arXiv research

Quantitative rigidity theorem for Alexandrov spaces with curvature bounds.

Alternative approach to rigidity of high-dimensional isometric immersions.

In this paper we show a quantitative rigidity result for the minimizer of the Willmore functional among all projective planes in $\mathbb{R}^n$ with $n\ge 4$. We also construct an explicit counterexample to a corresponding rigidity result in codimension one, by showing that an Enneper surface might split-off during a b…

The paper investigates quantitative rigidity using Colding's monotonicity formulas for Ricci curvature.

The study provides optimal estimates for surfaces close to constant mean curvature.

Study almost rigidity of super Ricci flow with non-negative Muller quantity.

This paper proves a new, more precise version of Cheng's maximal diameter theorem for manifolds with positive Ricci curvature.

Paper proves rigidity of certain Ricci shrinkers.

The paper proves a quantitative rigidity result for spaces with specific curvature bounds.

In this paper we prove several quantitative rigidity results for conformal immersions of surfaces in $\mathbb{R}^n$ with bounded total curvature. We show that (branched) conformal immersions which are close in energy to either a round sphere, a conformal Clifford torus, an inverted catenoid, an inverted Enneper's minim…

Proves a quantitative index theorem for positive scalar curvature metrics.

Sharp estimate shows maps with small energy defect are close to rational maps.

The paper proves rigidity estimates for varifolds almost minimizing the Willmore energy.

In this paper we show that in some cases the E.Hopf rigidity phenomenon admits quantitative interpretation. More precisely we estimate from above the measure of the set $\mathcal{M}$ swept by minimal orbits. These estimates are sharp, i.e. if $\mathcal{M}$ occupies the whole phase space we recover the E.Hopf rigidity. …

Let $M$ be a compact $n$-manifold of $\operatorname{Ric}_M\ge (n-1)H$ ($H$ is a constant). We are concerned with the following space form rigidity: $M$ is isometric to a space form of constant curvature $H$ under either of the following conditions: (i) There is $ρ>0$ such that for any $x\in M$, the open $ρ$-ball at $x^…

The paper proves surfaces close to spheres under specific conditions.

Study shows how compact shapes can be rigidly mapped into complete manifolds.

Study graph products of groups, classifying them up to measure equivalence and rigidity.

We review classical results where the method of the moving planes has been used to prove symmetry properties for overdetermined PDE's boundary value problems (such as Serrin's overdetermined problem) and for rigidity problems in geometric analysis (like Alexandrov soap bubble Theorem), and we give an overview of some r…

Study proves quantitative results for isoperimetric problem outside convex bodies in the plane.

Study on stability of half-harmonic maps from R to S, proving non-degeneracy and quantitative stability.

We prove a qualitative and a quantitative stability of the following rigidity theorem: an anisotropic totally umbilical closed hypersurface is the Wulff shape. Consider $n \geq 2$, $p\in (1, \, +\infty)$ and $Σ$ an $n$-dimensional, closed hypersurface in $\mathbb{R}^{n+1}$, boundary of a convex, open set. We show that …

Sharp Talenti-type comparison theorem for p-Laplacian on RCD(K,N) spaces.

Proves metric spaces with Euclidean heat kernel are isometric to Euclidean space.

Study circumcenters in Finsler unitary groups with optimal convexity bounds.

This is the second paper of two in a series under the same title ([CRX]); both study the quantitative volume space form rigidity conjecture: a closed $n$-manifold of Ricci curvature at least $(n-1)H$, $H=\pm 1$ or $0$ is diffeomorphic to a $H$-space form if for every ball of definite size on $M$, the lifting ball on th…

Study shows surfaces with similar length spectra are smoothly deformable.

Sharp stability of isometries on Heisenberg group proven.

The study examines hypersurfaces close to constant mean curvature and their proximity to spheres.

Localized curvature bounds ensure harmonic maps are constant.

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

Paper proves rigidity of Poincaré-Einstein manifolds with cylindrical conformal infinities.

The study shows how discrete graphs can resemble hypercube structures under certain curvature conditions.

Paper develops a framework for hyperbolic Monge-Ampère equation on strips, proving well-posedness and stability.

QTMRL uses RL with multi-indicators to improve trading adaptability.

In 2004, Taubes introduced the space of minimal hyperbolic germs with elements consisting of the first and second fundamental form of an equivariant immersed minimal disk in hyperbolic 3-space. Herein, we initiate a further study of this space by studying the behavior of a dynamically defined function which records the…

Study models opaque financial markets using multi-agent simulation.

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

The paper explores geometric relationships in manifolds with curvature constraints, proving new inequalities and rigidity results.

New inequality for eigenfunctions on curved spaces.

The paper analyzes harmonic maps to flat model spaces to prove geometric stability of the positive mass theorem.

Unique steady and expanding solitons with spherical links identified.

The purpose of this paper is to exhibit a quantitative stability result for the class of Möbius transformations of $\mathbb{S}^{n-1}$ when $n\geq 3$. The main estimate is of local nature and asserts that for a Lipschitz map that is apriori close to a Möbius transformation, an average conformal-isoperimetric type of def…

Study of elastic models in non-Euclidean spaces via Γ-convergence.

The paper studies SDP feasibility and sos ranks for specific polynomials.

In a seminal paper "Volumen und Oberfläche" (1903), Minkowski introduced the basic notion of mixed volumes and the corresponding inequalities that lie at the heart of convex geometry. The fundamental importance of characterizing the extremals of these inequalities was already emphasized by Minkowski himself, but has to…

In the first part of this short article, we define a renormalized F-functional for perturbations of non-compact steady Ricci solitons. This functional motivates a stability inequality which plays an important role in questions concerning the regularity of Ricci-flat spaces and the non-uniqueness of the Ricci flow with …

Study on stability of free boundary Willmore problem using new gradient inequality.