Related papers: Global geometry under isotropic Brownian flows
In the first part of this paper, we develop the theory of anisotropic curvature measures for convex bodies in the Euclidean space. It is proved that any convex body whose boundary anisotropic curvature measure equals a linear combination of…
We study the volume preserving mean curvature flow of a surface immersed in an asymptotically flat $3$-manifold modeling an isolated gravitating system in General Relativity. We show that, if the ambient manifold has positive ADM mass and…
We consider the mean curvature flow of the graph of a smooth map $f:\mathbb{R}^2\to\mathbb{R}^2$ between two-dimensional Euclidean spaces. If $f$ satisfies an area-decreasing property, the solution exists for all times and the evolving…
Mean curvature flow evolves isometrically immersed base manifolds $M$ in the direction of their mean curvatures in an ambient manifold $\bar{M}$. If the base manifold $M$ is compact, the short time existence and uniqueness of the mean…
We are interested in the impact of entropies on the geometry of a hypersurface of a Riemannian manifold. In fact, we will be able to compare the volume entropy of a hypersurface with that of the ambient manifold, provided some geometric…
Let (M,g) be a compact Riemannian manifold of hyperbolic type, i.e M is a manifold admitting another metric of strictly negative curvature. In this paper we study the geodesic flow restricted to the set of geodesics which are minimal on the…
We study the extrinsic geometry of isometric immersions into Riemannian manifolds of co-dimension one via a fourth-order geometric evolution of the shape operator. Motivated by bi-harmonic map theory and generalized Chen's conjecture, we…
Many physical situations are characterized by interfaces with a non trivial shape so that relevant geometric features, such as interfacial area, curvature or unit normal vector, can be used as main indicators of the topology of the…
In this paper we introduce a new geometric flow --- the hyperbolic gradient flow for graphs in the $(n+1)$-dimensional Euclidean space $\mathbb{R}^{n+1}$. This kind of flow is new and very natural to understand the geometry of manifolds. We…
We consider the evolution by mean curvature flow of a closed submanifold of the complex projective space. We show that, if the submanifold has small codimension and satisfies a suitable pinching condition on the second fundamental form,…
We construct bi-Lipschitz embeddings into Euclidean space for manifolds and orbifolds of bounded diameter and curvature. The distortion and dimension of such embeddings is bounded by diameter, curvature and dimension alone. Our results also…
In this paper, we introduce and study the conformal mean curvature flow of submanifolds of higher codimension in the Euclidean space $\bbr^n$. This kind of flow is a special case of a general modified mean curvature flow which is of various…
In this paper, we define a class of new geometric flows on a complete Riemannian manifold. The new flow is related to the generalized (third order) Landau-Lifishitz equation. On the other hand it could be thought of a special case of the…
We study the Wasserstein space (with quadratic cost) of Euclidean spaces as an intrinsic metric space. In particular we compute their isometry groups. Surprisingly, in the case of the line, there exists a (unique) "exotic" isometric flow.…
The mean curvature flow is an evolution process under which a submanifold deforms in the direction of its mean curvature vector. The hypersurface case has been much studied since the eighties. Recently, several theorems on regularity,…
Effects of geometric constraints on a steady flow potential are described by an elliptic-hyperbolic generalization of the harmonic map equations. Sufficient conditions are given for global triviality.
In this paper we provide new existence results for isoperimetric sets of large volume in Riemannian manifolds with nonnegative Ricci curvature and Euclidean volume growth. We find sufficient conditions for their existence in terms of the…
We consider a convex Euclidean hypersurface that evolves by a volume or area preserving flow with speed given by a general nonhomogeneous function of the mean curvature. For a broad class of possible speed functions, we show that any closed…
A geometric flow based in the Riemann-Christoffel curvature tensor that in two dimensions has some common features with the usual Ricci flow is presented. For $n$ dimensional spaces this new flow takes into account all the components of the…
We study the mean curvature flow of complete space-like submanifolds in pseudo-Euclidean space with bounded Gauss image, as well as that of complete submanifolds in Euclidean space with convex Gauss image. By using the confinable property…