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

In this paper, we study the anisotropic Minkowski problem. It is a problem of prescribing the anisotropic Gauss-Kronecker curvature for a closed strongly convex hypersurface in Euclidean space as a function on its anisotropic normals in…

We consider a Dirichlet problem for the mean curvature operator in the Minkowski spacetime, obtaining a necessary and sufficient condition for the existence of a spacelike solution, with prescribed mean curvature, which is the graph of a…

We establish a general formula for the enclosed volume of constant mean curvature (CMC) surfaces in Euclidean three space with translational periods forming a lattice. The formula relates the volume to the surface area, a…

We study boundary value problems for bounded uniform domains in $\mathbb{R}^n$, $n\geq 2$, with non-Lipschitz (and possibly fractal) boundaries. We prove Poincar\'e inequalities with trace terms and uniform constants for uniform…

We observe some higher order Poincare-type inequalities on a closed manifold, which is inspired by Hurwitz's proof of the Wirtinger's inequality using Fourier theory. We then give some geometric implication of these inequalities by applying…

In this paper, we propose a new assumption (1.2) that involves a small oscillation and $C^2$ norms for maps from smooth bounded domains into Euclidean spaces. Furthermore, by assuming that the domain has non-negative Ricci curvature, we…

We obtain certain inequalities involving several intrinsic invariants namely scalar curvature, Ricci curvature and $k$-Ricci curvature, and main extrinsic invariant namely squared mean curvature for submanifolds in a locally conformal…

On a compact Riemannian manifold with boundary, we study how Ricci curvature of the interior affects the geometry of the boundary. First we establish integral inequalities for functions defined solely on the boundary and apply them to…

In this paper, we prove a sharp anisotropic $L^p$ Minkowski inequality involving the total $L^p$ anisotropic mean curvature and the anisotropic $p$-capacity, for any bounded domains with smooth boundary in $\mathbb{R}^n$. As consequences,…

We study the problem of convergence of the normalized Ricci flow evolving on a compact manifold $\Omega$ without boundary. In \cite{KS10, KS15} we derived, via PDE techniques, global-in-time existence of the classical solution and…

In the present paper, we prove that a lower bound on the $1$-weighted Ricci curvature is equivalent to a convexity of entropies on the Wasserstein space. Based on such characterization, we provide some interpolation inequalities such as the…

A general integral inequality is established for compact spacelike submanifolds of codimension two in the Lorentz-Minkowski spacetime under the assumption that the mean curvature vector field is parallel. This inequality is then used to…

Given a compact Riemannian manifold with boundary, we prove that the space of embedded, which may be improper, free boundary minimal hypersurfaces with uniform area and Morse index upper bound is compact in the sense of smoothly graphical…

Consider an essentially nonbranching metric measure space with the measure contraction property of Ohta and Sturm, or with a Ricci curvature lower bound in the sense of Lott, Sturm and Villani. We prove a sharp upper bound on the inscribed…

The aim of this paper is to give two complete and simple characterizations of Minkowski norms N on an arbitrary topological real vector space such that the sublevel sets of N are strictly convex. We first show that this property is…

We introduce a scalar invariant on manifolds with density which is analogous to the renormalized volume coefficient $v_3$ in conformal geometry. We show that this invariant is variational and that shrinking gradient Ricci solitons are…

A well studied classical problem is the harmonicity of functions satisfying the restricted mean-value property (RMVP). While this has so far been studied mainly for domains in $\mathbb{R}^n$, we consider this problem in the general setting…

In this paper, we use the inverse mean curvature flow to establish an optimal Minkowski type inquality, weighted Alexandrov-Fenchel inequality for the mean convex star shaped hypersurfaces in Reissner-Nordstr\"om-anti-deSitter manifold and…

A property of smooth convex domains $\Omega \subset \mathbb{R}^n$ is that if two points on the boundary $x, y \in \partial \Omega$ are close to each other, then their normal vectors $n(x), n(y)$ point roughly in the same direction and this…

Let $m\in\mathbb{N},$ $m\geq 2,$ and let $\{p_j\}_{j=1}^m$ be a finite subset of $\mathbb{S}^2$ such that $0\in\mathbb{R}^3$ lies in its positive convex hull. In this paper we make use of the classical Minkowski problem, to show the…