arXiv research

Entropy rigidity proven for 3D and higher convex projective manifolds.

Proves rigidity for maps between manifolds using degree theory and current developments.

New volume comparison theorem for gradient Ricci almost solitons.

Sharp bounds on Alexandrov spaces' boundaries with rigidity analysis.

The paper proves estimates and theorems for Kähler manifolds.

We prove a Lipschitz-Volume rigidity theorem for the non-collapsed Gromov-Hausdorff limits of manifolds with Ricci curvature bounded from below. This is a counterpart of the Lipschitz-Volume rigidity in Alexandrov geometry.

A rigidity theorem for smooth Legendrian self-shrinkers is proven.

Paper shows regions close to negatively curved metrics are minimal fillings and rigid.

The Cayley hyperbolic space minimizes volume entropy among finite-volume metrics.

Quantitative rigidity theorem for Alexandrov spaces with curvature bounds.

Study circles to understand dynamics and rigidity in homogeneous spaces.

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

Motivated by Brendle-Marques-Neves' counterexample to the Min-Oo's conjecture, we prove a volume constrained scalar curvature rigidity theorem which applies to the hemisphere.

We extend the theory of Patterson-Sullivan measure to any regular covering of a compact manifold using the Busemann compactification and derive an integral formula for the volume entropy. As applications we prove some rigidity theorems for the volume entropy.

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

The paper proves convex bodies are minimal fillings and have Lipschitz-volume rigidity.

The paper proves rigidity results for Einstein manifolds with specific geometric constraints.

Two rigidity theorems for manifolds with nonnegative Ricci curvature and specific volume growth.

We study Riemannian manifolds with boundary under a lower Ricci curvature bound, and a lower mean curvature bound for the boundary. We prove a volume comparison theorem of Bishop-Gromov type concerning the volumes of the metric neighborhoods of the boundaries. We conclude several rigidity theorems. As one of them, we o…

We consider complete noncompact Riemannian manifolds with quadratically decaying lower Ricci curvature bounds and minimal volume growth. We first prove a rigidity result showing that ends with strongly minimal volume growth are isometric to warped product manifolds. Next we consider the almost rigid case in which manif…

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 …

Given a compact Alexadrov $n$-space $Z$ with curvature curv $\ge κ$, and let $f: Z\to X$ be a distance non-increasing onto map to another Alexandrov $n$-space with curv $\ge κ$. The relative volume rigidity conjecture says that if $X$ achieves the relative maximal volume i.e. $vol(Z)=vol(X)$, then $X$ is isometric to $…

Smoothly conjugate Anosov flows on 3D manifolds are actually smoothly conjugate.

In this note we give a short proof to the rigidity of volume entropy. The result says that for a closed manifold with Ricci curvature bounded from below, if the universal cover has maximal volume entropy, then it is the space form. This theorem was first proved by F. Ledrappier and X. Wang in [1].

Let $X$ and $Y$ be length metric spaces. Let $\mathcal H^n$ denote the $n$-dimensional Hausdorff measure. The Lipschitz-Volume Rigidity is a property that if there exists a 1-Lipschitz map $f\colon X\to Y$ and $0<\mathcal H^n(X)=\mathcal H^n(f(X))<\infty$, then $f$ preserves the length of path. This property holds for …

The study shows almost maximal volume entropy rigidity for certain manifolds with integral Ricci curvature.

For a closed, strictly convex projective manifold of dimension $n\geq 3$ that admits a hyperbolic structure, we show that the ratio of Hilbert volume to hyperbolic volume is bounded below by a constant that depends only on dimension. We also show that for such spaces, if topological entropy of the geodesic flow goes to…

In this paper, we prove that principal circle bundles over the complex projective space equipped with the standard Sasakian structures are volume rigid among all $K$-contact manifolds satisfying positivity conditions of tensors involing the Tanaka-Webster curvature.

Kahler manifolds with specific curvature properties are close to projective spaces.

Study on Kähler manifolds with nonnegative Ricci curvature, focusing on rigidity.

Second part of Q-curvature research focusing on volume comparison.

Axisymmetric Ricci solitons are rigid under non-axisymmetric perturbations.

Study rigidity and volume optimization of hyperbolic polyhedra.

Totally geodesic submanifolds in convex cores are properly immersed and have finite volume.

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^…

Study rigidity by logarithmic capacity and related functions.

The main subject of this expository paper is a connection between Gromov's filling volumes and a boundary rigidity problem of determining a Riemannian metric in a compact domain by its boundary distance function. A fruitful approach is to represent Riemannian metrics by minimal surfaces in a Banach space and to prove r…

In this paper, we firstly establish a new volume growth estimate for spacelike entire graphs in the pseudo-Euclidean space $\mathbb{R}^{m+n}_n$. Then by using this volume growth estimate and the Co-Area formula, we prove various rigidity results for spacelike entire self-shrinking graphs.

The paper proves volume comparison theorems for metrics near stable Einstein and Ricci flat manifolds.

We construct arithmetic Kleinian groups that are profinitely rigid in the absolute sense: each is distinguished from all other finitely generated, residually finite groups by its set of finite quotients. The Bianchi group $\mathrm{PSL}(2,\mathbb{Z}[ω])$ with $ω^2+ω+1=0$ is rigid in this sense. Other examples include th…

The main results of this paper consists of two parts. Firstly, we obtain an almost rigidity theorem which says that on a RCD(0, N) space, when a domain between two level sets of a distance function has almost maximal volume compared to that of a cylinder, then this portion is close to a cylinder as a metric space. Seco…

The study solves a problem in conformal geometry with applications to Q-curvature.

The paper studies Q-curvature and volume entropy on manifolds, proving polynomial growth polyharmonic function finiteness and rigidity.

This paper extends our earlier results to higher dimensions using a different approach, based on the rigidity of complex structures on certain domains.

Profinite rigidity of certain 3-manifolds detected through Dehn fillings.

The paper is centered around a new proof of the infinitesimal rigidity of smooth closed surfaces with everywhere positive Gauss curvature. We use a reformulation that replaces deformation of an embedding by deformation of the metric inside the body bounded by the surface. The proof is obtained by studying derivatives o…

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…

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