Related papers: Trudinger-Moser embedding on the hyperbolic space

The celebrated Nash Embedding Theorem asserts that every closed Riemannian manifold can be isometrically embedded into a sufficiently high-dimensional Euclidean space. In this paper, we prove an analogous result in the conformally compact…

We obtain new topological restrictions for complete Riemannian manifolds with nonnegative Ricci curvature and RCD(0,n) spaces. Our main results are a Betti number rigidity theorem which answers a question open since work of M.-T. Anderson…

We establish some important inequalities under the condition that the weighted Ricci curvature $\mathrm{Ric}_{\infty}\geq K$ for some constant $K >0$ by using improved Bochner inequality and its integrated form. Firstly, we obtain a sharp…

In this note, we prove an optimal upper bound for the first Dirac eigenvalue of some hypersurfaces in Euclidean space by combining a positive mass theorem and the construction of quasi-spherical metrics. As a direct consequence of this…

Based on a quantitative version of the classical Hopf-Rinow theorem in terms of the doubling property, we prove new precompactness principles in the (pointed) Gromov-Hausdorff topology for domains in (maybe incomplete) Riemannian manifolds…

We prove a sharp isoperimetric inequality for measured Finsler manifolds having non-negative Ricci curvature and Euclidean volume growth. We also prove a rigidity result for this inequality, under the additional hypotheses of boundedness of…

The paper establishes a sharp and rigid isoperimetric-type inequality in Lorentzian signature under the assumption of Ricci curvature bounded below in the timelike directions. The inequality is proved in the high generality of Lorentzian…

We obtain three types of results in this paper. Firstly we improve Leray's inequality by providing several types of reminder terms, secondly we introduce several Hilbert spaces based on these improved Leray inequalities and discuss their…

Using harmonic mean curvature flow, we establish a sharp Minkowski type lower bound for total mean curvature of convex surfaces with a given area in Cartan-Hadamard 3-manifolds. This inequality also improves the known estimates for total…

We use the intrinsic area to define a distance on the space of homothety classes of convex bodies in the $n$-dimensional Euclidean space, which makes it isometric to a convex subset of the infinite dimensional hyperbolic space. The ambient…

The paper is concerned with regularity properties of boundaries of causal pasts of points in a 3+1-dimensional Einstein-vacuum spacetime. In a Lorentzian manifold such boundaries play crucial role in propagation of linear and nonlinear…

We obtain new sharp isoperimetric inequalities on a Riemannian manifold equipped with a probability measure, whose generalized Ricci curvature is bounded from below (possibly negatively), and generalized dimension and diameter of the convex…

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 further research on the accelerated optimization phenomenon on Riemannian manifolds by introducing accelerated global first-order methods for the optimization of $L$-smooth and geodesically convex (g-convex) or $\mu$-strongly g-convex…

The main purpose of this short note, on the one hand, to is rigorize some part of the proof of Theorem 1.3 in [11] in a simple way, and on the other hand, to give an alternative argument from local inequalities to global ones.

We study the size of the isometry group Isom(M, g) of Riemannian manifolds (M, g) as g varies. For M not admitting a circle action, we show that the order of Isom(M, g) can be universally bounded in terms of the bounds on Ricci curvature,…

Tubular neighborhoods play an important role in differential topology. We have applied these constructions to geometry of almost Hermitian manifolds. At first, we consider deformations of tensor structures on a normal tubular neighborhood…

We establish the extrinsic Bonnet-Myers Theorem for compact Riemannian manifolds with positive Ricci curvature. And we show the almost rigidity for compact hypersurfaces, which have positive sectional curvature and almost maximal extrinsic…

We consider solutions (M,g(t)), 0 <= t <T, to Ricci flow on compact, four dimensional manifolds without boundary. We prove integral curvature estimates which are valid for any such solution. In the case that the scalar curvature is bounded…

We prove rigidity results involving the Hawking mass for surfaces immersed in a $3$-dimensional, complete Riemannian manifold $(M,g)$ with non-negative scalar curvature (resp. with scalar curvature bounded below by $-6$). Roughly, the main…