the

text: membrane,

text: such

text: as

text: changes

text: in

text: shape

text: or

text: mass,

text: are

text: not

text: reported.

text: Because

text: they

text: are

text: heavier

text: and

text: thus

text: move

text: more

text: slowly

text: in

text: thermal

text: equilibrium,

text: a

text: naive

text: expectation

text: might

text: be

text: that

text: more

text: massive

text: noble

text: gases

text: would

text: diﬀuse

text: through

text: materials

text: more

text: slowly

text: than

text: less

text: massive

text: ones.

text: This

text: is

text: observed

text: in

text: only

text: some

text: conditions.

text: For

text: example,

text: in

text: Xenon

text: permeability

text: through

text: Kapton

text: ﬁlms

text: is

text: shown

text: to

text: be

text: a

text: factor

text: times

text: lower

text: than

text: krypton

text: and

text: times

text: lower

text: than

text: Argon

text: at

text: the

text: same

text: temperature.

text: However,

text: careful

text: analysis

text: of

text: the

text: time

text: proﬁles

text: and

text: pressure

text: dependencies

text: of

text: noble

text: gas

text: permeation

text: as

text: reported

text: in

text: Ref

text: illuminates

text: more

text: nuanced

text: behaviour.

text: There,

text: two

text: distinct

text: absorption

text: mechanisms

text: are

text: inferred

text: in

text: oriented

text: polypropylene

text: (OPP)

text: from

text: the

text: two

text: time

text: constants

text: in

text: absorption

text: and

text: desorption

text: curves.

text: These

text: are

text: named

text: the

text: “normal

text: pathway”,

text: or

text: standard

text: diﬀusion

text: through

text: the

text: solid

text: polymer

text: matrix,

text: and

text: “cavity

text: condensation”,

text: which

text: occurs

text: only

text: for

text: the

text: heaviest

text: gases

text: and

text: only

text: at

text: the

text: highest

text: pressures.

text: Under

text: this

text: mechanism,

text: small

text: numbers

text: of

text: atoms

text: may

text: enter

text: material

text: voids,

text: where

text: interactions

text: with

text: the

text: walls

text: create

text: potential

text: wells

text: that

text: increase

text: the

text: local

text: eﬀective

text: pressure.

text: This

text: causes

text: condensation

text: of

text: the

text: gas

text: in

text: these

text: voids,

text: enhancing

text: the

text: absorption

text: rate.

text: Related

text: phenomena

text: have

text: been

text: observed

text: in

text: porous

text: carbon

text: media,

text: where

text: enhanced

text: production

text: of

text: Xenon

text: dimers

text: is

text: observed

text: and

text: eﬀective

text: pore

text: pressures

text: in

text: excess

text: of

text: bar

text: can

text: be

text: generated

text: This

text: process

text: is

text: postulated

text: to

text: also

text: be

text: active

text: in

text: the

text: amorphous

text: regions

text: of

text: polymer

text: matrices,

text: with

text: measurements

text: from

text: suggesting

text: that

text: it

text: accounts

text: for

text: of

text: diﬀusion

text: of

text: Xenon

text: at

text: bar

text: in

text: OPP.

text: The

text: characteristic

text: two-exponential

text: behaviour

text: is

text: reported

text: for

text: Xe

text: and

text: Kr

text: but

text: not

text: Ar,

text: He

text: or

text: Ne,

text: which

text: have

text: much

text: lower

text: boiling

text: points.

text: At

text: the

text: pressures

text: considered

text: in

text: this

text: work,

text: the

text: cavity

text: condensation

text: mechanism

text: is

text: expected

text: to

text: be

text: further

text: enhanced

text: relative

text: to

text: that

text: reported

text: at

text: bar

text: in

text: This

text: may

text: provide

text: an

text: explanation

text: for

text: the

text: striking

text: qualitative

text: diﬀerence

text: between

text: Xenon

text: and

text: Argon

text: absorption

text: in

text: bulk

text: HDPE,

text: UHMW

text: and

text: PTFE.

text: The

text: diﬀerence

text: between

text: these

text: materials

text: and

text: POM

text: and

text: PEEK

text: would

text: then

text: be

text: understood

text: to

text: lie

text: in

text: the

text: diﬀerent

text: density

text: of

text: microscopic

text: voids

text: in

text: these

text: materials

text: that

text: may

text: enable

text: the

text: condensation

text: mechanism.

text: (right)

text: displays

text: mass

text: versus

text: length

text: for

text: HDPE,

text: UHMW,

text: and

text: PTFE

text: soaked

text: in

text: Xenon

text: both

text: with

text: diﬀerent

text: pressures

text: and

text: diﬀerent

text: soaking

text: times.

text: There

text: is

text: a

text: clear

text: correlation

text: between

text: mass

text: and

text: length

text: change

text: ratio

text: for

text: each

text: of

text: the

text: materials,

text: with

text: PTFE

text: appearing

text: to

text: display

text: a

text: distinct

text: slope

text: compared

text: to

text: the

text: other

text: two.

text: The

text: dielectric

text: strength

text: of

text: these

text: materials

text: show

text: similar

text: behaviours

text: between

text: high

text: pressure

text: Argon

text: and

text: Xenon

text: gases.

text: More

text: scatter

text: observed

text: between

text: points

text: at

text: diﬀerent

text: pressures

text: in

text: Xenon.

text: This

text: may

text: be

text: because

text: dielectric

text: strength

text: of

text: Xenon

text: is

text: simply

text: higher,

text: and

text: thus

text: the

text: relevant

text: surfaces

text: carry

text: more

text: charge

text: which

text: may

text: ﬂuctuate

text: to

text: initiate

text: a

text: breakdown.

text: This

text: scatter

text: is

text: not

text: observed

text: for

text: HDPE,

text: which

text: maintains

text: a

text: reliable

text: and

text: predictable

text: breakdown

text: strength

text: even

text: to

text: the

text: highest

text: pressures.

text: In

text: both

text: cases,

text: POM

text: is

text: the

text: weakest

text: material,

text: apparently

text: carrying

text: surface

text: breakdowns

text: at

text: low

text: voltages

text: in

text: both

text: cases.

text: We

text: noted

text: that

text: under

text: higher

text: breakdown

text: ﬁelds,

text: some

text: pitting

text: occurred

text: in

text: the

text: electrodes,

text: and

text: a

text: dark

text: residue

text: was

text: observed

text: to

text: accumulate

text: on

text: the

text: HV

text: cable

text: insulation

text: and

text: posts,

text: which

text: we

text: suspect

text: may

text: be

text: the

text: ejected

text: material

text: from

text: the

text: spark.

text: This

text: may

text: potentially

text: explain

text: the

text: more

text: scattered

text: behaviour

text: in

text: xenon

text: gas,

text: as

text: this

text: ejected

text: material

text: could

text: introduce

text: new

text: but

text: sporadic

text: conductive

text: paths.


text: [7] NEXT Collaboration, P. Ferrario et al., First proof of topological signature in the high pressure xenon gas TPC with electroluminescence ampliﬁcation for the NEXT experiment, JHEP 01 (2016) 104, [arXiv:1507.0590]. [8] D. Fancher et al., Performance of a Time Projection Chamber, Nucl. Instrum. Meth. 161 (1979) 383. [9] F. T. Avignone, III, S. R. Elliott, and J. Engel, Double Beta Decay, Majorana Neutrinos, and Neutrino Mass, Rev. Mod. Phys. 80 (2008) 481–516, [arXiv:0708.1033]. [10] NEXT Collaboration, V. Alvarez et al., NEXT-100 Technical Design Report (TDR): Executive Summary, JINST 7 (2012) T06001, [arXiv:1202.0721]. [11] NEXT Collaboration, F. Granena et al., NEXT, a HPGXe TPC for neutrinoless double beta decay searches, arXiv:0907.4054. [12] NEXT Collaboration, J. J. Gomez-Cadenas et al., Present status and future perspectives of the NEXT experiment, Adv. High Energy Phys. 2014 (2014) 907067, [arXiv:1307.3914]. [13] NEXT Collaboration, V. Alvarez et al., Near-Intrinsic Energy Resolution for 30 to 662 keV Gamma Rays in a High Pressure Xenon Electroluminescent TPC, Nucl. Instrum. Meth. A708 (2013) 101–114, [arXiv:1211.4474]. [14] NEXT Collaboration, V. Alvarez et al., Initial results of NEXT-DEMO, a large-scale prototype of the NEXT-100 experiment, JINST 8 (2013) P04002, [arXiv:1211.4838]. [15] V. Alvarez et al., SiPMs coated with TPB : coating protocol and characterization for NEXT, JINST 7 (2012) P02010, [arXiv:1201.2018]. [16] C. M. B. Monteiro, Determination of argon and xenon absolute electroluminescence yields in Gas Proportional Scintillation Counters. PhD thesis, 2011. [17] S. Pan, AXEL: High pressure xenon gas Time Projection Chamber for neutrinoless double beta decay search, J. Phys. Conf. Ser. 718 (2016), no. 6 062047. [18] NEXT Collaboration, N. Lopez March, The NEXT high pressure xenon gas TPC for neutrinoless double beta decay searches, JINST 13 (2018), no. 01 C01048. [19] C. Cantini et al., First test of a high voltage feedthrough for liquid Argon TPCs connected to a 300 kV power supply, JINST 12 (2017), no. 03 P03021, [arXiv:1611.0208]. [20] B. J. Mount et al., LUX-ZEPLIN (LZ) Technical Design Report, arXiv:1703.0914. [21] XENON Collaboration, E. Aprile et al., Design and Performance of the XENON10 Dark Matter Experiment, Astropart. Phys. 34 (2011) 679–698, [arXiv:1001.2834]. [22] LUX Collaboration, D. S. Akerib et al., The Large Underground Xenon (LUX) Experiment, Nucl. Instrum. Meth. A704 (2013) 111–126, [arXiv:1211.3788]. [23] M. Auger et al., The EXO-200 detector, part I: Detector design and construction, JINST 7 (2012) P05010, [arXiv:1202.2192]. [24] S. Lockwitz and H. Jostlein, A Study of Dielectric Breakdown Along Insulators Surrounding Conductors in Liquid Argon, JINST 11 (2016), no. 03 P03026, [arXiv:1506.0418]. [25] ICARUS Collaboration, S. Amerio et al., Design, construction and tests of the ICARUS T600 detector, Nucl. Instrum. Meth. A527 (2004) 329–410. [26] MicroBooNE Collaboration, R. Acciarri et al., Design and Construction of the MicroBooNE Detector, JINST 12 (2017), no. 02 P02017, [arXiv:1612.0582]. – 16 –

text: NEXT

font: OVNYGS+TeXGyreTermes-Bold

text: Collaboration,

text: P.

text: Ferrario

text: et

text: al.,

text: First

font: DYEZPM+TeXGyreTermes-Italic

text: proof

text: of

text: topological

text: signature

text: in

text: the

text: high

text: pressure

text: xenon

text: gas

text: TPC

text: with

text: electroluminescence

text: ampliﬁcation

text: for

text: the

text: NEXT

text: experiment,

text: JHEP

font: VZCYZV+newtxtt

text: D.

text: Fancher

text: et

text: al.,

text: Performance

text: of

text: a

text: Time

text: Projection

text: Chamber,

text: Nucl.

text: Instrum.

text: Meth.

text: F.

text: T.

text: Avignone,

text: III,

text: S.

text: R.

text: Elliott,

text: and

text: J.

text: Engel,

text: Double

text: Beta

text: Decay,

text: Majorana

text: Neutrinos,

text: and

text: Neutrino

text: Mass,

text: Rev.

text: Mod.

text: Phys.

text: NEXT

text: Collaboration,

text: V.

text: Alvarez

text: et

text: al.,

text: Technical

text: Design

text: Report

text: (TDR):

text: Executive

text: Summary,

text: JINST

text: NEXT

text: Collaboration,

text: F.

text: Granena

text: et

text: al.,

text: NEXT,

text: a

text: HPGXe

text: TPC

text: for

text: neutrinoless

text: double

text: beta

text: decay

text: searches,

text: NEXT

text: Collaboration,

text: J.

text: J.

text: Gomez-Cadenas

text: et

text: al.,

text: Present

text: status

text: and

text: future

text: perspectives

text: of

text: the

text: NEXT

text: experiment,

text: Adv.

text: High

text: Energy

text: Phys.

text: NEXT

text: Collaboration,

text: V.

text: Alvarez

text: et

text: al.,

text: Near-Intrinsic

text: Energy

text: Resolution

text: for

text: to

text: keV

text: Gamma

text: Rays

text: in

text: a

text: High

text: Pressure

text: Xenon

text: Electroluminescent

text: TPC,

text: Nucl.

text: Instrum.

text: Meth.

text: NEXT

text: Collaboration,

text: V.

text: Alvarez

text: et

text: al.,

text: Initial

text: results

text: of

text: NEXT-DEMO,

text: a

text: large-scale

text: prototype

text: of

text: the

text: experiment,

text: JINST

text: V.

text: Alvarez

text: et

text: al.,

text: SiPMs

text: coated

text: with

text: TPB

text: coating

text: protocol

text: and

text: characterization

text: for

text: NEXT,

text: JINST

text: C.

text: M.

text: B.

text: Monteiro,

text: Determination

text: of

text: argon

text: and

text: xenon

text: absolute

text: electroluminescence

text: yields

text: in

text: Gas

text: Proportional

text: Scintillation

text: Counters.

text: PhD

text: thesis,

text: S.

text: Pan,

text: AXEL:

text: High

text: pressure

text: xenon

text: gas

text: Time

text: Projection

text: Chamber

text: for

text: neutrinoless

text: double

text: beta

text: decay

text: search,

text: J.

text: Phys.

text: Conf.

text: Ser.

text: no.

text: NEXT

text: Collaboration,

text: N.

text: Lopez

text: March,

text: The

text: NEXT

text: high

text: pressure

text: xenon

text: gas

text: TPC

text: for

text: neutrinoless

text: double

text: beta

text: decay

text: searches,

text: JINST

text: no.

text: C.

text: Cantini

text: et

text: al.,

text: First

text: test

text: of

text: a

text: high

text: voltage

text: feedthrough

text: for

text: liquid

text: Argon

text: TPCs

text: connected

text: to

text: a

text: kV

text: power

text: supply,

text: JINST

text: no.

text: B.

text: J.

text: Mount

text: et

text: al.,

text: LUX-ZEPLIN

text: (LZ)

text: Technical

text: Design

text: Report,

text: XENON

text: Collaboration,

text: E.

text: Aprile

text: et

text: al.,

text: Design

text: and

text: Performance

text: of

text: the

text: Dark

text: Matter

text: Experiment,

text: Astropart.

text: Phys.

text: LUX

text: Collaboration,

text: D.

text: S.

text: Akerib

text: et

text: al.,

text: The

text: Large

text: Underground

text: Xenon

text: (LUX)

text: Experiment,

text: Nucl.

text: Instrum.

text: Meth.

text: M.

text: Auger

text: et

text: al.,

text: The

text: detector,

text: part

text: I:

text: Detector

text: design

text: and

text: construction,

text: JINST

text: S.

text: Lockwitz

text: and

text: H.

text: Jostlein,

text: A

text: Study

text: of

text: Dielectric

text: Breakdown

text: Along

text: Insulators

text: Surrounding

text: Conductors

text: in

text: Liquid

text: Argon,

text: JINST

text: no.

text: ICARUS

text: Collaboration,

text: S.

text: Amerio

text: et

text: al.,

text: Design,

text: construction

text: and

text: tests

text: of

text: the

text: ICARUS

text: detector,

text: Nucl.

text: Instrum.

text: Meth.

text: MicroBooNE

text: Collaboration,

text: R.

text: Acciarri

text: et

text: al.,

text: Design

text: and

text: Construction

text: of

text: the

text: MicroBooNE

text: Detector,

text: JINST

text: no.


text: 1.012 1.07 7 day soak (day 0) 7 day 7 day soak (day 0) 9 day 1 day 9 day 1 day 13 day 1.06 1.010 2 day 13 day 2 day 25 day 4 day 25 day 4 day 1.05 1.008 1.04 1.006 1.03 1.004 1.02 1.002 1.01 1.000 1.00 0.998 0.99 PTFE HDPE PEEK POM UHMW PTFE HDPE PEEK POM UHMW Figure 8. Left:Ratios of lengths of posts after soaking for one week in Xenon at 15 bar and then put under vacuum divided by the original length of posts. Right: Ratios of masses of posts after soaking for one week in Xenon at 15 bar and then put under vacuum divided by the original, pre-vacuum masses of posts. and tightened to hold them in place. Both ball bearings are tapped and drilled 12 mm deep for the PEEK rod to securely thread into and the opposite side ground down a slight amount to provide a ﬂat spot for the insulating material posts to rest. Several preliminary frames were tested to hold the ball bearings before settling on this design, which we denote A-C. Frame A used drilled acrylic blocks to hold the electrodes with PEEK rods, but the weight of the steel balls caused the acrylic to warp when heating the vessel during the vacuum stage, leading to misalignment of the spheres. The acrylic also outgassed profusely, leading to a high Figure 9. Left: Setup for holding the insulating posts in a smooth electric ﬁeld where the left sphere is the high voltage, right is ground. Right: Typical posts used for these tests. From top to bottom the materials are UHMW, POM, PEEK, HDPE, and PTFE. – 9 – oitaR htgneL oitaR ssaM ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

font: default

text: day

text: soak

text: (day

text: day

text: day

text: soak

text: (day

text: day

text: day

text: day

text: day

text: day

text: day

text: day

text: day

text: day

text: day

text: day

text: day

text: PTFE

text: HDPE

text: PEEK

text: POM

text: UHMW

text: PTFE

text: HDPE

text: PEEK

text: POM

text: UHMW

text: Figure

text: Left:Ratios

text: of

text: lengths

text: of

text: posts

text: after

text: soaking

text: for

text: one

text: week

text: in

text: Xenon

text: at

text: bar

text: and

text: then

text: put

text: under

text: vacuum

text: divided

text: by

text: the

text: original

text: length

text: of

text: posts.

text: Right:

text: Ratios

text: of

text: masses

text: of

text: posts

text: after

text: soaking

text: for

text: one

text: week

text: in

text: Xenon

text: at

text: bar

text: and

text: then

text: put

text: under

text: vacuum

text: divided

text: by

text: the

text: original,

text: pre-vacuum

text: masses

text: of

text: posts.

text: and

text: tightened

text: to

text: hold

text: them

text: in

text: place.

text: Both

text: ball

text: bearings

text: are

text: tapped

text: and

text: drilled

text: mm

text: deep

text: for

text: the

text: PEEK

text: rod

text: to

text: securely

text: thread

text: into

text: and

text: the

text: opposite

text: side

text: ground

text: down

text: a

text: slight

text: amount

text: to

text: provide

text: a

text: ﬂat

text: spot

text: for

text: the

text: insulating

text: material

text: posts

text: to

text: rest.

text: Several

text: preliminary

text: frames

text: were

text: tested

text: to

text: hold

text: the

text: ball

text: bearings

text: before

text: settling

text: on

text: this

text: design,

text: which

text: we

text: denote

text: A-C.

text: Frame

text: A

text: used

text: drilled

text: acrylic

text: blocks

text: to

text: hold

text: the

text: electrodes

text: with

text: PEEK

text: rods,

text: but

text: the

text: weight

text: of

text: the

text: steel

text: balls

text: caused

text: the

text: acrylic

text: to

text: warp

text: when

text: heating

text: the

text: vessel

text: during

text: the

text: vacuum

text: stage,

text: leading

text: to

text: misalignment

text: of

text: the

text: spheres.

text: The

text: acrylic

text: also

text: outgassed

text: profusely,

text: leading

text: to

text: a

text: high

text: Figure

text: Left:

text: Setup

text: for

text: holding

text: the

text: insulating

text: posts

text: in

text: a

text: smooth

text: electric

text: ﬁeld

text: where

text: the

text: left

text: sphere

text: is

text: the

text: high

text: voltage,

text: right

text: is

text: ground.

text: Right:

text: Typical

text: posts

text: used

text: for

text: these

text: tests.

text: From

text: top

text: to

text: bottom

text: the

text: materials

text: are

text: UHMW,

text: POM,

text: PEEK,

text: HDPE,

text: and

text: PTFE.

text: oitaR

text: htgneL

text: oitaR

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text:                  c13c12 c13s12 s13e−iδ   1 0 0  − − − U = c s s s c eiδ c c s s s eiδ s c 0 eiγ 0 , (71) 23 12 23 13 12 23 12 23 13 12 23 13 − − − s s c s c eiδ s c c s s eiδ c c 0 0 1 23 12 23 13 12 23 12 23 13 12 23 13 ≡ ≡ where c cos θ and s sin θ (θ , θ , and θ being the three lepton mixing angles). ij ij ij ij 12 13 23 γ is the Majorana and δ is the Dirac phase. Since the lightest neutrino is massless in the GScM, there is only one physical Majorana phase. C Parametrisation of Yukawa couplings We want to express the Yukawa couplings in terms of neutrino masses and lepton mixing parameters. We follow the discussion in Ref. [86]. On the one hand the rank-two neutrino mass matrix can be expressed in terms of the two non-vanishing mass eigenvalues and two columns of (cid:88) (cid:88) the PMNS matrix in the ﬂavour basis M = m u∗u† = v vT . (72) ν i i i i i i i √ ≡ The vectors v m u∗ are linearly independent and span a two-dimensional vector space. i i i On the other hand the calculation of the neutrino mass matrix results in the following form M = x xT + x xT (73) ν 1 2 2 1 with (see Eqs. (12) and (13)) (cid:114) (cid:114) sin 2θ m sin 2θ m x = ψ F (m , m (cid:48) , m ) y , x = ψ F (m , m (cid:48) , m ) y (cid:48). (74) 1 32 π2 η0 η0 ψ Φ 2 32 π2 η0 η0 ψ Φ If sin 2θ < 0 the square roots will yield a complex number. The two linearly independent vectors x can be expressed in terms of the vectors v i i x = a v , (75) i ij j × − (cid:54) where (a ) forms an invertible 2 2 matrix, i.e. det a = a a a a = 0. Using the two ij 11 22 12 21 diﬀerent parametrizations of the neutrino mass matrix, we can ﬁnd possible solutions of a : ij (cid:88) (cid:88) (cid:88) M = x xT + x xT (76) ν 1 2 2 1 = (cid:88) a a v vT + a a v vT = (a a + a a )v vT (77) 1i 2j i j 2i 1j i j 1i 2j 2i 1j i j i,j i,j i,j = v vT . (78) i i i 31 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: U

text: c

text: s

text: s

text: s

text: c

text: eiδ

text: c

text: c

text: s

text: s

text: s

text: eiδ

text: s

text: c

text: eiγ

text: s

text: s

text: c

text: s

text: c

text: eiδ

text: s

text: c

text: c

text: s

text: s

text: eiδ

text: c

text: c

text: where

text: c

text: cos

text: θ

text: and

text: s

text: sin

text: θ

text: (θ

text: θ

text: and

text: θ

text: being

text: the

text: three

text: lepton

text: mixing

text: angles).

text: ij

text: ij

text: ij

text: ij

text: γ

text: is

text: the

text: Majorana

text: and

text: δ

text: is

text: the

text: Dirac

text: phase.

text: Since

text: the

text: lightest

text: neutrino

text: is

text: massless

text: in

text: the

text: GScM,

text: there

text: is

text: only

text: one

text: physical

text: Majorana

text: phase.

text: C

text: Parametrisation

text: of

text: Yukawa

text: couplings

text: We

text: want

text: to

text: express

text: the

text: Yukawa

text: couplings

text: in

text: terms

text: of

text: neutrino

text: masses

text: and

text: lepton

text: mixing

text: parameters.

text: We

text: follow

text: the

text: discussion

text: in

text: Ref.

text: On

text: the

text: one

text: hand

text: the

text: rank-two

text: neutrino

text: mass

text: matrix

text: can

text: be

text: expressed

text: in

text: terms

text: of

text: the

text: two

text: non-vanishing

text: mass

text: eigenvalues

text: and

text: two

text: columns

text: of

text: the

text: PMNS

text: matrix

text: in

text: the

text: ﬂavour

text: basis

text: M

text: m

text: u∗u†

text: v

text: vT

text: ν

text: i

text: i

text: i

text: i

text: i

text: i

text: i

text: The

text: vectors

text: v

text: m

text: u∗

text: are

text: linearly

text: independent

text: and

text: span

text: a

text: two-dimensional

text: vector

text: space.

text: i

text: i

text: i

text: On

text: the

text: other

text: hand

text: the

text: calculation

text: of

text: the

text: neutrino

text: mass

text: matrix

text: results

text: in

text: the

text: following

text: form

text: M

text: x

text: xT

text: x

text: xT

text: ν

text: with

text: (see

text: Eqs.

text: and

text: sin

text: 2θ

text: m

text: sin

text: 2θ

text: m

text: x

text: ψ

text: F

text: (m

text: m

text: m

text: y

text: x

text: ψ

text: F

text: (m

text: m

text: m

text: y

text: π2

text: η0

text: η0

text: ψ

text: Φ

text: π2

text: η0

text: η0

text: ψ

text: Φ

text: If

text: sin

text: 2θ

text: the

text: square

text: roots

text: will

text: yield

text: a

text: complex

text: number.

text: The

text: two

text: linearly

text: independent

text: vectors

text: x

text: can

text: be

text: expressed

text: in

text: terms

text: of

text: the

text: vectors

text: v

text: i

text: i

text: x

text: a

text: v

text: i

text: ij

text: j

text: where

text: (a

text: forms

text: an

text: invertible

text: matrix,

text: i.e.

text: det

text: a

text: a

text: a

text: a

text: a

text: Using

text: the

text: two

text: ij

text: diﬀerent

text: parametrizations

text: of

text: the

text: neutrino

text: mass

text: matrix,

text: we

text: can

text: ﬁnd

text: possible

text: solutions

text: of

text: a

text: ij

text: M

text: x

text: xT

text: x

text: xT

text: ν

text: a

text: a

text: v

text: vT

text: a

text: a

text: v

text: vT

text: (a

text: a

text: a

text: a

text: )v

text: vT

text: i

text: j

text: i

text: j

text: i

text: j

text: i,j

text: i,j

text: i,j

text: v

text: vT

text: i

text: i

text: i

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text: The Passarino–Veltman functions B and B [87] arise from two-point self-energies. In dimen- 22 0 sional regularisation, these two functions read [61] (cid:90) (cid:90) − − 1 1 − B (q2, m2, m2) = 1(∆ + 1)(m2 + m2 1q2) X ln(X i(cid:15)) dx , (96) 22 1 2 4 1 2 3 2 0 − 1 − B (q2, m2, m2) = ∆ ln(X i(cid:15)) dx (97) 0 1 2 0 with the deﬁnitions ≡ − − − ≡ 2 − X m2x + m2(1 x) q2x(1 x) , ∆ − + ln 4π γ (98) 1 2 4 d E (cid:39) in d space-time dimensions, where γ 0.577 is the Euler–Mascheroni constant. Note that E B and B are symmetric in their last two arguments. We use the compact analytic expressions 22 0 given in App. B of Ref. [62]. We conﬁrmed that the expressions agree with the ones in the inert doublet model [88] when taking the appropriate limit. References [1] Y. Cai, J. Herrero-Garc´ıa, M. A. Schmidt, A. Vicente and R. R. Volkas, From the trees to the forest: a review of radiative neutrino mass models, Front.in Phys. 5 (2017) 63, [1706.08524]. [2] E. Ma, I. Picek and B. Radovˇci´c, New Scotogenic Model of Neutrino Mass with U(1) D Gauge Interaction, Phys. Lett. B726 (2013) 744–746, [1308.5313]. [3] E. Ma, Veriﬁable radiative seesaw mechanism of neutrino mass and dark matter, Phys. Rev. D73 (2006) 077301, [hep-ph/0601225]. [4] A. G. Hessler, A. Ibarra, E. Molinaro and S. Vogl, Probing the scotogenic FIMP at the LHC, JHEP 01 (2017) 100, [1611.09540]. [5] A. Vicente and C. E. Yaguna, Probing the scotogenic model with lepton ﬂavor violating processes, JHEP 02 (2015) 144, [1412.2545]. [6] E. Molinaro, C. E. Yaguna and O. Zapata, FIMP realization of the scotogenic model, JCAP 1407 (2014) 015, [1405.1259]. [7] T. Toma and A. Vicente, Lepton Flavor Violation in the Scotogenic Model, JHEP 01 (2014) 160, [1312.2840]. [8] J. Racker, Mass bounds for baryogenesis from particle decays and the inert doublet model, JCAP 1403 (2014) 025, [1308.1840]. [9] D. Schmidt, T. Schwetz and T. Toma, Direct Detection of Leptophilic Dark Matter in a Model with Radiative Neutrino Masses, Phys. Rev. D85 (2012) 073009, [1201.0906]. [10] G. B. Gelmini, E. Osoba and S. Palomares-Ruiz, Inert-Sterile Neutrino: Cold or Warm Dark Matter Candidate, Phys. Rev. D81 (2010) 063529, [0912.2478]. 36 ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: The

text: Passarino–Veltman

text: functions

text: B

text: and

text: B

text: arise

text: from

text: two-point

text: self-energies.

text: In

text: dimen-

text: sional

text: regularisation,

text: these

text: two

text: functions

text: read

text: B

text: X

text: ln(X

text: dx

text: B

text: ln(X

text: dx

text: with

text: the

text: deﬁnitions

text: X

text: x)

text: x)

text: ln

text: 4π

text: γ

text: d

text: E

text: in

text: d

text: space-time

text: dimensions,

text: where

text: γ

text: is

text: the

text: Euler–Mascheroni

text: constant.

text: Note

text: that

text: E

text: B

text: and

text: B

text: are

text: symmetric

text: in

text: their

text: last

text: two

text: arguments.

text: We

text: use

text: the

text: compact

text: analytic

text: expressions

text: given

text: in

text: App.

text: B

text: of

text: Ref.

text: We

text: conﬁrmed

text: that

text: the

text: expressions

text: agree

text: with

text: the

text: ones

text: in

text: the

text: inert

text: doublet

text: model

text: when

text: taking

text: the

text: appropriate

text: limit.

text: References

text: Y.

text: Cai,

text: J.

text: Herrero-Garc´ıa,

text: M.

text: A.

text: Schmidt,

text: A.

text: Vicente

text: and

text: R.

text: R.

text: Volkas,

text: From

text: the

text: trees

text: to

text: the

text: forest:

text: a

text: review

text: of

text: radiative

text: neutrino

text: mass

text: models,

text: Front.in

text: Phys.

text: E.

text: Ma,

text: I.

text: Picek

text: and

text: B.

text: Radovˇci´c,

text: New

text: Scotogenic

text: Model

text: of

text: Neutrino

text: Mass

text: with

text: D

text: Gauge

text: Interaction,

text: Phys.

text: Lett.

text: E.

text: Ma,

text: Veriﬁable

text: radiative

text: seesaw

text: mechanism

text: of

text: neutrino

text: mass

text: and

text: dark

text: matter,

text: Phys.

text: Rev.

text: A.

text: G.

text: Hessler,

text: A.

text: Ibarra,

text: E.

text: Molinaro

text: and

text: S.

text: Vogl,

text: Probing

text: the

text: scotogenic

text: FIMP

text: at

text: the

text: LHC,

text: JHEP

text: A.

text: Vicente

text: and

text: C.

text: E.

text: Yaguna,

text: Probing

text: the

text: scotogenic

text: model

text: with

text: lepton

text: ﬂavor

text: violating

text: processes,

text: JHEP

text: E.

text: Molinaro,

text: C.

text: E.

text: Yaguna

text: and

text: O.

text: Zapata,

text: FIMP

text: realization

text: of

text: the

text: scotogenic

text: model,

text: JCAP

text: T.

text: Toma

text: and

text: A.

text: Vicente,

text: Lepton

text: Flavor

text: Violation

text: in

text: the

text: Scotogenic

text: Model,

text: JHEP

text: J.

text: Racker,

text: Mass

text: bounds

text: for

text: baryogenesis

text: from

text: particle

text: decays

text: and

text: the

text: inert

text: doublet

text: model,

text: JCAP

text: D.

text: Schmidt,

text: T.

text: Schwetz

text: and

text: T.

text: Toma,

text: Direct

text: Detection

text: of

text: Leptophilic

text: Dark

text: Matter

text: in

text: a

text: Model

text: with

text: Radiative

text: Neutrino

text: Masses,

text: Phys.

text: Rev.

text: G.

text: B.

text: Gelmini,

text: E.

text: Osoba

text: and

text: S.

text: Palomares-Ruiz,

text: Inert-Sterile

text: Neutrino:

text: Cold

text: or

text: Warm

text: Dark

text: Matter

text: Candidate,

text: Phys.

text: Rev.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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