Related papers: Dissipative Hyperbolic Geometric Flow
We show the existence of a global unique and analytic solution for the mean curvature flow, the surface diffusion flow and the Willmore flow of entire graphs for Lipschitz initial data with small Lipschitz norm. We also show the existence…
Explicit equations are given for describing the space-time evolution of non-ideal (viscous) relativistic fluids undergoing boost-invariant longitudinal and arbitrary transverse expansion. The equations are derived from the second-order…
We derive a new hyperbolic model describing the propagation of internal waves in a stratified shallow water with a non-hydrostatic pressure distribution. The construction of the hyperbolic model is based on the use of additional…
We consider relativistic hydrodynamics in the limit where the number of spatial dimensions is very large. We show that under certain restrictions, the resulting equations of motion simplify significantly. Holographic theories in a large…
In this article we investigate the dynamical properties of the geodesic flow for a proper metric space endowed with a proper action by isometries of a group with a contracting element. We show that the existence of a contracting isometry is…
We introduce a natural extension of the concept of gradient Ricci soliton: the Ricci almost soliton. We provide existence and rigidity results, we deduce a-priori curvature estimates and isolation phenomena, and we investigate some…
We study the Ricci flow on $\mathbb{R}^{4}$ starting at an SU(2)-cohomogeneity 1 metric $g_{0}$ whose restriction to any hypersphere is a Berger metric. We prove that if $g_{0}$ has no necks and is bounded by a cylinder, then the solution…
These expository notes present a proof of the Stable/Unstable Manifold Theorem (also known as the Hadamard--Perron Theorem). They also give examples of hyperbolic dynamics: geodesic flows on surfaces of negative curvature and dispersing…
In this paper we characterize non-collapsed limits of Ricci flows. We show that such limits are smooth away from a set of codimension $\geq 4$ in the parabolic sense and that the tangent flows at every point are given by gradient shrinking…
In this paper we study conformal solitons for the mean curvature flow in hyperbolic space $\mathbb{H}^{n+1}$. Working in the upper half-space model, we focus on horo-expanders, which relate to the conformal field $-\partial_0$. We classify…
This paper initiates a classification programme of flows of $\mathrm{SU}(2)$-structures on $4$-manifolds which have short-time existence and uniqueness. Our approach adapts a representation-theoretic method originally due to Bryant in the…
This paper investigates curve flows on the light-cone in the 3-dimensional Minkowski space. We derive the Harnack inequality for the heat flow and present a detailed classification of space-periodic solitons for a third-order curvature…
We give an application of a Huisken monotonicity-type formula for the mean curvature flow in a compact smooth manifold with a Riemannian metric that evolves by a shrinking self-similar solution of the extended Ricci flow. Our investigation…
We consider the inverse mean curvature flow by parallel hypersurfaces in space forms. We show that such a flow exists if and only if the initial hypersurface is isoparametric. The flow is characterized by an algebraic equation satisfied by…
This paper defines a parabolic frequency for solutions of the heat equation on a Ricci flow and proves it's monotonicity along the flow. Frequency monotonicity is known to have many useful consequences; here it is shown to provide a simple…
We first define Pseudo-Calabi flow, as {equation*} {{aligned}{{\partial \varphi}\over {\partial t}}&= -f(\varphi), \triangle_varphi f(\varphi) &= S(\varphi) - \ul S.{aligned}. \end{equation*} Then we prove the well-posedness of this flow…
The paper addresses the numerical approximation of two variants of hyperbolic mean curvature flow of surfaces in $\mathbb R^3$. For each evolution law we propose both a finite element method, as well as a finite difference scheme in the…
The elastic flow, which is the $L^2$-gradient flow of the elastic energy, has several applications in geometry and elasticity theory. We present stable discretizations for the elastic flow in two-dimensional Riemannian manifolds that are…
We consider the volume preserving flow of smooth, closed and convex hypersurfaces in the hyperbolic space $\mathbb{H}^{n+1} (n\geq 2)$ with the speed given by arbitrary positive power $\alpha$ of the Gauss curvature. We prove that if the…
This article presents new gradient estimates for positive solutions to the nonlinear fast diffusion equation on smooth metric measure spaces, involving the $f$-Laplacian. The gradient estimates of interest are mainly of…