相关论文: A Minkowski type inequality for manifolds with pos…

We prove a laplacian comparison theorem in the barrier sense for the function distance to the boundary of Riemannian manifolds with nonnegative Ricci curvature, area and mean curvature of the boundary bounded above. As an application we get…

The purpose of this paper is to derive volume and other geometric information for three-dimensional complete manifolds with positive scalar curvature. In the case that the Ricci curvature is nonnegative, it is shown that the volume of the…

The author gives an alternative and simple proof of the global existence of smooth solutions to the Cauchy problem for wave maps from the 1+2-dimensional Minkowski space to an arbitrary compact smooth Riemannian manifold without boundary,…

In this paper, we prove the existence of hypersurfaces in the Euclidean space with prescribed boundary and whose k-th Weingarten curvature equals a given function that depends on the normal of the hypersurface. The proof is based on the…

Hamilton's pinching conjecture, that three-dimensional complete non-compact manifolds with pinched Ricci curvature are flat, has recently been resolved using Ricci flow. In this paper we prove a direct analogue of that result in all…

Studying the geometric flow plays a powerful role in mathematics and physics. In this paper, we introduce the mean curvature flow on Finsler manifolds and give a number of examples of the mean curvature flow. For Minkowski spaces, a special…

The Brunn-Minkowski inequality states that for bounded measurable sets $A$ and $B$ in $\mathbb{R}^n$, we have $|A+B|^{1/n} \geq |A|^{1/n}+|B|^{1/n}$. Also, equality holds if and only if $A$ and $B$ are convex and homothetic sets in…

A pair of subsets of Euclidean space which nearly achieves equality in the Brunn-Minkowski inequality must nearly coincide with a pair of homothetic convex sets. The two-dimensional case was treated in a previous paper in this series by an…

For submanifolds tangent to the structure vector field in cosymplectic space forms, we establish a basic inequality between the main intrinsic invariants of the submanifold, namely its sectional curvature and scalar curvature on one side;…

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 collection of reverse Alexandrov-Fenchel type inequalities in anisotropic, Euclidean, spherical, and hyperbolic settings. The unifying principle is that the relevant deficit is controlled by curvature radius data, or equivalently…

We prove a curvature pinching result for the Ricci flow on asymptotically flat manifolds: if an asymptotically flat manifold of dimension $n\geq 3$ has scale-invariant integral norm of curvature sufficiently pinched relative to the inverse…

Let $M^n$ be an $n$-dimensional Riemannian manifold with boundary $\partial M$. Assume that Ricci curvature is bounded from below by $(n-1)k$, for $k\in \RR$, we give a sharp estimate of the upper bound of $\rho(x)=\dis(x, \partial M)$, in…

We establish a sharp Fenchel-Willmore inequality for closed submanifolds of arbitrary dimension and codimension immersed in a complete Riemannian manifold with non-negative intermediate Ricci curvature and Euclidean volume growth. In the…

We prove a sharp Log-Sobolev inequality for submanifolds of a complete non-compact Riemannian manifold with asymptotic non-negative intermediate Ricci curvature and Euclidean volume growth. Our work extends a result of Dong-Lin-Lu which…

Gromov's universal filling inequalities relate the filling radius and the filling volume of a Riemannian manifold to its volume. The main result of the present article is that in dimensions at least three the optimal constants in the…

In this paper, we show that the inverse anisotropic mean curvature flow in $\mathbb{R}^{n+1}$, initiating from a star-shaped, strictly $F$-mean convex hypersurface, exists for all time and after rescaling the flow converges exponentially…

We show that for a noncollapsing sequence of closed, connected, oriented Riemannian manifolds with Ricci curvature uniformly bounded from below and diameter uniformly bounded above, Gromov-Hausdorff convergence essentially agrees with…

In this paper, an n-dimensional complete open manifold with nonnegative Ricci curvature and collapsing volume has been investigated. If its radial sectional curvature bounded from below, it shows that such a manifold is of finite…

Quantum inequalities are lower bounds for local averages of quantum observables that have positive classical counterparts, such as the energy density or the Wick square. We establish such inequalities in general (possibly interacting)…