arXiv research

This article presents a proof of Pogorelov's result that there exists a $C^{2,1}$ metric with no local $C^2$ realization in $\mathbb{R}^3$. It also construct in a very elementary way a $C^{1,1}$ realization of this metric. Pogorelov's result is somewhat controversial among the community of researchers that study isomet…

Life and the mathematical legacy of the great mathematician A.V. Pogorelov.

Paper derives estimates for Hessian equations under concavity assumptions.

Researchers provide counterexamples to Pogorelov and Toponogov's questions about saddle surfaces.

The paper establishes Pogorelov type estimates for solutions to Hessian quotient equations in hyperbolic space.

Interior $C^{2}$ estimates for sum Hessian quotient equations on Riemannian manifolds

Non-compact convex sets in hyperbolic 3-space are rigid under isometries.

The paper proves a rigidity theorem for non-compact convex sets in hyperbolic 3-space.

We prove some Bernstein theorems for entire space-like submanifolds in pseudo-Euclidean spaces and, as a corollary, we obtain a new proof of the Calabi-Pogorelov theorem on global solutions of Monge-Ampere equations.

We present constructions inspired by the Ma-Schlenker example of~\cite{Ma:2012hl} that show the non-rigidity of spherical inversive distance circle packings. In contrast to the use in~\cite{Ma:2012hl} of an infinitesimally flexible Euclidean polyhedron, embeddings in de Sitter space, and Pogorelov maps, our elementary …

We prove that any simply connected special Kaehler manifold admits a canonical immersion as a parabolic affine hypersphere. As an application, we associate a parabolic affine hypersphere to any nondegenerate holomorphic function. Also we show that a classical result of Calabi and Pogorelov on parabolic spheres implies …

It is shown that any smooth strictly convex global solution of $\det(\frac{\partial^{2}u}{\partial ξ_{i}\partial ξ_{j}}) = \exp \left\{-\sum_{i=1}^n d_i \frac{\partial u}{\partial ξ_{i}} - d_0\right\},$ where $d_0$, $d_1$,...,$d_n$ are constants, must be a quadratic polynomial. This extends a well-known theorem of Jö…

In this paper we prove the existence of complete, noncompact convex hypersurfaces whose $p$-curvature function is prescribed on a domain in the unit sphere. This problem is related to the solvability of Monge-Ampère type equations subject to certain boundary conditions depending on the value of $p$. The special case of…

We show the uniqueness of strictly convex closed smooth self-similar solutions to the $α$-Gauss curvature flow with $(1/n) < α< 1+(1/n)$. We introduce a Pogorelov type computation, and then we apply the strong maximum principle. Our work combined with earlier works on the Gauss Curvature flow imply that the $α$-Gauss c…

The paper proves a Bonnesen-type inequality for the real projective plane.

Closed surfaces minimize total curvature in curved spaces.

A pseudo-edge graph of a convex polyhedron K is a 3-connected embedded graph in K whose vertices coincide with those of K, whose edges are distance minimizing geodesics, and whose faces are convex. We construct a convex polyhedron K in Euclidean 3-space with a pseudo-edge graph with respect to which K is not unfoldable…

Paper proves rigidity of convex hypersurfaces in various spaces.

Smooth solutions found for a curvature problem in hyperbolic space.

Revisits Weyl's problem on isometric immersions of spheres into 3D manifolds.

New geometric approach gives apriori estimate for optimal transport maps.

Solves a long-standing convex geometry problem about mixed volumes.

A classical theorem, mainly due to Aleksandrov and Pogorelov, states that any Riemannian metric on $S^2$ with curvature $K>-1$ is induced on a unique convex surface in $H^3$. A similar result holds with the induced metric replaced by the third fundamental form. We show that the same phenomenon happens with yet another …

Celebrated work of Alexandrov and Pogorelov determines exactly which metrics on the sphere are induced on the boundary of a compact convex subset of hyperbolic three-space. As a step toward a generalization for unbounded convex subsets, we consider convex regions of hyperbolic three-space bounded by two properly embedd…

A family of closed manifolds is called cohomologically rigid if a cohomology ring isomorphism implies a diffeomorphism for any two manifolds in the family. We establish cohomological rigidity for large families of 3-dimensional and 6-dimensional manifolds defined by 3-dimensional polytopes. We consider the class P of 3…

In this paper, we prove $W^{1,p}$ ($p>n$) and $C^{0,α}$ ($0 < α< 1$) precompactness for classes of Riemannian $n$-manifolds with boundary satisfying uniform $L^{\infty}$ bounds on curvature, mean curvature, diameter, and the $(n-1)$-volume of the boundary. In particular, we identify a class of convex manifolds and a cl…

The paper studies the topology and geometry of simple orbifolds, generalizing concepts from simple polytopes.

The Weyl problem is extended to hyperbolic and anti-de Sitter spaces, connecting geometry, analysis, and group theory.