Related papers: Isoperimetric inequalities for eigenvalues by inve…

We obtain inequalities for all Laplace eigenvalues of Riemannian manifolds with an upper sectional curvature bound, whose rudiment version for the first Laplace eigenvalue was discovered by Berger in 1979. We show that our inequalities…

We analyze a gradient flow of closed planar curves minimizing the anisoperimetric ratio. For such a flow the normal velocity is a function of the anisotropic curvature and it also depends on the total interfacial energy and enclosed area of…

In the first part of the paper we survey some nonlocal flows of convex plane curves ever studied so far and discuss properties of the flows related to enclosed area and length, especially the isoperimetric ratio and the isoperimetric…

Several inequalities for the isoperimetric ratio for plane curves are derived. In particular, we obtain interpolation inequalities between the deviation of curvature and the isoperimetric ratio. As applications, we study the large-time…

In this paper we complete the study started in [Pi2] of evolution by inverse mean curvature flow of star-shaped hypersurface in non-compact rank one symmetric spaces. We consider the evolution by inverse mean curvature flow of a closed,…

We study a variant of the mean curvature flow for closed, convex hypersurfaces where the normal velocity is a nonhomogeneous function of the principal curvatures. We show that if the initial hypersurface satisfies a certain pinching…

Let $M$ be an $n$-dimensional closed Riemannian manifold with metric $g$, $d\mu=e^{-\phi(x)}d\nu$ be the weighted measure and $\Delta_{p,\phi}$ be the weighted $p$-Laplacian. In this article we will investigate monotonicity for the first…

The paper is devoted to the study of some properties of the first eigenvalue of the anisotropic $p$-Laplace operator with Robin boundary condition involving a function $\beta$ which in general is not constant. In particular we obtain sharp…

We study some properties of Laplacian eigenvalues with negative Robin boundary conditions. We will show some monotonicity properties on annuli of the first eigenvalue by means of shape optimization techniques.

Various properties of isoperimetric, functional, Transport-Entropy and concentration inequalities are studied on a Riemannian manifold equipped with a measure, whose generalized Ricci curvature is bounded from below. First, stability of…

In this article, by following the method in \cite{PT}, combining Willmore energy with isoperimetric inequalities, we construct two examples of singularities under mean curvature flow in $\mathbb{H}^3$. More precisely, there exists a torus,…

It is shown that a hypersurface of a space form is the initial data for a solution to the mean curvature flow by parallel hypersurfaces if, and only if, it is isoparametric. By solving an ordinary differential equation, explicit solutions…

One isoperimetric inequality for the fundamental sloshing eigenvalue is derived under the assumption that containers have vertical side walls and either finite or infinite depth. It asserts that among all such containers, whose free…

We consider the inverse mean curvature flow in smooth Riemannian manifolds of the form $([R_{0},\infty)\times S^n,\bar{g})$ with metric $\bar{g}=dr^2+{\vartheta}^2(r){\sigma}$ and non-positive radial sectional curvature. We prove, that for…

We consider inverse curvature flows in the $(n+1)$-dimensional Euclidean space, $n\geq 2,$ expanding by arbitrary negative powers of a 1-homogeneous, monotone curvature function $F$ with some concavity properties. We obtain asymptotical…

In this paper we prove a sharp lower bound for the first nontrivial Neumann eigenvalue $\mu_1(\Omega)$ for the $p$-Laplace operator in a Lipschitz, bounded domain $\Omega$ in $\R^n$. Our estimate does not require any convexity assumption on…

We consider inverse curvature flows in warped product manifolds, which are constrained subject to local terms of lower order, namely the radial coordinate and the generalized support function. Under various assumptions we prove longtime…

We establish lower bounds for the first non-zero eigenvalue for the natural geometric sub-elliptic Laplacian operator defined on sub-Riemannian manifolds of step 2 that satisfy a positive curvature condition. The methods are very general…

We propose a notion of scalar curvature lower bounds in a three-dimensional Riemannian manifold endowed with a $C^0$ metric based on the monotonicity of the Hawking mass along the inverse mean curvature flow. We present a stability theorem…

In this note, we discuss the mean curvature flow of graphs of maps between Riemannian manifolds. Special emphasis will be placed on estimates of the flow as a non-linear parabolic system of differential equations. Several global existence…