Related papers: On Cheng's Eigenvalue Comparison Theorems

We study the biharmonic Steklov eigenvalue problem on a compact Riemannian manifold $\Omega$ with smooth boundary. We give a computable, sharp lower bound of the first eigenvalue of this problem, which depends only on the dimension, a lower…

We prove a rigidity theorem that shows that, under many circumstances, quasi-isometric embeddings of equal rank, higher rank symmetric spaces are close to isometric embeddings. We also produce some surprising examples of quasi-isometric…

It is shown that many results, previously believed to be properties of the Lichnerowicz Ricci curvature, hold for the Ricci curvature of all Gauduchon connections. We prove the existence of $t$--Gauduchon Ricci-flat metrics on the…

We consider rigidity properties of compact symmetric spaces $X$ with metric $g_0$ of rank one. Suppose $g$ is another Riemannian metric on $X$ with sectional curvature $\kappa$ bounded by $0 \leq \kappa \leq 1$. If $g$ equals $g_0$ outside…

By using optimal transport theory, we prove a sharp dimension-free isoperimetric inequality involving the volume entropy, in metric measure spaces with non-negative Ricci curvature in the sense of Lott--Sturm--Villani. We show that this…

In this note, we extend the rigidity of Cheng-Yau gradient estimate in \cite{HXY} to surfaces with lower Ricci curvature bound. Motivated by these sharp Cheng-Yau gradient estimates, pointwise Cheng-Yau gradient estimates for higher…

We study the qualitative stability of two classes of Sobolev inequalities on Riemannian manifolds. In the case of positive Ricci curvature, we prove that an almost extremal function for the sharp Sobolev inequality is close to an extremal…

This paper studies eigenvalues of some Steklov problems. Among other things, we show the following sharp estimtes. Let $\Omega$ be a bounded smooth domain in an $n(\geq 2)$-dimensional Hadamard manifold an let $0=\lambda_0 < \lambda_1\leq…

We use the Gauss-Bonnet theorem and the triangle comparison theorems of Rauch and Toponogov to show that on compact Riemann surfaces of negative curvature period integrals of eigenfunctions $e_\lambda$ over geodesics go to zero at the rate…

Let $M^n$ be a closed convex hypersurface lying in a convex ball $B(p,R)$ of the ambient $(n+1)$-manifold $N^{n+1}$. We prove that, by pinching Heintze-Reilly's inequality via sectional curvature upper bound of $B(p,R)$, 1st eigenvalue and…

In this article we prove a differentiable rigidity result. Let $(Y, g)$ and $(X, g_0)$ be two closed $n$-dimensional Riemannian manifolds ($n\geqslant 3$) and $f:Y\to X$ be a continuous map of degree $1$. We furthermore assume that the…

For each degree p, we construct on any closed manifold a family of Riemannian metrics, with fixed volume such that any positive eigenvalues of the rough and Hodge Laplacians acting on differential p-forms converge to zero. In particular, on…

The comparison theory for the Riccati equation satisfied by the shape operator of parallel hypersurfaces is generalized to semi-Riemannian manifolds of arbitrary index, using one-sided bounds on the Riemann tensor which in the Riemannian…

We study harmonic and quasi-harmonic discs in metric spaces admitting a uniformly local quadratic isoperimetric inequality for curves. The class of such metric spaces includes compact Lipschitz manifolds, metric spaces with upper or lower…

Sharp comparison theorems are derived for all eigenvalues of the (weighted) Laplacian, for various classes of weighted-manifolds (i.e. Riemannian manifolds endowed with a smooth positive density). Examples include Euclidean space endowed…

Examples show that Riemannian manifolds with almost-Euclidean lower bounds on scalar curvature and Perelman entropy need not be close to Euclidean space in any metric space sense. Here we show that if one additionally assumes an…

It is known that the so-called rotation minimizing (RM) frames allow for a simple and elegant characterization of geodesic spherical curves in Euclidean, hyperbolic, and spherical spaces through a certain linear equation involving the…

We obtain some rigidity results for metrics whose Schouten tensor is bounded from below after conformal transformations. Liang Cheng recently proved that a complete, nonflat, locally conformally flat manifold with Ricci pinching condition…

Consider a compact manifold $M$ with smooth boundary $\partial M$. Suppose that $g$ and $\tilde{g}$ are two Riemannian metrics on $M$. We construct a family of metrics on $M$ which agrees with $g$ outside a neighborhood of $\partial M$ and…

In the conformal class of Euclidean space, we give some volume comparison theorems with help of Q-curvature. Meanwhile, for compact four dimensional manifolds with non-negative scalar curvature, we give a volume rigidity theorem with…