Related papers: Topological rigidity of compact manifolds supporti…

Let $F^{n+p}(c)$ be an $(n+p)$-dimensional simply connected space form with nonnegative constant curvature $c$. We prove that if $M^n(n\geq4)$ is a compact submanifold in $F^{n+p}(c)$, and if $Ric_M>(n-2)(c+H^2),$ where $H$ is the mean…

In this paper, we consider asymptotically flat Riemannnian manifolds $(M^n,g)$ with $C^0$ metric $g$ and $g$ is smooth away from a closed bounded subset $\Sigma$ and the scalar curvature $R_g\ge 0$ on $M\setminus \Sigma$. For given $n\le…

We prove that a locally compact space with an upper curvature bound is a topological manifold if and only if all of its spaces of directions are homotopy equivalent and not contractible. We discuss applications to homology manifolds, limits…

We prove that for any Riemannian metric $g$ on a closed orientable surface $\Sigma$ and any spacelike embedding $f:\Sigma \rightarrow M$ in a pseudo-Riemannian manifold $(M,h)$, the embedding $f$ can be $C^{0}$-approximated by a smooth…

Using elementary comparison geometry, we prove: Let $(M,g)$ be a simply-connected complete Riemannian manifold of dimension $\ge 3$. Suppose that the sectional curvature $K$ satisfies $ -1-s(r) \le K \le -1$, where $r$ denotes distance to a…

It is well known that the Euclidean Sobolev inequality holds on any Cartan-Hadamard manifold of dimension $ n\ge 3 $, i.e. any complete, simply connected Riemannian manifold with nonpositive sectional curvature. As a byproduct of the…

We show the following result: Let $(M,g_0)$ be a compact manifold of dimension $n\geq 12$ with positive isotropic curvature. Then $M$ is diffeomorphic to a spherical space form, or a quotient manifold of $\mathbb{S}^{n-1}\times \mathbb{R}$…

Let $(M,g)$ be an asymptotically flat Riemannian manifold of dimension $n\geq 3$ with positive mass. We give a short proof based on Lyapunov-Schmidt reduction of the existence of an asymptotic foliation of $(M, g)$ by stable constant mean…

For $n\geq 2$, $p\in(1,n)$, the "best $p$-Sobolev inequality" on an open set $\Omega\subset\mathbb{R}^n$ is identified with a family $\Phi_\Omega$ of variational problems with critical volume and trace constraints. When $\Omega$ is bounded…

We prove that if $M$ is a three-manifold with scalar curvature greater than or equal to -2 and $\Sigma\subset M$ is a two-sided compact embedded Riemann surface of genus greater than 1 which is locally area-minimizing, then the area of…

In this paper we prove that, given a compact four dimensional smooth Riemannian manifold (M,g) with smooth boundary there exists a metric conformal to g with constant T-curvature, zero Q-curvature and zero mean curvature under generic and…

We present several rigidity results for Riemannian manifolds $(M^n,g)$ with scalar curvature $S \ge -n(n-1)$ (or $S\ge 0$), and having compact boundary $N$ satisfying a related mean curvature inequality. The proofs make use of results on…

Let $(M, \bar g)$ be an $n$-dimensional complete Riemannian manifold. In this paper, we considers the following conformal scalar curvature rigidity problem: Given a compact smooth domain $\Omega$ with $\partial \Omega$, can one find a…

Let $(X, g_0)$ be a complete, simply connected Riemannian manifold with sectional curvatures $K_{g_0}$ satisfying $-b^2 \leq K_{g_0} \leq -1$ for some $b \geq 1$. Let $g_1$ be a Riemannian metric on $X$ such that $g_1 = g_0$ outside a…

This paper generalizes a rigidity result of complex hyperbolic spaces by M. Herzlich. We prove that an almost Hermitian spin manifold $(M,g)$ of real dimension $4n+2$ which is strongly asymptotic to $\hyp{\C}^{2n+1}$ and satisfies a certain…

This note is concerned with some essential properties (optimal isoperimetry, first variation, and monotonicity formula) of the so-called $[0,1)\ni\gamma$-torsional rigidity $\mathcal{T}_{\gamma,\mathsf{g}}$ on a complete Riemannian…

In this paper, we show that a closed $n$-dimensional generalized $(\lambda, n+m)$-Einstein manifold of constant scalar curvature with weakly radially zero Ricci curvature is isometric to either a sphere ${\Bbb S}^n$, or a product ${\Bbb…

We provide a somewhat geometric proof of a rigidity theorem by M. Ledoux and C. Xia concerning complete manifolds with non-negative Ricci curvature supporting an Euclidean-type Sobolev inequality with (almost) best Sobolev constant. Using…

We prove the following result: Let $(M,g_0)$ be a compact manifold of dimension $n\geq 12$ with positive isotropic curvature. Then $M$ is diffeomorphic to a spherical space form, or the total space of an orbifiber bundle over $\mathbb{S}^1$…

Let $(M,g)$ be a smooth Riemannian manifold and $\mathsf{G}$ a compact Lie group acting on $M$ effectively and by isometries. It is well known that a lower bound of the sectional curvature of $(M,g)$ is again a bound for the curvature of…