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?



What is known about sufficient conditions for the rigidity of a convex surface?
Answer to question 1: proper compact subsets of convex surfaces are infinitesimally flexible.


In his book Pogorelov proves infinitesimal flexibility of the graph of a convex function over a strictly convex compact subset of the plane. (Basically the idea is that in order to find an infinitesimal bending of a surface with positive Gauss curvature one has to solve an elliptic PDE.) This implies infinitesimal flexibility of any proper compact subset of a strictly convex surface. Indeed, infinitesimal rigidity is projectively invariant (an amazing property given that the definition is in terms of distance). If you send to infinity a plane that cuts the surface in a neighborhood of one point, then the complement of this neighborhood becomes a graph of a convex function.


Answer to question 2: closed convex surfaces without flat pieces are infinitesimally rigid.


This is proved in the cited book of Pogorelov (Theorem 4 in Section 4.7). Clearly, any flat piece allows infinitesimal deformations: take any vector field supported inside this piece and orthogonal to its plane.


As to recent research in the field, one may consult a series of surveys by Ivanova-Karatopraklieva and Sabitov. A good textbook on the topic is the first half of volume 5 of Spivak's "Comprehensive introduction...".


One direction in which one can generalize rigidity results for convex surfaces is rigidity of hyperbolic manifolds with convex boundary. (Indeed, rigidity of a convex surface translates as rigidity of a Euclidean ball enclosed by this surface.) In this article:


*Schlenker, Jean-Marc*, [**Hyperbolic manifolds with convex boundary**](http://dx.doi.org/10.1007/s00222-005-0456-x), Invent. Math. 163, No. 1, 109-169 (2006). [ZBL1091.53019](https://zbmath.org/?q=an:1091.53019).


infinitesimal rigidity of hyperbolic manifolds with convex boundary is proved (using Pogorelov's theorem, by the way).


