arXiv research

We study the local Killing Lie algebra of meromorphic almost rigid geometric structures on complex manifolds. This leads to classification results for compact complex manifolds bearing holomorphic rigid geometric structures.

We analyze sub-Riemannian and lightlike metrics from the point of view of their rigidity as geometric structures. Following Cartan's and Gromov's formal definitions, they are never rigid, yet, in generic cases, they naturally give rise to rigid geometric structures!?

New method proves length spectrum rigidity in various geometric settings.

The paper examines torsional rigidity bounds under geometric flows.

Study geometric rigidity of surfaces in negative curvature manifolds.

The paper proposes conjectures about moduli space rigidity.

Profinite rigidity proven for many hyperbolic manifolds.

Confirming a conjecture, new CAT(0) spaces of higher rank are rigid.

In a recent paper Hodgson and Kerckhoff prove a local rigidity theorem for finite volume, three dimensional hyperbolic cone-manifolds. In this paper we extend this result to geometrically finite cone-manifolds. Our methods also give a new proof of a local version of the classical rigidity theorem for geometrically fini…

Study on mappings in Carnot groups, proving rigidity results.

Study on geometric flows and rigidity of solitons.

Study proves rigidity of specific self-shrinkers under certain geometric conditions.

The aim of this note is to give a geometric proof for classical local rigidity of lattices in semisimple Lie groups. We are reproving well known results in a more geometric (and hopefully clearer) way.

In \cite{BK02}, M. Bonk and B. Kleiner proved a rigidity theorem for expanding quasi-Möbius group actions on Ahlfors $n$-regular metric spaces with topological dimension $n$. This led naturally to a rigidity result for quasi-convex geometric actions on CAT$(-1)$-spaces that can be seen as a metric analog to the "entrop…

The paper studies rigidity and continuity in nonlinear elasticity on manifolds and hypersurfaces.

Closed hyperbolic manifolds and manifolds with nonpositive sectional curvature are geometrically rigid under certain curvature conditions.

We classify the torsion pairs in a tube category and show that they are in bijection with maximal rigid objects in the extension of the tube category containing the Pruefer and adic modules. We show that the annulus geometric model for the tube category can be extended to the larger category and interpret torsion pairs…

The paper classifies 3D spherical Sasakian manifolds using geometric and algebraic methods.

We prove that rigid representations of the fundamental group of a surface into the group of oreintation-preserving homeomorphisms of the circle are geometric, thereby establishing a converse statement of a theorem by the first author.

The study proves conditions for CAT(0) spaces with higher rank rigidity.

Rigorous model for 2-gerbes simplifies calculations in physics.

Schoen-Yau's zero mass theorem stability remains an open question.

We prove that certain volume preserving actions of Lie groups and their lattices do not preserve rigid geometric structures in the sense of Gromov. The actions considered are the "exotic" examples obtained by Katok and Lewis and the first author, by blowing up closed orbits in the well known actions on homogeneous spac…

Paper investigates rigidity phenomena for weighted Ricci curvature bounds with Laplacian comparison theorem.

Geometric structures modeled on rational homogeneous manifolds are studied to characterize rational homogeneous manifolds and to prove their deformation rigidity. To generalize these characterizations and deformation rigidity results to quasihomogeneous varieties, we first study horospherical varieties and geometric st…

Study geometric properties of generalized vacuum static spaces.

We study the rigidity results for self-shrinkers in Euclidean space by restriction of the image under the Gauss map. The geometric properties of the target manifolds carry into effect. In the self-shrinking hypersurface situation Theorem 3.1 and Theorem 3.2 not only improve the previous results, but also are optimal. I…

Green functions for GJMS operators on spheres derived, linking geometry and rigidity.

The paper bounds eigenvalues of the Jacobi operator and derives rigidity results for CMC hypersurfaces.

We prove global rigidity for compact hyperbolic and spherical cone-3-manifolds with cone-angles $\leq π$ (which are not Seifert fibered in the spherical case), furthermore for a class of hyperbolic cone-3-manifolds of finite volume with cone-angles $\leq π$, possibly with boundary consisting of totally geodesic hyperbo…

The study examines rigidity properties of noncompact manifolds with nonnegative Ricci curvature.

The paper explores rigid geometric structures near surfaces with equality in area-charge inequalities.

Topologically and geometrically engaging actions have proved to be useful to obtain rigidity results for semisimple Lie group actions. We show that the action of a simple noncompact Lie group on a compact manifold preserving a unimodular rigid geometric structure of algebraic type (e.g. a connection together with a vol…

Paper shows non-arithmetic surface with unique geometric property.

We prove the rigidity and vanishing of several indices of "geometrically natural" twisted Dirac operators on almost even-Clifford Hermitian manifolds admitting circle actions by automorphisms.

Sharp pinching conditions restrict the geometry and topology of submanifolds.

Develops a framework for stuck knots with rigid constraints and invariants.

In this paper, we study some basic geometric properties of pseudohermitian submanifolds of the Heisenberg groups. In particular, we obtain the uniqueness and existence theorems, and some rigidity theorems.

We propose a unified computational framework for the problem of deformation and rigidity of submanifolds in a homogeneous space under geometric constraint. A notion of 1-rigidity of a submanifold under admissible deformations is introduced. It measures how a deformation deviates from a one parameter family of motions u…

The paper shows symmetries of a geometric space for Coxeter groups.

In this paper we study the rigidity of infinite volume 3-manifolds with sectional curvature $-b^2\le K\le -1$ and finitely generated fundamental group. In-particular, we generalize the Sullivan's quasi-conformal rigidity for finitely generated fundamental group with empty dissipative set to negative variable curvature …

High-dimensional ConvNets detect patterns in 32+ dimensions for geometric registration.

New rigidity results for specific hypersurfaces in spacetimes.

Study on topological rigidity of ALE vector bundles with specific conditions.

Graphically discrete groups have strong rigidity properties.

Groups acting on product trees are boundary rigid.

We prove rigidity for hypersurfaces with boundary in the unit $(n+1)$-sphere with scalar curvature bounded below by $n(n-1)$. Under appropriate boundary conditions, the hypersurfaces are shown to be part of the equatorial spheres. The lower bound $n(n-1)$ is critical in the sense that the hypersurface may contain geode…

We prove a topological rigidity result for simple, thick, hyperbolic P-manifolds of dimension 2: isomorphism of the fundamental groups implies homeomorphism of the P-manifolds. An immediate application is a diagram rigidity theorem for certain amalgamations of free groups: the direct limits of two such diagrams are iso…