arXiv research

New theorem proves rigidity of circle packings in hyperbolic geometry.

The paper proves rigidity of bordered polyhedral surfaces using variational principles.

We prove rigidity theorems for shrinking gradient Ricci solitons supporting the Heisenberg-Pauli-Weyl uncertainty principle with the sharp constant in $\mathbb{R}^n$. In addtion, we partially give analogous rigidity results of the Caffarelli-Kohn-Nirenberg inequalities on shrinking Ricci solitons.

We show that all PL manifolds of dimension $\geq 3$ have spines similar to Bing's house with two rooms. Beyond this we explore approximation rigidity and an $h$-principle.

Study shows critical width for rigidity of equatorial zones on spheres.

Plane triangulations remain rigid under discrete conformal changes.

In the present paper, we revisit the rigidity of hypersurfaces in Euclidean space. We highlight Darboux equation and give new proof of rigidity of hypersurfaces by energy method and maximal principle.

New proof of shrinking gradient Ricci soliton rigidity.

New rigidity results for submanifolds in GRW spacetimes.

The study proves rigidity and non-rigidity of spherical caps in mean curvature.

Study proves spacelike self-shrinkers are hyperplanes under certain conditions.

We study the rigidity of polyhedral surfaces using variational principle. The action functionals are derived from the cosine laws. The main focus of this paper is on the cosine law for a non-triangular region bounded by three possibly disjoint geodesics. Several of these cosine laws were first discovered and used by Fe…

The paper proposes conjectures about moduli space rigidity.

The paper proves uncertainty principles on Finsler measure spaces.

Survey on rigidity results for graphs with prescribed mean curvature.

Plane Delaunay triangulations are rigid under Luo's discrete conformal change.

Two Anosov metrics with same boundary distance are isometric.

Paper proves rigidity of discrete conformal structures on polyhedral surfaces.

The study examines Perelman singular manifolds and their properties.

Since $n$-dimensional $λ$-hypersurfaces in the Euclidean space $\mathbb {R}^{n+1}$ are critical points of the weighted area functional for the weighted volume-preserving variations, in this paper, we study the rigidity properties of complete $λ$-hypersurfaces. We give a gap theorem of complete $λ$-hypersurfaces with po…

Unified approach classifies stable and minimal elastic curves.

Study of complete space-like self-expanders in Minkovski space.

Based on works by Hopf, Weinberger, Hamilton and Evans, we state and prove the strong elliptic maximum principle for smooth sections in vector bundles over Riemannian manifolds and give some applications in Differential Geometry. Moreover, we use this maximum principle to obtain various rigidity theorems and Bernstein …

Inversive distance circle packing metric was introduced by P Bowers and K Stephenson \cite{BS} as a generalization of Thurston's circle packing metric \cite{T1}. They conjectured that the inversive distance circle packings are rigid. For nonnegative inversive distance, Guo \cite{Guo} proved the infinitesimal rigidity a…

We study rigidity of polyhedral surfaces and the moduli space of polyhedral surfaces using variational principles. Curvature like quantities for polyhedral surfaces are introduced. Many of them are shown to determine the polyhedral metric up to isometry. The action functionals in the variational approaches are derived …

New proof for global rigidity of vertex scaling on polyhedral surfaces.

A Euclidean (or hyperbolic) circle packing on a closed triangulated surface with prescribed inversive distance is locally determined by its cone angles. We prove this by applying a variational principle.

This paper investigates several global rigidity issues for polyhedral surfaces including inversive distance circle packings. Inversive distance circle packings are polyhedral surfaces introduced by P. Bowers and K. Stephenson as a generalization of Andreev-Thurston's circle packing. They conjectured that inversive dist…

Study on geometric flows and rigidity of solitons.

Proves rigidity of circle packings in the plane, generalizing previous work.

Paper proves a discrete Schwarz-Pick lemma for generalized circle packings.

The paper studies circle packings on surfaces with boundary and their total geodesic curvatures.

We prove that any properly oriented $C^{2,1}$ isometric immersion of a positively curved Riemannian surface M into Euclidean 3-space is uniquely determined, up to a rigid motion, by its values on any curve segment in M. A generalization of this result to nonnegatively curved surfaces is presented as well under suitable…

Using a maximum principle for self-shrinkers of the mean curvature flow, we give new proofs of a rigidity theorem for rotationally symmetric compact self-shrinkers and a result about the asymptotic behavior of self-shrinkers. This comparison argument also implies a linear bound for the second fundamental form of self-s…

The Hawking energy is nonnegative and rigid on area-constrained surfaces in general relativity.

New discrete conformal structures on surfaces with boundary, proving global rigidity and constructing hyperbolic metrics.

We use the weighted Hsiung-Minkowski integral formulas and Brendle's inequality to show new rigidity results. First, we prove Alexandrov type results for closed embedded hypersurfaces with radially symmetric higher order mean curvature in a large class of Riemannian warped product manifolds, including the Schwarzschild…

Maps between positively curved manifolds with non-increasing area are rigid.

We study the strong maximum principle for horizontal (p-) mean curvature operator and p-(sub)laplacian operator on subriemannian manifolds including, in particular, Heisenberg groups and Heisenberg cylinders. Under a certain Hormander type condition on vector fields, we show the strong maximum principle holds in higher…

Our purpose in this paper is to apply some maximum principles in order to study the rigidity of complete spacelike hypersurfaces immersed in a spatially weighted generalized Robertson-Walker (GRW) spacetime, which is supposed to obey the so called strong null convergence condition. Under natural constraints on the weig…

Paper proves rigidity theorems for AE Q-singular spaces.

New uncertainty principle for Schrödinger equations on hyperbolic manifolds.

Thurston's Circle Pattern Theorem studies existence and rigidity of circle patterns of a given combinatorial type and the given non-obtuse exterior intersection angles. Using topological degree theory, variational principle, Teichmuller theory, and Sard's Theorem, this paper generalizes Circle Pattern Theorem to the ca…

Localized curvature bounds ensure harmonic maps are constant.

The orthogonal decomposition of the Webster curvature provides us a way to characterize some canonical metrics on a pseudo-Hermitian manifold. We derive some subelliptic differential inequalities from the Weitzenböck formulas for the traceless pseudo-Hermitian Ricci tensor and the Chern-Moser tensor of Sasakian manifol…

The fundamental theorem of the theory of optimal control, the Pontryagin maximum principle (PMP), is extended to the setting of almost Lie (AL) algebroids, geometrical objects generalizing Lie algebroids. This formulation of the PMP yields, in particular, a scheme comprising reductions of optimal control problems simil…

Paper presents a new Pohozaev-Schoen identity for non-compact manifolds.

We study some basic problems of translating solitons: the volume growth, generalized maximum principle, Gauss maps and certain functions related to the Gauss maps, finally we carry out point-wise estimates and integral estimates for the squared norm of the second fundamental form. Those estimates give rigidity theorems…