arXiv research

Unified rigidity theorem for Plateau surfaces in $\mathbb{B}^n$.

Paper shows examples of hyperbolic cone structures degenerating with decreasing cone angles.

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…

Study on rigidity of warped cones and expanders.

Study on sine-cones' spectra and stability under Ricci-de Turck flow.

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 Serrin's problem in convex cones with rigidity results and geometric inequalities.

Rigidity theorem for scalar curvature on odd-dimensional singular manifolds.

Local rigidity shown for certain spherical conical metrics.

We show that any compact orientable hyperbolic 3-cone-manifold with cone angle at most πcan be continuously deformed to a complete hyperbolic manifold homeomorphic to the complement of the singularity. This together with the local rigidity by Hodgson and Kerckhoff implies the global rigidity for compact orientable hype…

Characterizes rigid and flexible hyperbolic cone metrics and billiards.

The present paper gives an example of a rigid spherical cone-manifold and that of a flexible one which are both Seifert fibred.

Paper proves rigidity and index of Y-cones in unit ball.

Minimal Lagrangian diffeomorphisms between spherical surfaces are rigid.

Study on $p$-Laplace equation in convex cones, proving rigidity under specific conditions.

Flexible metrics found on a genus 2 surface.

In this paper, we show the rigidity of isometric immersions for a Riemannian manifold of dimension $n-1$ into the light cone of $n+1$ dimensional Minkowski, de Sitter and anti-de Sitter spacetimes for $n\geq 3$.

Rigidity of cones with bounded Ricci curvature proven.

Combining several previously known arguments, we prove marked length spectrum rigidity for surfaces with nonpositively curved Riemannian metrics away from a finite set of cone-type singularities with cone angles $>2π$. With an additional condition, we can weaken the requirement on one metric to `no conjugate points.'

We investigate the local deformation space of 3-dimensional cone-manifold structures of constant curvature $κ\in \{-1,0,1\}$ and cone-angles $\leq π$. Under this assumption on the cone-angles the singular locus will be a trivalent graph. In the hyperbolic and the spherical case our main result is a vanishing theorem fo…

The Stoker problem, first formulated in 1968, consists in understanding to what extent a convex polyhedron is determined by its dihedral angles. By means of the double construction, this problem is intimately related to rigidity issues for 3-dimensional cone-manifolds. In a former paper, two such rigidity results were …

Given a closed orientable Euclidean cone 3-manifold C with cone angles less than or equal to pi, and which is not almost product, we describe the space of constant curvature cone structures on C with cone angles less than pi. We establish a regeneration result for such Euclidean cone manifolds into spherical or hyperbo…

Given a geodesic inside a simply-connected, complete, non-positively curved Riemannian (NPCR) manifold M, we get an associated geodesic inside the asymptotic cone Cone(M). Under mild hypotheses, we show that if the latter is contained inside a bi-Lipschitz flat, then the original geodesic supports a non-trivial, orthog…

Two rigidity results for Legendrian singularities in complex-analytic category.

This paper proves a rigidity result for annuli in $RCD(K, N)$-spaces.

We develop the deformation theory of hyperbolic cone-3-manifolds with cone-angles less than $2π$, i.e. contained in the interval $(0,2π)$. In the present paper we focus on deformations keeping the topological type of the cone-manifold fixed. We prove local rigidity for such structures. This gives a positive answer to a…

Characterizes eigenfunctions of Lawson-Osserman cone and proves its integrability.

Sharp isoperimetric inequality for Finsler manifolds with non-negative Ricci curvature.

New inequality shows all special submanifolds in light cone are totally umbilical spheres.

Researchers extend monotonicity formulas for harmonic functions in RCD(0,N) spaces.

New approach to convexity and monotonicity on metric spaces.

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

Abstract cone operators prove scalar curvature comparisons on singular manifolds.

Weiss and, independently, Mazzeo and Montcouquiol recently proved that a 3--dimensional hyperbolic cone-manifold (possibly with vertices) with all cone angles less than $2π$ is infinitesimally rigid. On the other hand, Casson provided 1998 an example of an infinitesimally flexible cone-manifold with some of the cone an…

Study spherical metrics on flat torus with cone singularities.

The study classifies and proves rigidity of Legendrian self-shrinkers in 3D and 5D.

The deformation theory of hyperbolic and Euclidean cone-manifolds with all cone angles less then 2π plays an important role in many problems in low dimensional topology and in the geometrization of 3-manifolds. Furthermore, various old conjectures dating back to Stoker about the moduli of convex hyperbolic and Euclidea…

In this paper we prove that the space of flat metrics (nonpositively curved Euclidean cone metrics) on a closed, oriented surface is marked length spectrally rigid. In other words, two flat metrics assigning the same lengths to all closed curves differ by an isometry isotopic to the identity. The novel proof suggests a…

Let O be a three-dimensional Nil-orbifold, with branching locus a knot Sigma transverse to the Seifert fibration. We prove that O is the limit of hyperbolic cone manifolds with cone angle in (pi-epsilon, pi). We also study the space of Dehn filling parameters of O-Sigma. Surprisingly it is not diffeomorphic to the defo…

Study Busemann spaces with measures under MCP, proving rigidity and structure theorems.

The study proves rigidity of Einstein manifolds with specific curvature conditions.

We study the geometry of hyperbolic cone surfaces, possibly with cusps or geodesic boundaries. We prove that any hyperbolic cone structure on a surface of non-exceptional type is determined up to isotopy by the geodesic lengths of a finite specific homotopy classes of non-peripheral simple closed curves. As an applicat…

The entropy-degree theorem applies to Alexandrov spaces with curvature constraints.

Study rigid classes on hyperkahler manifolds, showing general ones are rigid.

Length spectral rigidity is the question of under what circumstances the geometry of a surface can be determined, up to isotopy, by knowing only the lengths of its closed geodesics. It is known that this can be done for negatively curved Riemannian surfaces, as well as for negatively-curved cone surfaces. Steps are tak…

The paper introduces flat grafting to deform quadratic differentials on surfaces.

The paper proves rigidity and ε-regularity theorems for Ricci shrinkers.

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.