Related papers: Rigidity of compact Riemannian spin Manifolds with…

We prove a scalar-mean rigidity theorem for compact Riemannian manifolds with boundary in dimension less than five by developing a dimension reduction argument for mean curvature, which extends Schoen-Yau's dimension reduction argument for…

In this paper, we prove some rigidity theorems for compact Bach-flat $n$-manifold with the positive constant scalar curvature. In particular, our conditions in Theorem 1.4 have the additional properties of being sharp.

We study the boundary and lens rigidity problems on domains without assuming the convexity of the boundary. We show that such rigidities hold when the domain is a simply connected compact Riemannian surface without conjugate points. For the…

A rigidity result for weakly asymptotically hyperbolic manifolds with lower bounds on Ricci curvature is proved without assuming that the manifolds are spin. The argument makes use of a quasi-local mass characterization of Euclidean balls…

Gromov conjectured that the total mean curvature of the boundary of a compact Riemannian manifold can be estimated from above by a constant depending only on the boundary metric and on a lower bound for the scalar curvature of the fill-in.…

This paper studies three-dimensional compact static manifolds with boundary and positive scalar curvature. We prove that, under a suitable bound on the Ricci curvature, the orientable quotient of the Nariai static manifold with boundary…

We prove a generalization of Gromov's conjecture on scalar curvature rigidity of convex polytopes to arbitrary convex Riemannian polytope type domains via harmonic spinors on convex domians with boundary condition constructed by Brendle. In…

In this paper we generalize the main result of [4] for manifolds that are not necessarily Einstein. In fact, we obtain an upper bound for the volume of a locally volume-minimizing closed hypersurface $\Sigma$ of a Riemannian 5-manifold $M$…

In this paper, we proved a compactness result about Riemannian manifolds with an arbitrary pointwisely pinched Ricci curvature tensor.

For complete Riemannian manifolds with vanishing Bach tensor and positive constant scalar curvature, we provide a rigidity theorem characterized by some pointwise inequalities. Furthermore, we prove some rigidity results under an inequality…

For compact submanifolds in Euclidean and Spherical space forms with Ricci curvature bounded below by a function $\alpha(n,k,H,c)$ of mean curvature, we prove that the submanifold is either isometric to the Einstein Clifford torus, or a…

In this paper, we prove that a Riemannian $n$-manifold $M$ with sectional curvature bounded above by $1$ that contains a minimal $2$-sphere of area $4\pi$ which has index at least $n-2$ has constant sectional curvature $1$. The proof uses…

The goal of this paper is to investigate some rigidity properties of stable solutions of elliptic equations set on manifolds with boundary. We provide several types of results, according to the dimension of the manifold and the sign of its…

The inscribed radius of a compact manifold with boundary is bounded above if its Ricci curvature and mean curvature are bounded from below. The rigidity result implies that the upper bound can be achieved only in space form. In this paper,…

We show that a compact Riemannian $3$-manifold $M$ with strictly convex simply connected boundary and sectional curvature $K\leq a\leq 0$ is isometric to a convex domain in a complete simply connected space of constant curvature $a$,…

We prove the rigidity of rectifiable boundaries with constant distributional mean curvature in the Brendle class of warped product manifolds (which includes important models in General Relativity, like the deSitter--Schwarzschild and…

In this paper, we consider a compact Riemannian manifold whose boundary is endowed with a Riemannian flow. Under a suitable curvature assumption depending on the O'Neill tensor of the flow, we prove that any solution of the basic Dirac…

This article establishes a low-regularity Riemannian positive mass theorem for non-spin manifolds whose metrics are only $C^0 \cap W_{\mathrm{loc}}^{1,n}$ and smooth outside a compact set. The main theorem asserts that asymptotically flat…

In this paper, as a continuation of [30], we consider the Gromov-Hausdorff convergence and collapsing in the family of compact Riemannian manifolds with boundary satisfying lower bounds on the sectional curvatures of interior manifolds,…

When the Ricci curvature of a Riemannian manifold is not lower bounded by a constant, but lower bounded by a continuous function, we give a new characterization of this lower bound through the convexity of relative entropy on the…