arXiv research

Paper proves stability of quermassintegral inequalities using inverse curvature flow.

The paper improves inequalities for nearly spherical sets using quermassintegrals.

Unified Minkowski problem discussed for (p,q)-mixed quermassintegrals.

Paper explores curvature flows on spheres to prove inequalities.

The study proves stability of quermassintegral inequalities in hyperbolic space.

New $L_p$-Steiner quermassintegrals defined from Steiner formula.

Study flows to analyze sphere quermassintegrals.

New convexity concept applied to sphere yields quermassintegral inequalities.

Paper proves a generalized Alexandrov-Fenchel inequality for convex hypersurfaces with capillary boundary.

We give a simple proof of the insoperimetric inequality for quermassintegrals of non-convex starshaped domains, using a reslut of Gerhardt \cite{G} and Urbas \cite{U} on an expanding geometric curvature flow.

Flow preserves quermassintegrals, converging to a geodesic sphere.

The paper proves inequalities for convex hypersurfaces in spheres and hyperbolic spaces.

In this paper, we solve various isoperimetric problems for the quermassintegrals and the curvature integrals in the hyperbolic space $\H^n$, by using quermassintegral preserving curvature flows. As a byproduct, we obtain hyperbolic Alexandrov-Fenchel inequalities.

The paper proves stability of inequalities for nearly spherical sets in various spaces.

In this article, we introduce a new type of mean curvature flow for bounded star-shaped domains in space forms and prove its longtime existence, exponential convergence without any curvature assumption. Along this flow, the enclosed volume is a constant and the surface area evolves monotonically. Moreover, for a bounde…

Proves inequalities for hypersurfaces in the sphere, solving a long-standing problem.

The paper proves inequalities for hypersurfaces in a unit ball with specific boundary conditions.

The article proves inequalities for capillary hypersurfaces in hyperbolic space.

New weighted geometric inequalities for hypersurfaces in R^n proved.

The paper proves geometric inequalities in sphere using locally constrained flows.

The paper proves geometric inequalities and their stabilities for curves in hyperbolic space.

In this paper we first establish an optimal Sobolev type inequality for hypersurfaces in $\H^n$(see Theorem \ref{mainthm1}). As an application we obtain hyperbolic Alexandrov-Fenchel inequalities for curvature integrals and quermassintegrals. Precisely, we prove a following geometric inequality in the hyperbolic space …

We consider the quermassintegral preserving flow of closed \emph{h-convex} hypersurfaces in hyperbolic space with the speed given by any positive power of a smooth symmetric, strictly increasing, and homogeneous of degree one function $f$ of the principal curvatures which is inverse concave and has dual $f_*$ approachi…

In this paper, we first study the locally constrained curvature flow of hypersurfaces in hyperbolic space, which was introduced by Brendle, Guan and Li [7]. This flow preserves the $m$th quermassintegral and decreases $(m+1)$th quermassintegral, so the convergence of the flow yields sharp Alexandrov-Fenchel type inequa…

Paper solves anisotropic capillary Minkowski problem for p ≥ 1.

The dual Minkowski problem for even data asks what are the necessary and sufficient conditions on an even prescribed measure on the unit sphere for it to be the $q$-th dual curvature measure of an origin-symmetric convex body in $\mathbb{R}^n$. A full solution to this is given when $1 < q < n$. The necessary and suffic…

We extend the classical Aleksandrov-Fenchel inequality for mixed volumes to functionals arising naturally in hermitian integral geometry. As a consequence, we obtain Brunn-Minkowski and isoperimetric inequalities for hermitian quermassintegrals.

Through the study of some elliptic and parabolic fully nonlinear PDEs, we establish conformal versions of quermassintegral inequality, the Sobolev inequality and the Moser-Trudinger inequality for the geometric quantities associated to the Schouten tensor on locally conformally flat manifolds.

In this paper, we establish a generalised Blaschke-Santalò inequality for convex bodies in $\mathbb R^{n+1}$. This inequality gives an upper bound estimate for the product of dual quermassintegrals of convex body and its polar set. Our argument is based on induction on dimensions.

This paper continues the study of Alexandrov-Fenchel inequalities for quermassintegrals for $k$-convex domains. It focuses on the application to the Michael-Simon type inequalities for $k$-curvature operators. The proof uses optimal transport maps as a tool to relate curvature quantities defined on the boundary of a do…

Study on curvature flow in Minkowski space for cocompact hypersurfaces.

In this paper, the dual Orlicz curvature measure is proposed and its basic properties are provided. A variational formula for the dual Orlicz-quermassintegral is established in order to give a geometric interpretation of the dual Orlicz curvature measure. Based on the established variational formula, a solution to the …

Study anisotropic flows without global terms and solve dual Orlicz Christoffel-Minkowski problems.

The paper proves geometric inequalities for pinched convex hypersurfaces in de Sitter space.

Paper solves inequalities for capillary hypersurfaces in half-spaces.

In this paper we first introduce quermassintegrals for free boundary hypersurfaces in the $(n+1)$-dimensional Euclidean unit ball. Then we solve some related isoperimetric type problems for convex free boundary hypersurfaces, which lead to new Alexandrov-Fenchel inequalities. In particular, for $n=2$ we obtain a Minkow…

Study anisotropic flow for capillary hypersurfaces, proving new inequalities.

In this article, we will use inverse mean curvature flow to establish an optimal Sobolev-type inequality for hypersurfaces $Σ$ with nonnegative sectional curvature in $\mathbb{H}^n$. As an application, we prove the hyperbolic Alexandrov-Fenchel inequalities for hypersurfaces with nonnegative sectional curvature in $\ma…

In this paper we prove the following geometric inequality in the hyperbolic space $\H^n$ ($n\ge 5)$, which is a hyperbolic Alexandrov-Fenchel inequality, \[\begin{array}{rcl} \ds \int_Σ\s_4 d μ\ge \ds\vs C_{n-1}^4ω_{n-1}\left\{\left(\frac{|Σ|}{ω_{n-1}} \right)^\frac 12 + \left(\frac{|Σ|}{ω_{n-1}} \right)^{\frac 12\frac…

Study of star-shaped hypersurfaces with capillary boundary using constrained mean curvature flow.

The paper proves new inequalities for convex hypersurfaces using centro-affine geometry.

In this paper, we study flows of hypersurfaces in hyperbolic space, and apply them to prove geometric inequalities. In the first part of the paper, we consider volume preserving flows by a family of curvature functions including positive powers of $k$-th mean curvatures with $k=1,\cdots,n$, and positive powers of $p$-t…

The paper finds the largest eigenvalue for a specific type of domain in hyperbolic space.

The paper establishes sharp geometric inequalities for hypersurfaces in warped product manifolds.

We consider the class of $λ$-concave bodies in $\mathbb R^{n+1}$; that is, convex bodies with the property that each of their boundary points supports a tangent ball of radius $1/λ$ that lies locally (around the boundary point) inside the body. In this class we solve a reverse isoperimetric problem: we show that the co…

The ABP method is used to prove geometric inequalities for submanifolds and tensors.

Flow of convex hypersurfaces in hyperbolic space converges to geodesic spheres.

We prove an analogue of the classical Steiner formula for the $L_p$ affine surface area of a Minkowski outer parallel body for any real parameters $p$. We show that the classical Steiner formula and the Steiner formula of Lutwak's dual Brunn Minkowski theory are special cases of this new Steiner formula. This new Stein…