text: Quality Assessment Metrics for Edge Detection and Edge-aware Filtering: A Tutorial Review Diana Sadykova, and Alex Pappachen James School of Engineering Nazarbayev University, Astana Email: Diana.Sadykova@nu.edu.kz; apj@ieee.org Abstract— whole source of information is provided, however, this might The quality assessment of edges in an image is an important lead to vast data transmission. RR comparison can solve this topic as it helps to benchmark the performance of edge detectors, problem, but regions of interests should be carefully selected and edge-aware ﬁlters that are used in a wide range of image [8]. The least used method is NR because it is entirely based processing tasks. The most popular image quality metrics such as Mean squared error (MSE), Peak signal-to-noise ratio (PSNR) on information from a distorted image [9]. and Structural similarity (SSIM) metrics for assessing and Commonly used metrics are MSE, PSNR and SSIM. MSE justifying the quality of edges. However, they do not address is basically a weighted function of deviations in images, or the structural and functional accuracy of edges in images with a square difference between compared images [10]. In Eq. 1, wide range of natural variabilities. In this review, we provide an overview of all the most relevant performance metrics that can be M and N stands for image size, while I1(s, t) and I2(s, t) used to benchmark the quality performance of edges in images. for locations. (cid:88) (cid:88) We identify four major groups of metrics and also provide a critical insight into the evaluation protocol and governing 1 M N − equations. MSE = (I (i, j) I (i, j))2 (1) MN 1 2 Index Terms—Edge preservation, image metrics, edge quality i=1 j=1 measures, edge detection, sobel ﬁlters, mean square error, PSNR, SSIM Another measure, which is strongly related to MSE is PSNR, deﬁned by Eq. 2. It indicates the level of losses or I. INTRODUCTION signals integrity [11]. Edges in the images from the most basic visual cues max(I)2 for the identiﬁcation and recognition objects. Application of PSNR = 10log( ) (2) MSE conventional image ﬁlters results in blurring of the edges due to inhomogeneity in pixel intensity values. Various edge aware Although these measures are widely used, experimental ﬁlters have been introduced as a promising solution, including results showed, that images with a different level of distortion anisotropic diffusion, bilateral ﬁlter, domain transform ﬁlter, might result in similar PSNR and MSE values [12]. guided ﬁlter, wavelet transform based ﬁltering and G-neighbor SSIM is more closely related to the human visual system classiﬁcation ﬁltering [1]–[5]. It is commonly accepted that as it extracts useful information as luminance (l), contrast efﬁciency of the ﬁlters is graded according to Mean squared (c) and structure (s). It can be applied to evaluate structure error (MSE), Peak signal-to-noise ratio (PSNR) and Structural preservation and noise removal [13]. similarity index (SSIM) metrics [6]. However, these types of assessment orients only on overall image quality and therefore are not completely suitable for understanding the actual quality SSIM = function(l(I , I ), c(I , I ), s(I , I )) (3) 1 2 1 2 1 2 of edges. In this work, we provide a critical review of most promising edge-oriented metrics suitable for analysis and The main limitation of SSIM measure is inability to measure measurement of edge quality. highly blurred images [14] successfully. All the three most There are two main groups of image quality assessment common metrics MSE, PSNR, and SSIM, are limited in their for edges: subjective and objective. Subjective evaluation can use for benchmarking the performance of edge in the images. be represented by mean opinion score (MOS) or an average of human visual assessment. This type of the evaluation is II. EDGE-AWARE PERFORMANCE METRICS considered to be more exact and accurate [7]. The second Figure 1 shows the overview of the metrics that can be approach is applied compared with a set standards and uti- useful for benchmarking the performance of edge quality of lization of an original or an ideal image. Objective techniques objects in the images after edge detection or after applying an are divided into three categories: (1) full reference (FR), (2) edge-aware ﬁlter. Overall, we can group the metrics into four reduced reference (RR) and (3) no reference (NR) evaluation. distinct categories, i.e. that based on (a) Sobel ﬁltering, (2) The FR evaluation results in a better performance, as the SSIM, (3) MSE and (4) Gaussian kernel functions. ##LTLine## ##LTLine##

text: Quality

text: Assessment

text: Metrics

text: for

text: Edge

text: Detection

text: and

text: Edge-aware

text: Filtering:

text: A

text: Tutorial

text: Review

text: Diana

text: Sadykova,

text: and

text: Alex

text: Pappachen

text: James

text: School

text: of

text: Engineering

text: Nazarbayev

text: University,

text: Astana

text: Email:

text: Diana.Sadykova@nu.edu.kz;

text: apj@ieee.org

text: Abstract—

text: whole

text: source

text: of

text: information

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text: however,

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text: vast

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text: RR

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text: performance

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text: ##LTLine##

text: ##LTLine##


text: (cid:88) (cid:88) (cid:112) 1 M N − EBIQA = (I I )2 (6) MN 1 2 i=1 j=1 Merits: The main merits of using the EBIQA metric are: • Application of Sobel ﬁlter considered to be time efﬁcient, effective and simple. • Provides better performance than PSNR and SSIM in FR evaluation. • Assessment of edge similarity showed results similar to MOS i.e. had a good response to image degradation. • Might be further utilized for RR or NR. Demerits: • The need to have large data transmission . Fig. 1. Overview of edge based IQA B. Sobel-based reduced reference evaluation The work which was done by Martini et. al [17] also draws A. EBIQA: Edge based image quality assessment upon the estimation of change in edge-based features. Similar Edge preservation one of the most important aspect during to EBIQA Sobel ﬁltering is applied for edge identiﬁcation the human visual assessment. Attar, Shahbahrami, and Rad [15]. In this ﬁlter, weighted pixel difference equation is applied proposed EBIQA technique that aims to operate on the human × for 3 3 windows, and then the pixel is considered to be perception of the features [15]. The metric consists of the four an edge if it is bigger than a given threshold. The overall major steps, where original and distorted images are compared: algorithm for the implementation includes: 1) Edges locations are identiﬁed utilizing Sobel edge de- × 1) The image is sub-sampled to have 16 16 blocks. tector technique in both images. × 2) Then a selection of 12 out of all blocks is done using 2) The 16 16 pixel window size vectors are formed region of interest with centered symmetry as it is illus- at each image based on Eq.4 and Eq. 5, where I is 1 trated in Fig.2. reference image and I is the image under test [16]. 2 I = (O, AL, PL, N, V HO) (4) 1 I = (O, AL, PL, N, V HO) (5) 2 Here, each vector uses information about ﬁve edge oriented characteristics: • ’O’ stands for edge orientation in the image, which Fig. 2. Example of 12 blocks considered for RR evaluation is a total number of edges. • ’AL’ or value of the average length of all edges. 3) Sobel ﬁlter is applied separately to each of the selected • ’PL’ estimates the number of pixels with a similar block, rather than to the full image as in the case of level of intensity values. EBIQA. • ’N’ responds for sum of pixels, which form edges. 4) Number of edges evaluated by low pass ﬁltering edges • ’VH’ corresponds for sum of pixels, which form by application of thresholds, referred as t and t . Then 1 2 edges in either vertically or horizontally located the proposed parameter is estimated using Eq. 7, where edges. × w ,w are constant and M N represent the total 1 2 Note that initially in 2011 authors proposed only four number of pixels. (cid:88) edge related features for vector creation, but later in 2016 one more criterion was included for better per- MN 1 w t + w t formance, which is ’VH’ [16]. I = 1 1 2 1,2 (7) MN MN 3) Finally, we estimate EBIQA by Eq. 6, where average i=1 Euclidean distance of proposed vectors is estimated. ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

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text: where

text: average

text: Euclidean

text: distance

text: of

text: proposed

text: vectors

text: is

text: estimated.

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: √ √ Solving yields ui = Ck + 1 (Ck + Ci) and uj = Ck + 1 (Ck − Cj). Note that √ 0 3v 0 3v uj < Cj should hold, which is only true for Ck < Cj. Combining this restriction 0 with the conservation of mass we obtain Cj < Ci < 2Cj. Case 5, U-S-S Cj, Ck ≥ 0 given. From the conservation of mass we obtain (cid:16) (cid:17) Ci = Cj + Ck. The corresponding values of uj and uj are obtained from 0 0 √ 1 (cid:16) (cid:17) −v Ci − Ci = vuj − Cj + vuk − Ck 2 0 0 1 (cid:16) √ (cid:17) Cj + vuj = Ci − v Ci + vuk − Ck 0 2 0 1 √ Ck + vuk = Ci − v Ci + vuj − Cj . 0 2 0 We obtain −1 √ −1 √ uj = (Ci + Cj) − Ci , uk = (Ci + Ck) − Ci. 0 3v 0 3v Case 6, S-S-S Ci, Cj, Ck ≥ 0 given(cid:16). Then the conservatio(cid:17)n of mass requires Ci = Cj + Ck. Then 1 (cid:0) (cid:1) vui − Ci = vuj − Cj + vuk − Ck 0 2 0 0 (cid:16) (cid:17) 1 Cj + vuj = Ci + vui + vuk − Ck 0 2 0 0 1 Ck + vuk = Ci + vui + vuj − Cj , 0 2 0 0 which is a linear system for ui , uj, uk, but with rank 2. The values of ui , uj, uk are 0 0 0 0 0 0 not uniquely determined. Assuming the states ui , uj , uk to be given, we have to determine the new states B B B ui , uj and uk at the node. We have to consider eight diﬀerent conﬁgurations. For K K K each of them all possible combinations with stable or unstable layer solutions have to be discussed. Not admissible combinations are not listed. Case 1, RP1-1-1 ui ∈ (−∞, 0] : (U) or ((S) with Ci = 0) K uj ∈ [0, ∞) : (U) or ((S) with Cj = 0) K uk ∈ [0, ∞) : (U) or ((S) with Ck = 0) K SUU with Ci = 0 ⇒ Cj = Ck = 0 and ui = 0. 0 SSU with Ci = Cj = 0 ⇒ Ck = 0 and ui = uj = 0. 0 0 SUS with Ci = Ck = 0 ⇒ Cj = 0 and ui = uk = 0. 0 0 USS with Cj = Ck = 0 ⇒ Ci = 0 and uj = uk = 0. 0 0 SSS with Ci = Cj = Ck = 0 ⇒ ui = uj = uk = 0. 0 0 0 This gives ui = 0 uj = 0 uk = 0 K K K ui = 0 uj = 0 uk = 0 . 0 0 0 13 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Solving

text: yields

text: ui

text: Ck

text: (Ck

text: Ci)

text: and

text: uj

text: Ck

text: (Ck

text: Cj).

text: Note

text: that

text: uj

text: Cj

text: should

text: hold,

text: which

text: is

text: only

text: true

text: for

text: Ck

text: Cj.

text: Combining

text: this

text: restriction

text: with

text: the

text: conservation

text: of

text: mass

text: we

text: obtain

text: Cj

text: Ci

text: Case

text: U-S-S

text: Cj,

text: Ck

text: given.

text: From

text: the

text: conservation

text: of

text: mass

text: we

text: obtain

text: Ci

text: Cj

text: Ck.

text: The

text: corresponding

text: values

text: of

text: uj

text: and

text: uj

text: are

text: obtained

text: from

text: −v

text: Ci

text: Ci

text: vuj

text: Cj

text: vuk

text: Ck

text: Cj

text: vuj

text: Ci

text: v

text: Ci

text: vuk

text: Ck

text: Ck

text: vuk

text: Ci

text: v

text: Ci

text: vuj

text: Cj

text: We

text: obtain

text: −1

text: −1

text: uj

text: (Ci

text: Cj)

text: Ci

text: uk

text: (Ci

text: Ck)

text: Ci.

text: Case

text: S-S-S

text: Ci,

text: Cj,

text: Ck

text: Then

text: the

text: of

text: mass

text: requires

text: Ci

text: Cj

text: Ck.

text: Then

text: vui

text: Ci

text: vuj

text: Cj

text: vuk

text: Ck

text: Cj

text: vuj

text: Ci

text: vui

text: vuk

text: Ck

text: Ck

text: vuk

text: Ci

text: vui

text: vuj

text: Cj

text: which

text: is

text: a

text: linear

text: system

text: for

text: ui

text: uj,

text: uk,

text: but

text: with

text: rank

text: The

text: values

text: of

text: ui

text: uj,

text: uk

text: are

text: not

text: uniquely

text: determined.

text: Assuming

text: the

text: states

text: ui

text: uj

text: uk

text: to

text: be

text: given,

text: we

text: have

text: to

text: determine

text: the

text: new

text: states

text: B

text: B

text: B

text: ui

text: uj

text: and

text: uk

text: at

text: the

text: node.

text: We

text: have

text: to

text: consider

text: eight

text: diﬀerent

text: conﬁgurations.

text: For

text: K

text: K

text: K

text: each

text: of

text: them

text: all

text: possible

text: combinations

text: with

text: stable

text: or

text: unstable

text: layer

text: solutions

text: have

text: to

text: be

text: discussed.

text: Not

text: admissible

text: combinations

text: are

text: not

text: listed.

text: Case

text: ui

text: (U)

text: or

text: ((S)

text: with

text: Ci

text: K

text: uj

text: (U)

text: or

text: ((S)

text: with

text: Cj

text: K

text: uk

text: (U)

text: or

text: ((S)

text: with

text: Ck

text: K

text: SUU

text: with

text: Ci

text: Cj

text: Ck

text: and

text: ui

text: SSU

text: with

text: Ci

text: Cj

text: Ck

text: and

text: ui

text: uj

text: SUS

text: with

text: Ci

text: Ck

text: Cj

text: and

text: ui

text: uk

text: USS

text: with

text: Cj

text: Ck

text: Ci

text: and

text: uj

text: uk

text: SSS

text: with

text: Ci

text: Cj

text: Ck

text: ui

text: uj

text: uk

text: This

text: gives

text: ui

text: uj

text: uk

text: K

text: K

text: K

text: ui

text: uj

text: uk

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: Case 6, RP2-1-2 ui > 0, uj > 0, uk < 0. Then we have 3 subcases. √ B B B If ui ≥ − 2uk , then C = Ci , C = Ci and B B j 2 k 2 1 1 ui = ui > 0 uj = √ ui > 0 uk = √ ui > 0 K B K 2 B K 2 B √ (cid:113) If − 2uk > ui ≥ −uk , then C = C − C and B B B j i k ui = ui > 0 uj = (ui )2 − (uk )2 > 0 uk = uk < 0 K B K B B K B If ui < −uk , then C = C , C = 0 and B B i k j ui = uk < 0 uj = 0 uk = uk < 0 K B K K B Case 7, RP2-2-1 ui > 0, uj < 0, uk > 0. We have again 3 subcases. √ B B B If ui ≥ − 2uj , then C = Ci , C = Ci and B B j 2 k 2 1 1 ui = ui > 0 uj = √ ui > 0 uk = √ ui > 0 K B K 2 B K 2 B √ (cid:113) If − 2uj > ui ≥ −uj then C = C − C and B B B k i j ui = ui > 0 uj = uj < 0 uk = (ui )2 − (uj )2 > 0 K B K B K B B If ui < −uj , then C = C , C = 0 and B B i j k ui = uj < 0 uj = uj < 0 uk = 0 K B K B K Case 8, RP2-2-2 ui > 0, uj < 0, uk < 0. We have 4 subcases. √ B √ B B If ui ≥ − 2uj , ui ≥ − 2uk then C = Ci , C = Ci and B B B B j 2 k 2 1 1 ui = ui > 0 (cid:113) uj = √ ui > 0 uk = √ ui > 0 K B K 2 B K 2 B √ (cid:113) If − 2uj > ui ≥ (uj )2 + (uk )2, then C = C − C and B B B B k i j (cid:113) ui = ui > 0 uj = uj < 0 uk = (ui )2 − (uj )2 > 0 K B K B K B B √ (cid:113) If − 2uk > ui ≥ (uj )2 + (uk )2, then C = C − C and B B B B j i k (cid:113) ui = ui > 0 uj = (ui )2 − (uk )2 > 0 uk = uk < 0 K B K B B K B (cid:113) If ui ≤ (uj )2 + (uk )2, then C = C + C and B B B i j k ui = − (uj )2 + (uk )2 < 0 uj = uj < 0 uk = uk > 0 K B B K B K B 19 ##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##

text: Case

text: ui

text: uj

text: uk

text: Then

text: we

text: have

text: subcases.

text: B

text: B

text: B

text: If

text: ui

text: then

text: C

text: Ci

text: C

text: Ci

text: and

text: B

text: B

text: j

text: k

text: ui

text: ui

text: uj

text: ui

text: uk

text: ui

text: K

text: B

text: K

text: B

text: K

text: B

text: If

text: ui

text: −uk

text: then

text: C

text: C

text: C

text: and

text: B

text: B

text: B

text: j

text: i

text: k

text: ui

text: ui

text: uj

text: (ui

text: (uk

text: uk

text: uk

text: K

text: B

text: K

text: B

text: B

text: K

text: B

text: If

text: ui

text: −uk

text: then

text: C

text: C

text: C

text: and

text: B

text: B

text: i

text: k

text: j

text: ui

text: uk

text: uj

text: uk

text: uk

text: K

text: B

text: K

text: K

text: B

text: Case

text: ui

text: uj

text: uk

text: We

text: have

text: again

text: subcases.

text: B

text: B

text: B

text: If

text: ui

text: then

text: C

text: Ci

text: C

text: Ci

text: and

text: B

text: B

text: j

text: k

text: ui

text: ui

text: uj

text: ui

text: uk

text: ui

text: K

text: B

text: K

text: B

text: K

text: B

text: If

text: ui

text: −uj

text: then

text: C

text: C

text: C

text: and

text: B

text: B

text: B

text: k

text: i

text: j

text: ui

text: ui

text: uj

text: uj

text: uk

text: (ui

text: (uj

text: K

text: B

text: K

text: B

text: K

text: B

text: B

text: If

text: ui

text: −uj

text: then

text: C

text: C

text: C

text: and

text: B

text: B

text: i

text: j

text: k

text: ui

text: uj

text: uj

text: uj

text: uk

text: K

text: B

text: K

text: B

text: K

text: Case

text: ui

text: uj

text: uk

text: We

text: have

text: subcases.

text: B

text: B

text: B

text: If

text: ui

text: ui

text: then

text: C

text: Ci

text: C

text: Ci

text: and

text: B

text: B

text: B

text: B

text: j

text: k

text: ui

text: ui

text: uj

text: ui

text: uk

text: ui

text: K

text: B

text: K

text: B

text: K

text: B

text: If

text: ui

text: (uj

text: (uk

text: then

text: C

text: C

text: C

text: and

text: B

text: B

text: B

text: B

text: k

text: i

text: j

text: ui

text: ui

text: uj

text: uj

text: uk

text: (ui

text: (uj

text: K

text: B

text: K

text: B

text: K

text: B

text: B

text: If

text: ui

text: (uj

text: (uk

text: then

text: C

text: C

text: C

text: and

text: B

text: B

text: B

text: B

text: j

text: i

text: k

text: ui

text: ui

text: uj

text: (ui

text: (uk

text: uk

text: uk

text: K

text: B

text: K

text: B

text: B

text: K

text: B

text: If

text: ui

text: (uj

text: (uk

text: then

text: C

text: C

text: C

text: and

text: B

text: B

text: B

text: i

text: j

text: k

text: ui

text: (uj

text: (uk

text: uj

text: uj

text: uk

text: uk

text: K

text: B

text: B

text: K

text: B

text: K

text: B

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: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: RIGIDITY 17 (cid:54) Since we assume that grad f (p ) = 0, let s = 1 and s = 0 we have K 0 2 1 ≥ f (g (p )) > f (p ) 0. K 1 0 K 0 Therefore there exists a small enough constant r > 0 such that 0 inf f (g (q)) > sup f (q) ∈ K 1 ∈ K q B(p0,r0) q B(p0,r0) ⊂ where B(p , r ) M is the geodesic ball centered at p of radius r . 0 0 0 0 In particular we have ∩ ∅ (3.3) B(p , r ) g (B(p , r )) = . 0 0 1 0 0 Inequality (3.2) and equation (3.3) give that ∩ ∅ ∀ ∈ Z (3.4) B(p , r ) g (B(p , r )) = n +. 0 0 n 0 0 Which also implies ∩ ∅ ∀ (cid:54) ∈ Z (3.5) g (B(p , r )) g (B(p , r )) = n = m +. n 0 0 m 0 0 ≥ ∈ Otherwise there exist two positive integers n > m 1 and q , q 0 0 1 2 B(p , r ) such that g (q ) = g (q ). Since g is a ﬂow, g − (q ) = q 0 0 n0 1 m0 2 t n0 m0 1 2 which contradicts equation (3.4). On the other hand, for any t > 0 (we use Proposition 18.18 in [Lee13]), 0 (cid:90) we have d Vol(g (B(p , r )))| (cid:90) d | ∗ t 0 0 (3.6) = g (dVol) dt t=t0 dt t=t0 t B(p0,r0) (cid:90) ∗ = g (L(cid:32) (dVol)) t0 gradfK B(p0,r0) (cid:90) ∗ = g (Div(grad(f )) dVol) t0 K B(p0,r0) = ∆f dVol . K gt0(B(p0,r0)) ##LTLine## ##LTLine##

text: RIGIDITY

font: default

text: Since

text: we

text: assume

text: that

text: grad

text: f

text: (p

text: let

text: s

text: and

text: s

text: we

text: have

text: K

text: f

text: (g

text: (p

text: f

text: (p

text: K

text: K

text: Therefore

text: there

text: exists

text: a

text: small

text: enough

text: constant

text: r

text: such

text: that

text: inf

text: f

text: (g

text: (q))

text: sup

text: f

text: (q)

text: K

text: K

text: q

text: q

text: where

text: B(p

text: r

text: M

text: is

text: the

text: geodesic

text: ball

text: centered

text: at

text: p

text: of

text: radius

text: r

text: In

text: particular

text: we

text: have

text: B(p

text: r

text: g

text: (B(p

text: r

text: Inequality

text: and

text: equation

text: give

text: that

text: Z

text: B(p

text: r

text: g

text: (B(p

text: r

text: n

text: n

text: Which

text: also

text: implies

text: Z

text: g

text: (B(p

text: r

text: g

text: (B(p

text: r

text: n

text: m

text: n

text: m

text: Otherwise

text: there

text: exist

text: two

text: positive

text: integers

text: n

text: m

text: and

text: q

text: q

text: B(p

text: r

text: such

text: that

text: g

text: (q

text: g

text: (q

text: Since

text: g

text: is

text: a

text: ﬂow,

text: g

text: (q

text: q

text: t

text: which

text: contradicts

text: equation

text: On

text: the

text: other

text: hand,

text: for

text: any

text: t

text: (we

text: use

text: Proposition

text: in

text: we

text: have

text: d

text: Vol(g

text: (B(p

text: r

text: d

text: t

text: g

text: (dVol)

text: dt

text: dt

text: t

text: g

text: (dVol))

text: gradfK

text: g

text: (Div(grad(f

text: dVol)

text: K

text: ∆f

text: dVol

text: K

text: ##LTLine##

text: ##LTLine##


text: RIGIDITY 21 In this subsection we will ﬁnish the proof of Proposition 3.2. Proof of Proposition 3.2. We argue by contradiction. Assume that Aut (D) contains a nontrivial compact factor I. 0 By Proposition 3.1 we know that Aut (D) is semisimple. Then we may ⊂ 0 assume that K Aut (D) is the maximal compact factor containing I. On 0 the level of Lie algebras, the Lie algebra k of K is a factor of g. Since K ⊂ N → is characteristic in Aut (D), Γ (K). So the map ad (γ) : k k is ∈0 K well-deﬁned for any γ Γ. Since the Killing form is a canonical bi-invariant ∈ metric on k, for any γ Γ → ad (γ) : k k K is an isometry. Similar to the argument as in step-1 in the proof of Proposition 3.1 we { } let X ≤ ≤ be an orthonormal basis for k where k = dim(k). And deﬁne i 1 i k a function → R f : D K (cid:88) by k | | f (x) = X (x) 2 K i i=1 · | where X (x) = d (exp(tX ) x) . i ∈ dt i t=0 For any γ Γ, · d · − · · · | X (γ x) = (γ (γ 1 exp(tX ) γ x)) i i t=0 dt · · = dγ(x) ad (γ) X (x). K i As above we know that ad (γ) acts on k as an isometry. By Theorem 2.1 K ∈ we also know that γ acts on D as an isometry. Thus, we have for any γ Γ ∈ and x D, · f (γ x) = f (x). K K So f descends to a function, still denoted by f , on M = D/Γ. K K ∈ Then we apply [Fra89, Lemma 10.15] to get that for all p D, ≥ ≥ ∆f (p) f (p) 0. K K Again we get a nonnegtaive subharmonic function on M. Recall that Theorem 2.1 tells that D is complete. In particular M is complete and has ﬁnite volume. Then we can apply the same argument as in step-1 in the proof of Proposition 3.1 to get ≡ f 0 on M K ##LTLine## ##LTLine##

text: RIGIDITY

text: In

text: this

text: subsection

text: we

text: will

text: ﬁnish

text: the

text: proof

text: of

text: Proposition

text: Proof

text: of

text: Proposition

text: We

text: argue

text: by

text: contradiction.

text: Assume

text: that

text: Aut

text: (D)

text: contains

text: a

text: nontrivial

text: compact

text: factor

text: I.

text: By

text: Proposition

text: we

text: know

text: that

text: Aut

text: (D)

text: is

text: semisimple.

text: Then

text: we

text: may

text: assume

text: that

text: K

text: Aut

text: (D)

text: is

text: the

text: maximal

text: compact

text: factor

text: containing

text: I.

text: On

text: the

text: level

text: of

text: Lie

text: algebras,

text: the

text: Lie

text: algebra

text: k

text: of

text: K

text: is

text: a

text: factor

text: of

text: g.

text: Since

text: K

text: N

text: is

text: characteristic

text: in

text: Aut

text: (D),

text: Γ

text: (K).

text: So

text: the

text: map

text: ad

text: (γ)

text: k

text: k

text: is

text: ∈0

text: K

text: well-deﬁned

text: for

text: any

text: γ

text: Γ.

text: Since

text: the

text: Killing

text: form

text: is

text: a

text: canonical

text: bi-invariant

text: metric

text: on

text: k,

text: for

text: any

text: γ

text: Γ

text: ad

text: (γ)

text: k

text: k

text: K

text: is

text: an

text: isometry.

text: Similar

text: to

text: the

text: argument

text: as

text: in

text: in

text: the

text: proof

text: of

text: Proposition

text: we

text: let

text: X

text: be

text: an

text: orthonormal

text: basis

text: for

text: k

text: where

text: k

text: dim(k).

text: And

text: deﬁne

text: i

text: i

text: k

text: a

text: function

text: R

text: f

text: D

text: K

text: by

text: k

text: f

text: (x)

text: X

text: (x)

text: K

text: i

text: where

text: X

text: (x)

text: d

text: (exp(tX

text: x)

text: i

text: dt

text: i

text: For

text: any

text: γ

text: Γ,

text: d

text: X

text: (γ

text: x)

text: (γ

text: (γ

text: exp(tX

text: γ

text: x))

text: i

text: i

text: dt

text: dγ(x)

text: ad

text: (γ)

text: X

text: (x).

text: K

text: i

text: As

text: above

text: we

text: know

text: that

text: ad

text: (γ)

text: acts

text: on

text: k

text: as

text: an

text: isometry.

text: By

text: Theorem

text: K

text: we

text: also

text: know

text: that

text: γ

text: acts

text: on

text: D

text: as

text: an

text: isometry.

text: Thus,

text: we

text: have

text: for

text: any

text: γ

text: Γ

text: and

text: x

text: D,

text: f

text: (γ

text: x)

text: f

text: (x).

text: K

text: K

text: So

text: f

text: descends

text: to

text: a

text: function,

text: still

text: denoted

text: by

text: f

text: on

text: M

text: D/Γ.

text: K

text: K

text: Then

text: we

text: apply

text: Lemma

text: to

text: get

text: that

text: for

text: all

text: p

text: D,

text: ∆f

text: (p)

text: f

text: (p)

text: K

text: K

text: Again

text: we

text: get

text: a

text: nonnegtaive

text: subharmonic

text: function

text: on

text: M.

text: Recall

text: that

text: Theorem

text: tells

text: that

text: D

text: is

text: complete.

text: In

text: particular

text: M

text: is

text: complete

text: and

text: has

text: ﬁnite

text: volume.

text: Then

text: we

text: can

text: apply

text: the

text: same

text: argument

text: as

text: in

text: in

text: the

text: proof

text: of

text: Proposition

text: to

text: get

text: f

text: on

text: M

text: K

text: ##LTLine##

text: ##LTLine##


text: RIGIDITY 25 the orbit Aut (y) at y is J-invariant where J is the complex structure tensor 0 ∈ for D. At the point x D we have the following direct sum decomposition of real tangent vector spaces as · ⊕ T (D) = T (Aut (D) x) T (F ). x x 0 x Since K acts trivially on the second summand the nontrivially on the ﬁrst, x we see that the summands are J-invariant since the action of K on T (D) x x commutes with the action of J. Therefore, we have that the map H is biholomorphic. From [Nad90, · · Lemma 6.1] we know that the orbit Aut (D) x = (Aut (D)/K ) x is 0 0 x D biholomorphic to the unit disk . Since D is contractible, F is simply- connected. The uniformization theorem of Riemann surfaces implies that F C D ≥ must be biholomorphic to P1, or . Thus, Aut (F ) 2. Therefore, we 0 have ≥ D dimR Aut (D) dimR Aut ( ) + dimR Aut (F ) > 3 0 0 0 which contradicts to (4.2). Now we consider Case 2. As above we know that only assertion (i) of Theorem 4.2 may happen. → Thus, D D/ Aut (D) is a locally trivial ﬁbration. Since D is con- 0 tractible, it follows from the standard exact homotopy sequence of the ﬁbra- → tion D D/ Aut (D) that D/ Aut (D) is contractible (One may also apply 0 0 the work in [Oli76] in a more general setting to conclude that D/ Aut (D) { · } 0 is contractible). Thus, by our assumptions that dimR Aut (D) x = 3 for ∈ 0 all x D and dimR(D) = 4 we have D/ Aut (D) is homeomorphic to the 0 real line. That is, ∼ R D/ Aut (D) = . 0 ∩ Recall that Γ = π (M) and Γ = Γ Aut (D). The following eﬀective 1 0 0 action of the group Γ/Γ on D/ Aut (D) is well deﬁned: 0 0 × → Γ/Γ D/ Aut (D) D/ Aut (D) 0 0 0 (cid:55)→ (γΓ , [x]) [γ(x)]. 0 Set T = (D/ Aut (D))/(Γ/Γ ). 0 0 It is not hard to see that T is a manifold and the action above induces a natural map → θ : M T ∈ deﬁned by θ(p) = [p˜] where p˜ D is a lift point of p, which is a locally trivial ﬁbration (one may see [Sha77, Page 140] for more details). Since

text: RIGIDITY

text: the

text: orbit

text: Aut

text: (y)

text: at

text: y

text: is

text: J-invariant

text: where

text: J

text: is

text: the

text: complex

text: structure

text: tensor

text: for

text: D.

text: At

text: the

text: point

text: x

text: D

text: we

text: have

text: the

text: following

text: direct

text: sum

text: decomposition

text: of

text: real

text: tangent

text: vector

text: spaces

text: as

text: T

text: (D)

text: T

text: (Aut

text: (D)

text: x)

text: T

text: (F

text: x

text: x

text: x

text: Since

text: K

text: acts

text: trivially

text: on

text: the

text: second

text: summand

text: the

text: nontrivially

text: on

text: the

text: ﬁrst,

text: x

text: we

text: see

text: that

text: the

text: summands

text: are

text: J-invariant

text: since

text: the

text: action

text: of

text: K

text: on

text: T

text: (D)

text: x

text: x

text: commutes

text: with

text: the

text: action

text: of

text: J.

text: Therefore,

text: we

text: have

text: that

text: the

text: map

text: H

text: is

text: biholomorphic.

text: From

text: Lemma

text: we

text: know

text: that

text: the

text: orbit

text: Aut

text: (D)

text: x

text: (Aut

text: (D)/K

text: x

text: is

text: x

text: D

text: biholomorphic

text: to

text: the

text: unit

text: disk

text: Since

text: D

text: is

text: contractible,

text: F

text: is

text: simply-

text: connected.

text: The

text: uniformization

text: theorem

text: of

text: Riemann

text: surfaces

text: implies

text: that

text: F

text: C

text: D

text: must

text: be

text: biholomorphic

text: to

text: or

text: Thus,

text: Aut

text: (F

text: Therefore,

text: we

text: have

text: D

text: dimR

text: Aut

text: (D)

text: dimR

text: Aut

text: dimR

text: Aut

text: (F

text: which

text: contradicts

text: to

text: Now

text: we

text: consider

text: Case

text: As

text: above

text: we

text: know

text: that

text: only

text: assertion

text: (i)

text: of

text: Theorem

text: may

text: happen.

text: Thus,

text: D

text: D/

text: Aut

text: (D)

text: is

text: a

text: locally

text: trivial

text: ﬁbration.

text: Since

text: D

text: is

text: con-

text: tractible,

text: it

text: follows

text: from

text: the

text: standard

text: exact

text: homotopy

text: sequence

text: of

text: the

text: ﬁbra-

text: tion

text: D

text: D/

text: Aut

text: (D)

text: that

text: D/

text: Aut

text: (D)

text: is

text: contractible

text: (One

text: may

text: also

text: apply

text: the

text: work

text: in

text: in

text: a

text: more

text: general

text: setting

text: to

text: conclude

text: that

text: D/

text: Aut

text: (D)

text: is

text: contractible).

text: Thus,

text: by

text: our

text: assumptions

text: that

text: dimR

text: Aut

text: (D)

text: x

text: for

text: all

text: x

text: D

text: and

text: dimR(D)

text: we

text: have

text: D/

text: Aut

text: (D)

text: is

text: homeomorphic

text: to

text: the

text: real

text: line.

text: That

text: is,

text: R

text: D/

text: Aut

text: (D)

text: Recall

text: that

text: Γ

text: π

text: (M)

text: and

text: Γ

text: Γ

text: Aut

text: (D).

text: The

text: following

text: eﬀective

text: action

text: of

text: the

text: group

text: Γ/Γ

text: on

text: D/

text: Aut

text: (D)

text: is

text: well

text: deﬁned:

text: Γ/Γ

text: D/

text: Aut

text: (D)

text: D/

text: Aut

text: (D)

text: (γΓ

text: [x])

text: [γ(x)].

text: Set

text: T

text: (D/

text: Aut

text: (D))/(Γ/Γ

text: It

text: is

text: not

text: hard

text: to

text: see

text: that

text: T

text: is

text: a

text: manifold

text: and

text: the

text: action

text: above

text: induces

text: a

text: natural

text: map

text: θ

text: M

text: T

text: deﬁned

text: by

text: θ(p)

text: [p˜]

text: where

text: p˜

text: D

text: is

text: a

text: lift

text: point

text: of

text: p,

text: which

text: is

text: a

text: locally

text: trivial

text: ﬁbration

text: (one

text: may

text: see

text: Page

text: for

text: more

text: details).

text: Since


text: RIGIDITY 29 (cid:48) (cid:48) ∩ where Γ = Γ Aut (D) which is a ﬁnite index subgroup of Γ . 0 0 0 Proof. Consider the exact sequence { } → → → → { } (5.1) e Γ Γ Γ/Γ e . 0 0 (cid:48) Our aim is to show that after replacing Γ by a ﬁnite index subgroup Γ if necessary, the exact sequence above splits as a direct product. It is well-known [ML95, Chapter IV, Theorem 8.8] that such an extension like equation (5.1) is determined by → (i) a representation ρ : Γ/Γ Out(Γ ), and 0 0 (ii) a cohomology class in H2(Γ/Γ ; Z(Γ )) where Z(Γ )) is a Γ/Γ - 0 0 ρ 0 ρ 0 module via ρ. In particular, if the representation ρ and the center Z(Γ ) are both trivial, × 0 we get the trivial extension. That is, Γ = Γ Γ/Γ . 0 0 First from Proposition 3.1, 3.2 and Lemma 3.3 we know that Γ (or any 0 ﬁnite index subgroup of Γ ) is a lattice in a semisimple Lie group Aut (D) 0 0 without compact factors. From Proposition 5.1 we know that Aut (D) has 0 ﬁnite center. So the center Z(Γ ) is ﬁnite. By Lemma 2.3 we know that Γ 0 0 is torsion-free. So the center Z(Γ ) is trivial. Thus, it suﬃces to show that 0 (cid:48) after replacing Γ by a ﬁnite index subgroup Γ if necessary, the representation → ρ : Γ/Γ Out(Γ ) 0 0 is trivial. Consider the exact sequence { } → → → → { } (5.2) e Aut (D) < Aut (D), Γ > Γ/Γ e 0 0 0 where < Aut (D), Γ > is the smallest subgroup of Aut(D) containing 0 Aut (D) and Γ. The conjugation action of Γ on Aut (D) induces a repre- 0 0 sentation → ρ : Γ/Γ Out(Aut (D)). 1 0 0 From Proposition 3.1 we know that Aut (D) is semisimple. By [Hel01, 0 Chapter IX, Theorem 5.4] we know that Out(Aut (D)) is ﬁnite. Up to a ﬁ- 0 nite index subgroup of Γ if necessary, we may assume that the representation ρ is trivial. This gives a representation 1 → ρ : Γ/Γ Aut (D)/Z(Aut (D)). 2 0 0 0

text: RIGIDITY

text: where

text: Γ

text: Γ

text: Aut

text: (D)

text: which

text: is

text: a

text: ﬁnite

text: index

text: subgroup

text: of

text: Γ

text: Proof.

text: Consider

text: the

text: exact

text: sequence

text: e

text: Γ

text: Γ

text: Γ/Γ

text: e

text: Our

text: aim

text: is

text: to

text: show

text: that

text: after

text: replacing

text: Γ

text: by

text: a

text: ﬁnite

text: index

text: subgroup

text: Γ

text: if

text: necessary,

text: the

text: exact

text: sequence

text: above

text: splits

text: as

text: a

text: direct

text: product.

text: It

text: is

text: well-known

text: Chapter

text: IV,

text: Theorem

text: that

text: such

text: an

text: extension

text: like

text: equation

text: is

text: determined

text: by

text: (i)

text: a

text: representation

text: ρ

text: Γ/Γ

text: Out(Γ

text: and

text: (ii)

text: a

text: cohomology

text: class

text: in

text: Z(Γ

text: where

text: Z(Γ

text: is

text: a

text: Γ/Γ

text: ρ

text: ρ

text: module

text: via

text: ρ.

text: In

text: particular,

text: if

text: the

text: representation

text: ρ

text: and

text: the

text: center

text: Z(Γ

text: are

text: both

text: trivial,

text: we

text: get

text: the

text: trivial

text: extension.

text: That

text: is,

text: Γ

text: Γ

text: Γ/Γ

text: First

text: from

text: Proposition

text: and

text: Lemma

text: we

text: know

text: that

text: Γ

text: (or

text: any

text: ﬁnite

text: index

text: subgroup

text: of

text: Γ

text: is

text: a

text: lattice

text: in

text: a

text: semisimple

text: Lie

text: group

text: Aut

text: (D)

text: without

text: compact

text: factors.

text: From

text: Proposition

text: we

text: know

text: that

text: Aut

text: (D)

text: has

text: ﬁnite

text: center.

text: So

text: the

text: center

text: Z(Γ

text: is

text: ﬁnite.

text: By

text: Lemma

text: we

text: know

text: that

text: Γ

text: is

text: torsion-free.

text: So

text: the

text: center

text: Z(Γ

text: is

text: trivial.

text: Thus,

text: it

text: suﬃces

text: to

text: show

text: that

text: after

text: replacing

text: Γ

text: by

text: a

text: ﬁnite

text: index

text: subgroup

text: Γ

text: if

text: necessary,

text: the

text: representation

text: ρ

text: Γ/Γ

text: Out(Γ

text: is

text: trivial.

text: Consider

text: the

text: exact

text: sequence

text: e

text: Aut

text: (D)

text: Aut

text: (D),

text: Γ

text: Γ/Γ

text: e

text: where

text: Aut

text: (D),

text: Γ

text: is

text: the

text: smallest

text: subgroup

text: of

text: Aut(D)

text: containing

text: Aut

text: (D)

text: and

text: Γ.

text: The

text: conjugation

text: action

text: of

text: Γ

text: on

text: Aut

text: (D)

text: induces

text: a

text: repre-

text: sentation

text: ρ

text: Γ/Γ

text: Out(Aut

text: (D)).

text: From

text: Proposition

text: we

text: know

text: that

text: Aut

text: (D)

text: is

text: semisimple.

text: By

text: Chapter

text: IX,

text: Theorem

text: we

text: know

text: that

text: Out(Aut

text: (D))

text: is

text: ﬁnite.

text: Up

text: to

text: a

text: ﬁ-

text: nite

text: index

text: subgroup

text: of

text: Γ

text: if

text: necessary,

text: we

text: may

text: assume

text: that

text: the

text: representation

text: ρ

text: is

text: trivial.

text: This

text: gives

text: a

text: representation

text: ρ

text: Γ/Γ

text: Aut

text: (D)/Z(Aut

text: (D)).


text: RIGIDITY 33 In the proofs of Theorem 1.7 and 1.11 the key step is to show that Γ is 0 a lattice of a semisimple Lie group without compact factors of ﬁnite center. It is unclear for the relation between the HHR/USq manifold D and the semisimple Lie group Aut (D). The following question is interesting. 0 Question 5.3. Let D be a contractible HHR/USq complex manifold which holomorphically covers a manifold of ﬁnite-volume with the fundamental group Γ. If Γ is isomorphic to a lattice in an irreducible Hermitian sym- metric space N of noncompact type other than the hyperbolic plane, is D (anti)biholomorphic to N? Remark 5.4. If the K¨ahler-Einstein metric on D has nonpositive sectional curvature, [BE87, Theorem D] of Ballmann-Eberlein tells that D and N are isometric with respect to the K¨ahler-Einstein metrics. We end this section by the following result whose proof is a combination of several known results. It gives a positive answer to Question 5.3. Proposition 5.5 (Holomorphicity Rigidity). Let D be a contractible HHR/USq complex manifold which holomorphically covers a manifold of ﬁnite-volume whose fundamental group is Γ. If Γ is isomorphic to a lattice in an irre- ducible Hermitian symmetric space N of noncompact type without Euclidean de Rham factor other than the hyperbolic plane, then D is (anti)biholomorphic to N. Proof. From Theorem 2.2 we know that D/Γ has bounded geometry (in the sense of K¨ahler-Einstein metric). It is clear that N/Γ also has bounded geometry. Meanwhile, by Proposition 2.4 we know that the Euler charac- teristic number (cid:54) χ(Γ) = 0. Since both D/Γ and N/Γ are aspherical of bounded geometry, we apply [CG86, Corollary 5.2] to get dim(D) = dim(N) (cid:54) because χ(Γ) = 0. Since D is a HHR/USq complex manifold and D/Γ has ﬁnite volume, it follows from [Yeu09, Corollary 2] that D/Γ is quasi-projective variety of log-general type. Finally, thanks to Jost-Zuo [JZ97, Theorem 2.1] we get that D is (anti)biholomorphic to N. ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: RIGIDITY

text: In

text: the

text: proofs

text: of

text: Theorem

text: and

text: the

text: key

text: step

text: is

text: to

text: show

text: that

text: Γ

text: is

text: a

text: lattice

text: of

text: a

text: semisimple

text: Lie

text: group

text: without

text: compact

text: factors

text: of

text: ﬁnite

text: center.

text: It

text: is

text: unclear

text: for

text: the

text: relation

text: between

text: the

text: HHR/USq

text: manifold

text: D

text: and

text: the

text: semisimple

text: Lie

text: group

text: Aut

text: (D).

text: The

text: following

text: question

text: is

text: interesting.

text: Question

text: Let

text: D

text: be

text: a

text: contractible

text: HHR/USq

text: complex

text: manifold

text: which

text: holomorphically

text: covers

text: a

text: manifold

text: of

text: ﬁnite-volume

text: with

text: the

text: fundamental

text: group

text: Γ.

text: If

text: Γ

text: is

text: isomorphic

text: to

text: a

text: lattice

text: in

text: an

text: irreducible

text: Hermitian

text: sym-

text: metric

text: space

text: N

text: of

text: noncompact

text: type

text: other

text: than

text: the

text: hyperbolic

text: plane,

text: is

text: D

text: (anti)biholomorphic

text: to

text: N?

text: Remark

text: If

text: the

text: K¨ahler-Einstein

text: metric

text: on

text: D

text: has

text: nonpositive

text: sectional

text: curvature,

text: Theorem

text: D]

text: of

text: Ballmann-Eberlein

text: tells

text: that

text: D

text: and

text: N

text: are

text: isometric

text: with

text: respect

text: to

text: the

text: K¨ahler-Einstein

text: metrics.

text: We

text: end

text: this

text: section

text: by

text: the

text: following

text: result

text: whose

text: proof

text: is

text: a

text: combination

text: of

text: several

text: known

text: results.

text: It

text: gives

text: a

text: positive

text: answer

text: to

text: Question

text: Proposition

text: (Holomorphicity

text: Rigidity).

text: Let

text: D

text: be

text: a

text: contractible

text: HHR/USq

text: complex

text: manifold

text: which

text: holomorphically

text: covers

text: a

text: manifold

text: of

text: ﬁnite-volume

text: whose

text: fundamental

text: group

text: is

text: Γ.

text: If

text: Γ

text: is

text: isomorphic

text: to

text: a

text: lattice

text: in

text: an

text: irre-

text: ducible

text: Hermitian

text: symmetric

text: space

text: N

text: of

text: noncompact

text: type

text: without

text: Euclidean

text: de

text: Rham

text: factor

text: other

text: than

text: the

text: hyperbolic

text: plane,

text: then

text: D

text: is

text: (anti)biholomorphic

text: to

text: N.

text: Proof.

text: From

text: Theorem

text: we

text: know

text: that

text: D/Γ

text: has

text: bounded

text: geometry

text: (in

text: the

text: sense

text: of

text: K¨ahler-Einstein

text: metric).

text: It

text: is

text: clear

text: that

text: N/Γ

text: also

text: has

text: bounded

text: geometry.

text: Meanwhile,

text: by

text: Proposition

text: we

text: know

text: that

text: the

text: Euler

text: charac-

text: teristic

text: number

text: χ(Γ)

text: Since

text: both

text: D/Γ

text: and

text: N/Γ

text: are

text: aspherical

text: of

text: bounded

text: geometry,

text: we

text: apply

text: Corollary

text: to

text: get

text: dim(D)

text: dim(N)

text: because

text: χ(Γ)

text: Since

text: D

text: is

text: a

text: HHR/USq

text: complex

text: manifold

text: and

text: D/Γ

text: has

text: ﬁnite

text: volume,

text: it

text: follows

text: from

text: Corollary

text: that

text: D/Γ

text: is

text: quasi-projective

text: variety

text: of

text: log-general

text: type.

text: Finally,

text: thanks

text: to

text: Jost-Zuo

text: Theorem

text: we

text: get

text: that

text: D

text: is

text: (anti)biholomorphic

text: to

text: N.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 8 KEFENG LIU AND YUNHUI WU Theorem 1.16 and 1.17 by using Theorem 1.6 and 1.7. Acknowledgement. The authors would like to thank Prof. W. Ballmann, S. Krantz, B. Wong, S. T. Yau and K. Zuo for their interests. We especially would like to thank to Prof. B. Wong and S. T. Yau for their invaluable comments and suggestions which greatly improve this article. The ﬁrst author is partially supported by the NSFC, Grant No. 11531012 and a NSF grant. And the second author is partially supported by a grant from Tsinghua university. 2. Notations and Preliminaries This section contains general facts and necessary propositions for the proofs in subsequent sections. The general notation we use is as follows: (i) D is a bounded pseudoconvex domain or a HHR/USq complex man- ifold; (ii) M is a ﬁnite-volume manifold quotient of D; (iii) Γ := π (M); 1 (iv) Aut(D) is the automorphism group of D (clearly containing Γ); (v) Aut (D) is the identity component of Aut(D); 0 ∩ (vi) Γ := Γ Aut (D). 0 0 Our work highly relies on the K¨ahler-Einstein metric. We summarize the results needed. Let D be a bounded pseudoconvex domain. One may refer to the book [Dem] for general theory for bounded pseudoconvex domains. Cheng-Yau [CY80] showed that there always exists a complete K¨ahler-Einstein metric on a bounded pseudoconvex domain of C2-smooth boundary. Later Mok- Yau [MY83] removed the assumption on C2-smoothness for the boundary. More precisely, Theorem 2.1 (Cheng-Mok-Yau). Let D be a bounded pseudoconvex do- main. Then there exists a complete K¨ahler metric ω on D such that − (i) The Ricci curvature Ric = 1; ω (ii) The automorphism group Aut(D) acts on (D, ω) by isometries. Throughout the article we always assume that the complex manifold D is endowed with this K¨ahler-Einstein metric. We write D for (D, ω) for simplicity.

text: KEFENG

text: LIU

text: AND

text: YUNHUI

text: WU

text: Theorem

text: and

text: by

text: using

text: Theorem

text: and

text: Acknowledgement.

text: The

text: authors

text: would

text: like

text: to

text: thank

text: Prof.

text: W.

text: Ballmann,

text: S.

text: Krantz,

text: B.

text: Wong,

text: S.

text: T.

text: Yau

text: and

text: K.

text: Zuo

text: for

text: their

text: interests.

text: We

text: especially

text: would

text: like

text: to

text: thank

text: to

text: Prof.

text: B.

text: Wong

text: and

text: S.

text: T.

text: Yau

text: for

text: their

text: invaluable

text: comments

text: and

text: suggestions

text: which

text: greatly

text: improve

text: this

text: article.

text: The

text: ﬁrst

text: author

text: is

text: partially

text: supported

text: by

text: the

text: NSFC,

text: Grant

text: No.

text: and

text: a

text: NSF

text: grant.

text: And

text: the

text: second

text: author

text: is

text: partially

text: supported

text: by

text: a

text: grant

text: from

text: Tsinghua

text: university.

text: Notations

text: and

text: Preliminaries

text: This

text: section

text: contains

text: general

text: facts

text: and

text: necessary

text: propositions

text: for

text: the

text: proofs

text: in

text: subsequent

text: sections.

text: The

text: general

text: notation

text: we

text: use

text: is

text: as

text: follows:

text: (i)

text: D

text: is

text: a

text: bounded

text: pseudoconvex

text: domain

text: or

text: a

text: HHR/USq

text: complex

text: man-

text: ifold;

text: (ii)

text: M

text: is

text: a

text: ﬁnite-volume

text: manifold

text: quotient

text: of

text: D;

text: (iii)

text: Γ

text: π

text: (M);

text: (iv)

text: Aut(D)

text: is

text: the

text: automorphism

text: group

text: of

text: D

text: (clearly

text: containing

text: Γ);

text: (v)

text: Aut

text: (D)

text: is

text: the

text: identity

text: component

text: of

text: Aut(D);

text: (vi)

text: Γ

text: Γ

text: Aut

text: (D).

text: Our

text: work

text: highly

text: relies

text: on

text: the

text: K¨ahler-Einstein

text: metric.

text: We

text: summarize

text: the

text: results

text: needed.

text: Let

text: D

text: be

text: a

text: bounded

text: pseudoconvex

text: domain.

text: One

text: may

text: refer

text: to

text: the

text: book

text: [Dem]

text: for

text: general

text: theory

text: for

text: bounded

text: pseudoconvex

text: domains.

text: Cheng-Yau

text: showed

text: that

text: there

text: always

text: exists

text: a

text: complete

text: K¨ahler-Einstein

text: metric

text: on

text: a

text: bounded

text: pseudoconvex

text: domain

text: of

text: boundary.

text: Later

text: Mok-

text: Yau

text: removed

text: the

text: assumption

text: on

text: for

text: the

text: boundary.

text: More

text: precisely,

text: Theorem

text: (Cheng-Mok-Yau).

text: Let

text: D

text: be

text: a

text: bounded

text: pseudoconvex

text: do-

text: main.

text: Then

text: there

text: exists

text: a

text: complete

text: K¨ahler

text: metric

text: ω

text: on

text: D

text: such

text: that

text: (i)

text: The

text: Ricci

text: curvature

text: Ric

text: ω

text: (ii)

text: The

text: automorphism

text: group

text: Aut(D)

text: acts

text: on

text: (D,

text: ω)

text: by

text: isometries.

text: Throughout

text: the

text: article

text: we

text: always

text: assume

text: that

text: the

text: complex

text: manifold

text: D

text: is

text: endowed

text: with

text: this

text: K¨ahler-Einstein

text: metric.

text: We

text: write

text: D

text: for

text: (D,

text: ω)

text: for

text: simplicity.


text: 15 FIG. 12. Dependence of the average waiting time and trip time on commuter’s willingness to share, both in undersaturated and oversaturated regions similar to Fig.(8) but with the road network of Singapore. network. As we can see from Fig.(13), if we do not have TRS, there are always a large number of empty taxis in and around the city center, since we are in the under-saturated phase. When most people are open to TRS, the number of available taxis increases signiﬁcantly, but in and around the city center, most of the available taxis are actually occupied taxis with commuters who are open to TRS; there is a conspicuous lack of empty taxis in and around the city center. Given the high density of nodes in and around the city center, the average waiting time is strongly aﬀected by people booking from this region. Even though there are many available but occupied taxis, the chances that the routes of these taxis can be matched to the route of the new booking are always quite low, so most of the occupied taxis open to TRS are eﬀectively unavailable. On the other hand, empty taxis do not have this constraint of route matching. Thus empirically, the waiting time for the commuters in and around the city center could be longer, since the eﬀective available taxis, after the ﬁltering of route matching, is actually lower than the case when no one is allowed to taxi-sharing. It is thus quite interesting that with the road network of Singapore, taxi-sharing tends to push a lot of the empty taxis to the city fringe (including the industrial areas). Note this is independent of the spatial variation of the booking demand patterns, since each node is treated equally when random generation of the OD pairs are concerned; it is thus a purely network eﬀect. Since in our simulations empty taxis roam randomly, when taxis drop oﬀ commuters at the city fringe and becomes empty, they tend to stay around there because of the lack of nearby demands as a result of the low density of nodes (see Fig.(10)). On the other hands, vehicles in and around the city centers get less chance of being empty, because taxi-sharing and route matching lead to vehicles moving around more with commuters on board. While in reality such eﬀects could be suppressed by a non-trivial R (since after dropping oﬀ commuters near e city fringe the empty taxis naturally tend to move towards the city center) thus probably unobservable, what the simulation reveals here is one aspect of potential “penalty” for taxis carrying commuters towards less popular areas especially from the perspective of TRS, and could be suggestive on how to optimise the routing strategies for the drivers, especially when TRS becomes prevalent. VI. SIMULATIONS WITH EMPIRICAL OD DATA In the previous section, we analyzed some of the qualitative features of the taxi dynamics based on simple models with no empirical input. For more quantitative simulations and predictions on the detailed interactions between commuters and taxis, we need empirical input on the actual number of taxis on the street, the empirical spatiotemporal ##LTFigure## ##LTFigure##

text: FIG.

text: Dependence

text: of

text: the

text: average

text: waiting

text: time

text: and

text: trip

text: time

text: on

text: commuter’s

text: willingness

text: to

text: share,

text: both

text: in

text: undersaturated

text: and

text: oversaturated

text: regions

text: similar

text: to

text: but

text: with

text: the

text: road

text: network

text: of

text: Singapore.

text: network.

text: As

text: we

text: can

text: see

text: from

text: if

text: we

text: do

text: not

text: have

text: TRS,

text: there

text: are

text: always

text: a

text: large

text: number

text: of

text: empty

text: taxis

text: in

text: and

text: around

text: the

text: city

text: center,

text: since

text: we

text: are

text: in

text: the

text: under-saturated

text: phase.

text: When

text: most

text: people

text: are

text: open

text: to

text: TRS,

text: the

text: number

text: of

text: available

text: taxis

text: increases

text: signiﬁcantly,

text: but

text: in

text: and

text: around

text: the

text: city

text: center,

text: most

text: of

text: the

text: available

text: taxis

text: are

text: actually

text: occupied

text: taxis

text: with

text: commuters

text: who

text: are

text: open

text: to

text: TRS;

text: there

text: is

text: a

text: conspicuous

text: lack

text: of

text: empty

text: taxis

text: in

text: and

text: around

text: the

text: city

text: center.

text: Given

text: the

text: high

text: density

text: of

text: nodes

text: in

text: and

text: around

text: the

text: city

text: center,

text: the

text: average

text: waiting

text: time

text: is

text: strongly

text: aﬀected

text: by

text: people

text: booking

text: from

text: this

text: region.

text: Even

text: though

text: there

text: are

text: many

text: available

text: but

text: occupied

text: taxis,

text: the

text: chances

text: that

text: the

text: routes

text: of

text: these

text: taxis

text: can

text: be

text: matched

text: to

text: the

text: route

text: of

text: the

text: new

text: booking

text: are

text: always

text: quite

text: low,

text: so

text: most

text: of

text: the

text: occupied

text: taxis

text: open

text: to

text: TRS

text: are

text: eﬀectively

text: unavailable.

text: On

text: the

text: other

text: hand,

text: empty

text: taxis

text: do

text: not

text: have

text: this

text: constraint

text: of

text: route

text: matching.

text: Thus

text: empirically,

text: the

text: waiting

text: time

text: for

text: the

text: commuters

text: in

text: and

text: around

text: the

text: city

text: center

text: could

text: be

text: longer,

text: since

text: the

text: eﬀective

text: available

text: taxis,

text: after

text: the

text: ﬁltering

text: of

text: route

text: matching,

text: is

text: actually

text: lower

text: than

text: the

text: case

text: when

text: no

text: one

text: is

text: allowed

text: to

text: taxi-sharing.

text: It

text: is

text: thus

text: quite

text: interesting

text: that

text: with

text: the

text: road

text: network

text: of

text: Singapore,

text: taxi-sharing

text: tends

text: to

text: push

text: a

text: lot

text: of

text: the

text: empty

text: taxis

text: to

text: the

text: city

text: fringe

text: (including

text: the

text: industrial

text: areas).

text: Note

text: this

text: is

text: independent

text: of

text: the

text: spatial

text: variation

text: of

text: the

text: booking

text: demand

text: patterns,

text: since

text: each

text: node

text: is

text: treated

text: equally

text: when

text: random

text: generation

text: of

text: the

text: OD

text: pairs

text: are

text: concerned;

text: it

text: is

text: thus

text: a

text: purely

text: network

text: eﬀect.

text: Since

text: in

text: our

text: simulations

text: empty

text: taxis

text: roam

text: randomly,

text: when

text: taxis

text: drop

text: oﬀ

text: commuters

text: at

text: the

text: city

text: fringe

text: and

text: becomes

text: empty,

text: they

text: tend

text: to

text: stay

text: around

text: there

text: because

text: of

text: the

text: lack

text: of

text: nearby

text: demands

text: as

text: a

text: result

text: of

text: the

text: low

text: density

text: of

text: nodes

text: (see

text: On

text: the

text: other

text: hands,

text: vehicles

text: in

text: and

text: around

text: the

text: city

text: centers

text: get

text: less

text: chance

text: of

text: being

text: empty,

text: because

text: taxi-sharing

text: and

text: route

text: matching

text: lead

text: to

text: vehicles

text: moving

text: around

text: more

text: with

text: commuters

text: on

text: board.

text: While

text: in

text: reality

text: such

text: eﬀects

text: could

text: be

text: suppressed

text: by

text: a

text: non-trivial

text: R

text: (since

text: after

text: dropping

text: oﬀ

text: commuters

text: near

text: e

text: city

text: fringe

text: the

text: empty

text: taxis

text: naturally

text: tend

text: to

text: move

text: towards

text: the

text: city

text: center)

text: thus

text: probably

text: unobservable,

text: what

text: the

text: simulation

text: reveals

text: here

text: is

text: one

text: aspect

text: of

text: potential

text: “penalty”

text: for

text: taxis

text: carrying

text: commuters

text: towards

text: less

text: popular

text: areas

text: especially

text: from

text: the

text: perspective

text: of

text: TRS,

text: and

text: could

text: be

text: suggestive

text: on

text: how

text: to

text: optimise

text: the

text: routing

text: strategies

text: for

text: the

text: drivers,

text: especially

text: when

text: TRS

text: becomes

text: prevalent.

text: VI.

text: SIMULATIONS

text: WITH

text: EMPIRICAL

text: OD

text: DATA

text: In

text: the

text: previous

text: section,

text: we

text: analyzed

text: some

text: of

text: the

text: qualitative

text: features

text: of

text: the

text: taxi

text: dynamics

text: based

text: on

text: simple

text: models

text: with

text: no

text: empirical

text: input.

text: For

text: more

text: quantitative

text: simulations

text: and

text: predictions

text: on

text: the

text: detailed

text: interactions

text: between

text: commuters

text: and

text: taxis,

text: we

text: need

text: empirical

text: input

text: on

text: the

text: actual

text: number

text: of

text: taxis

text: on

text: the

text: street,

text: the

text: empirical

text: spatiotemporal

text: ##LTFigure##

text: ##LTFigure##


text: 18 lllllllllllllllllllllll pppsss === 001.5 l l lll l pppsss === 001.5 l lllllllllllllllllll lllllllllllllllllllllll 4000 5000 6000 7000 8000 4000 5000 6000 7000 8000 Number of Taxis Number of Taxis FIG. 16. Travel time plot at diﬀerent p , comparing the dynamics with empirical OD pair (left) and randomly generated OD s pair (right) with the same average rate of Pα generation. t VII. LIMITATIONS AND UNIVERSALITY OF NUMERICAL SIMULATIONS All models and simulations are approximations of the real world, and this is especially true for the taxi dynamics, embedded in a highly complex urban system. It is thus important to discuss in details the limits of our simulation analysis, as well as its universalities. For the former, we need to understand that certain detailed behaviours of the taxi dynamics cannot be captured by our simulations, and their quantitative (and even qualitative) results may not be trusted when compared to the empirical data. For the latter, we need to justify that many of the qualitative results from the numerical analysis are universal and can be generally applied to the empirical data of various urban systems, in spite of various simpliﬁcations we employed in our simulations. For commuter pick-up’s, it is important to note that our simulation follows a booking algorithm, in which taxis near the pick-up points know the exact position of the pick-up, and the nearest available taxi will always accept the booking and start moving to the pick-up point right away. We thus do not focus on accommodating taxi-hailing on the street, and we do not accurately model the time it takes for each commuter to board the taxi before heading towards the destination. The drivers also do not have the choice of rejecting the assigned commuter. While other realistic details may aﬀect the microscopic experience of individual taxis and commuters, the qualitative eﬀects of supply and demand of the taxis on average waiting or trip times, etc. should not be aﬀected, which we tested with extensive (though not necessarily thorough) simulations of various diﬀerent microscopic details. When comparing to the empirical data, one should note that the rate of commuter generation and the number of taxis in the simulation are both parameters indicating the eﬀective demand and supply, and should not be interpreted literarily. For example, the actual optimal number of taxis in a city would deviate from the N∗ obtained from the simulation, since occasionally taxi drivers may refuse to accept the assigned booking or be unable to ﬁnd the commuter (which tends to increase N∗), while road-side taxi-hailing may increase the eﬃciency of the pick-up’s (which tends to lower N∗). Nevertheless, the characteristic dependence of the taxi dynamics on the increase/decrease of taxi supply/demand based on simulation should still be useful for predicting similar characteristics of the empirical data. The movement of the empty taxis after they drop oﬀ the last commuter is an important part of the taxi dynamics, and in reality diﬀerent taxi drivers may have diﬀerent strategies (mostly based on experience) in maximizing the possibility of picking up the next commuter as early as possible. Such strategies can depend on diﬀerent time of the day as well as diﬀerent locations within the city. In our simulation, this part of the taxi dynamics is simpliﬁed. Empty taxis will just roam randomly from the destination of the last trip. Assuming in reality the drivers are on average more experienced, the simpliﬁcation in the simulation implies the simulated average waiting time could be longer, and the simulation could over-estimate the waiting time and N∗. On the other hand, we do not expect the actual routing strategy of the empty taxis to alter the qualitative behaviours of most of the simulated results, as long as the strategies are not vastly superior than a random routing. Many aspects of the dynamics of taxi-sharing strongly depend on the actual booking and route-matching algorithm, that determine whether or not two trips will be shared. In our simulation, a simple greedy algorithm is used, so that two routes will be shared if the detour time for each of the shared party will not exceed a certain cut-oﬀ time based on the shortest path routing, and the cut-oﬀ time is the only parameter in our algorithm. In addition to that, we also introduce the parameter of the commuters’ willingness to share their ride, which is the probability that the trip )elacS goL( emit levarT 8.3 6.3 4.3 2.3 0.3 8.2 )elacS goL( emit levarT 5.4 0.4 5.3 0.3 ##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##

text: lllllllllllllllllllllll

text: pppsss

text: l

text: l

text: lll

text: l

text: pppsss

text: l

text: lllllllllllllllllll

text: lllllllllllllllllllllll

text: Number

text: of

text: Taxis

text: Number

text: of

text: Taxis

text: FIG.

text: Travel

text: time

text: plot

text: at

text: diﬀerent

text: p

text: comparing

text: the

text: dynamics

text: with

text: empirical

text: OD

text: pair

text: (left)

text: and

text: randomly

text: generated

text: OD

text: s

text: pair

text: (right)

text: with

text: the

text: same

text: average

text: rate

text: of

text: Pα

text: generation.

text: t

text: VII.

text: LIMITATIONS

text: AND

text: UNIVERSALITY

text: OF

text: NUMERICAL

text: SIMULATIONS

text: All

text: models

text: and

text: simulations

text: are

text: approximations

text: of

text: the

text: real

text: world,

text: and

text: this

text: is

text: especially

text: true

text: for

text: the

text: taxi

text: dynamics,

text: embedded

text: in

text: a

text: highly

text: complex

text: urban

text: system.

text: It

text: is

text: thus

text: important

text: to

text: discuss

text: in

text: details

text: the

text: limits

text: of

text: our

text: simulation

text: analysis,

text: as

text: well

text: as

text: its

text: universalities.

text: For

text: the

text: former,

text: we

text: need

text: to

text: understand

text: that

text: certain

text: detailed

text: behaviours

text: of

text: the

text: taxi

text: dynamics

text: cannot

text: be

text: captured

text: by

text: our

text: simulations,

text: and

text: their

text: quantitative

text: (and

text: even

text: qualitative)

text: results

text: may

text: not

text: be

text: trusted

text: when

text: compared

text: to

text: the

text: empirical

text: data.

text: For

text: the

text: latter,

text: we

text: need

text: to

text: justify

text: that

text: many

text: of

text: the

text: qualitative

text: results

text: from

text: the

text: numerical

text: analysis

text: are

text: universal

text: and

text: can

text: be

text: generally

text: applied

text: to

text: the

text: empirical

text: data

text: of

text: various

text: urban

text: systems,

text: in

text: spite

text: of

text: various

text: simpliﬁcations

text: we

text: employed

text: in

text: our

text: simulations.

text: For

text: commuter

text: pick-up’s,

text: it

text: is

text: important

text: to

text: note

text: that

text: our

text: simulation

text: follows

text: a

text: booking

text: algorithm,

text: in

text: which

text: taxis

text: near

text: the

text: pick-up

text: points

text: know

text: the

text: exact

text: position

text: of

text: the

text: pick-up,

text: and

text: the

text: nearest

text: available

text: taxi

text: will

text: always

text: accept

text: the

text: booking

text: and

text: start

text: moving

text: to

text: the

text: pick-up

text: point

text: right

text: away.

text: We

text: thus

text: do

text: not

text: focus

text: on

text: accommodating

text: taxi-hailing

text: on

text: the

text: street,

text: and

text: we

text: do

text: not

text: accurately

text: model

text: the

text: time

text: it

text: takes

text: for

text: each

text: commuter

text: to

text: board

text: the

text: taxi

text: before

text: heading

text: towards

text: the

text: destination.

text: The

text: drivers

text: also

text: do

text: not

text: have

text: the

text: choice

text: of

text: rejecting

text: the

text: assigned

text: commuter.

text: While

text: other

text: realistic

text: details

text: may

text: aﬀect

text: the

text: microscopic

text: experience

text: of

text: individual

text: taxis

text: and

text: commuters,

text: the

text: qualitative

text: eﬀects

text: of

text: supply

text: and

text: demand

text: of

text: the

text: taxis

text: on

text: average

text: waiting

text: or

text: trip

text: times,

text: etc.

text: should

text: not

text: be

text: aﬀected,

text: which

text: we

text: tested

text: with

text: extensive

text: (though

text: not

text: necessarily

text: thorough)

text: simulations

text: of

text: various

text: diﬀerent

text: microscopic

text: details.

text: When

text: comparing

text: to

text: the

text: empirical

text: data,

text: one

text: should

text: note

text: that

text: the

text: rate

text: of

text: commuter

text: generation

text: and

text: the

text: number

text: of

text: taxis

text: in

text: the

text: simulation

text: are

text: both

text: parameters

text: indicating

text: the

text: eﬀective

text: demand

text: and

text: supply,

text: and

text: should

text: not

text: be

text: interpreted

text: literarily.

text: For

text: example,

text: the

text: actual

text: optimal

text: number

text: of

text: taxis

text: in

text: a

text: city

text: would

text: deviate

text: from

text: the

text: N∗

text: obtained

text: from

text: the

text: simulation,

text: since

text: occasionally

text: taxi

text: drivers

text: may

text: refuse

text: to

text: accept

text: the

text: assigned

text: booking

text: or

text: be

text: unable

text: to

text: ﬁnd

text: the

text: commuter

text: (which

text: tends

text: to

text: increase

text: N∗),

text: while

text: road-side

text: taxi-hailing

text: may

text: increase

text: the

text: eﬃciency

text: of

text: the

text: pick-up’s

text: (which

text: tends

text: to

text: lower

text: N∗).

text: Nevertheless,

text: the

text: characteristic

text: dependence

text: of

text: the

text: taxi

text: dynamics

text: on

text: the

text: increase/decrease

text: of

text: taxi

text: supply/demand

text: based

text: on

text: simulation

text: should

text: still

text: be

text: useful

text: for

text: predicting

text: similar

text: characteristics

text: of

text: the

text: empirical

text: data.

text: The

text: movement

text: of

text: the

text: empty

text: taxis

text: after

text: they

text: drop

text: oﬀ

text: the

text: last

text: commuter

text: is

text: an

text: important

text: part

text: of

text: the

text: taxi

text: dynamics,

text: and

text: in

text: reality

text: diﬀerent

text: taxi

text: drivers

text: may

text: have

text: diﬀerent

text: strategies

text: (mostly

text: based

text: on

text: experience)

text: in

text: maximizing

text: the

text: possibility

text: of

text: picking

text: up

text: the

text: next

text: commuter

text: as

text: early

text: as

text: possible.

text: Such

text: strategies

text: can

text: depend

text: on

text: diﬀerent

text: time

text: of

text: the

text: day

text: as

text: well

text: as

text: diﬀerent

text: locations

text: within

text: the

text: city.

text: In

text: our

text: simulation,

text: this

text: part

text: of

text: the

text: taxi

text: dynamics

text: is

text: simpliﬁed.

text: Empty

text: taxis

text: will

text: just

text: roam

text: randomly

text: from

text: the

text: destination

text: of

text: the

text: last

text: trip.

text: Assuming

text: in

text: reality

text: the

text: drivers

text: are

text: on

text: average

text: more

text: experienced,

text: the

text: simpliﬁcation

text: in

text: the

text: simulation

text: implies

text: the

text: simulated

text: average

text: waiting

text: time

text: could

text: be

text: longer,

text: and

text: the

text: simulation

text: could

text: over-estimate

text: the

text: waiting

text: time

text: and

text: N∗.

text: On

text: the

text: other

text: hand,

text: we

text: do

text: not

text: expect

text: the

text: actual

text: routing

text: strategy

text: of

text: the

text: empty

text: taxis

text: to

text: alter

text: the

text: qualitative

text: behaviours

text: of

text: most

text: of

text: the

text: simulated

text: results,

text: as

text: long

text: as

text: the

text: strategies

text: are

text: not

text: vastly

text: superior

text: than

text: a

text: random

text: routing.

text: Many

text: aspects

text: of

text: the

text: dynamics

text: of

text: taxi-sharing

text: strongly

text: depend

text: on

text: the

text: actual

text: booking

text: and

text: route-matching

text: algorithm,

text: that

text: determine

text: whether

text: or

text: not

text: two

text: trips

text: will

text: be

text: shared.

text: In

text: our

text: simulation,

text: a

text: simple

text: greedy

text: algorithm

text: is

text: used,

text: so

text: that

text: two

text: routes

text: will

text: be

text: shared

text: if

text: the

text: detour

text: time

text: for

text: each

text: of

text: the

text: shared

text: party

text: will

text: not

text: exceed

text: a

text: certain

text: cut-oﬀ

text: time

text: based

text: on

text: the

text: shortest

text: path

text: routing,

text: and

text: the

text: cut-oﬀ

text: time

text: is

text: the

text: only

text: parameter

text: in

text: our

text: algorithm.

text: In

text: addition

text: to

text: that,

text: we

text: also

text: introduce

text: the

text: parameter

text: of

text: the

text: commuters’

text: willingness

text: to

text: share

text: their

text: ride,

text: which

text: is

text: the

text: probability

text: that

text: the

text: trip

text: )elacS

text: goL(

text: emit

text: levarT

text: )elacS

text: goL(

text: emit

text: levarT

text: ##LTFigure##

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