TITLE: What is known about sufficient conditions for the rigidity of a convex surface?
QUESTION [16 upvotes]: A convex surface is a connected open subset of the boundary of a convex body in $\mathbb{R}^3$.
An "infinitesimal bending" of a convex surface $S$ is a deformation of $S$ given by a velocity field $v:S\rightarrow \mathbb{R}^3$ such that the length of every curve on the surface is preserved. Vector field $v$ is called the "bending field." A bending field is "trivial" if it corresponds to a simple translation or rotation of the surface. A surface is called "rigid" if all of it's bending fields are trivial.
A regular convex surface is one where every point has a neighborhood which can be parametrized by $r(u, v)$ with $r_u \times r_v \neq 0$.
Blaschke proved that a closed regular convex surface is rigid. Cauchy proved that all convex polyhedra (which are irregular) are rigid.
I've begun dabbling in A.V. Pogorelov's "Extrinsic Geometry of Convex Surfaces," where he proves and discusses both of these results in one of the chapters. However, the translation was published in 1973, so I'm curious about more recent research in this area. Namely, what is known about the rigidity of 1) regular convex surfaces which are not closed and 2) irregular convex surfaces which are not polyhedral?

REPLY [6 votes]: Answer to question 1: If a convex surface is not closed, then generally it is far from rigid as it might admit infinitely many isometric deformations; however, if the surface has $\mathcal{C}^{2,1}$ regularity and positive curvature, then it becomes rigid as soon as one fixes an arbitrarily small curve segment on that surface. This has been proved very recently in a joint work with Joel Spruck, which has been accepted for publication in International Math. Research Notices (IMRN):

Rigidity of nonnegatively curved surfaces relative to a curve,
    arXiv:1805.02580.

This result may also be extended to nonnegatively curved surfaces under some additional conditions. The proof uses Hormander's unique continuation principle for elliptic PDEs. Our methods also yield a very short proof of Cohn-Vossen's rigidity theorem for smooth closed convex surfaces, via Hopf's maximum principle, which is included in the appendix to the paper.