Related papers: The harmonic map heat flow on conic manifolds
In this paper, we present a general approach to obtain numerical schemes with good mesh properties for problems with moving boundaries, that is for evolving submanifolds with boundaries. This includes moving domains and surfaces with…
We introduce a heat flow associated to half-harmonic maps, which have been introduced by Da Lio and Rivi\`ere. Those maps exhibit integrability by compensation in one space dimension and are related to harmonic maps with free boundary. We…
We present two approaches to the heat flow on a Finsler manifold $(M,F)$: either as gradient flow on $L^2(M,m)$ for the energy; or as gradient flow on the reverse $L^2$-Wasserstein space $\mathcal{P}_2(M)$ of probability measures on $M$ for…
In this paper we investigate the convergence for the mean curvature flow of closed submanifolds with arbitrary codimension in space forms. Particularly, we prove that the mean curvature flow deforms a closed submanifold satisfying a…
An analytical model of high frequency oscillations of the kinetic and potential energies in a one-dimensional harmonic crystal with a substrate potential is obtained by introducing the nonlocal energies [1]. A generalization of the kinetic…
Mean curvature flow is the most natural evolution equation in extrinsic geometry, and shares many features with Hamilton's Ricci flow from intrinsic geometry. In this lecture series, I will provide an introduction to the mean curvature flow…
For any Riemannian foliation F on a closed manifold M with an arbitrary bundle-like metric, leafwise heat flow of differential forms is proved to preserve smoothness on M at infinite time. This result and its proof have consequences about…
We study surfaces evolving by mean curvature flow (MCF). For an open set of initial data that are $C^3$-close to round, but without assuming rotational symmetry or positive mean curvature, we show that MCF solutions become singular in…
Let (M,g) be a compact oriented Riemannian manifold with an incomplete edge singularity. This article shows that it is possible to evolve g by the Yamabe flow within a class of singular edge metrics. As the main analytic step we establish…
The understanding of the underlying dynamical mechanisms which determine the macroscopic laws of heat conduction is a long standing task of non-equilibrium statistical mechanics. A better understanding of the mechanism of heat conduction…
In this paper, we will study the existence of finite time singularity to harmonic heat flow and their formation patterns. After works of Coron-Ghidaglia, Ding and Chen-Ding, one knows blow-up solutions under smallness of initial energy for…
In this paper, we prove that the solution of the Landau-Lifshitz flow $u(t,x)$ from $\mathbb{H}^2$ to $\mathbb{H}^2$ converges to some harmonic map as $t\to\infty$. The essential observation is that although there exist infinite numbers of…
In this paper, we investigate the Gauss maps of a Ricci-mean curvature flow. A Ricci-mean curvature flow is a coupled equation of a mean curvature flow and a Ricci flow on the ambient manifold. Ruh and Vilms proved that the Gauss map of a…
The second law of thermodynamics dictates that heat flows spontaneously from a high-temperature entity to a lower-temperature one. Yet, recent advances have demonstrated that quantum correlations between a system and its thermal environment…
Inspired by Wilkin's work [23, 24] on Morse theory for the moduli space of Higgs bundles, we study the moduli space of gauged holomorphic maps by a heat flow approach in the spirit of Atiyah and Bott in a series of papers. In this paper,…
We prove the existence of closed convex ancient solutions to curvature flows which become more and more oval for large negative times. The speed function is a general symmetric function of the principal curvatures, homogeneous of degree…
We derive localized and global noncompact versions of Hamilton's gradient estimate for positive solutions to the heat equation on Riemannian manifolds with Ricci curvature bounded below. Our estimates are essentially optimal and…
In this paper, we extend the Hamilton's gradient estimates \cite{har93} and a monotonicity formula of entropy \cite{ni04} for heat flows from smooth Riemannian manifolds to (non-smooth) metric measure spaces with appropriate Riemannian…
The eventual concavity properties are useful to characterize geometric properties of the final state of solutions to parabolic equations. In this paper we give characterizations of the eventual concavity properties of the heat flow for…
In this paper we develop an approach to conformal geometry of piecewise flat metrics on manifolds. In particular, we formulate the combinatorial Yamabe problem for piecewise flat metrics. In the case of surfaces, we define the combinatorial…