Related papers: Well-posedness for the heat flow of polyharmonic m…

The main objective of this article is to study the mean curvature flow into an ambient compact smooth manifold M with boundary and with a Riemannian metric that evolves by a self-similar solution of the Ricci flow coupled with the harmonic…

In this work, we obtain a short time existence result for harmonic map heat flow coupled with a smooth family of complete metrics in the domain manifold. Our results generalize short time existence results for harmonic map heat flow by…

In this paper, we consider critical maps of a horizontal energy functional for maps from a sub-Riemannian manifold to a Riemannian manifold. These critical maps are referred to as subelliptic harmonic maps. In terms of the subelliptic…

We investigate a parabolic-elliptic system for maps $(u,v)$ from a compact Riemann surface $M$ into a Lorentzian manifold $N\times{\mathbb{R}}$ with a warped product metric. That system turns the harmonic map type equations into a parabolic…

We prove that for any two closed Riemannian manifolds $M^{2m}$ ($m\geq 1$) and $N$, there exists a minimizing (extrinsic) $m$-polyharmonic map for every free homotopy class in $[M^{2m}, N]$, provided that the homotopy group $\pi_{2m}(N)$ is…

Using the interpretation of the half-Laplacian on $S^1$ as the Dirichlet-to-Neumann operator for the Laplace equation on the ball $B$, we devise a classical approach to the heat flow for half-harmonic maps from $S^1$ to a closed target…

We investigate the existence of weak expanding solutions of the harmonic map flow for maps with values into a smooth closed Riemannian manifold. We prove the existence of such solutions in case the target manifold is isometrically embedded…

We establish effective existence and uniqueness for the heat flow on time-dependent Riemannian manifolds, under minimal assumptions tailored towards the study of Ricci flow through singularities. The main point is that our estimates only…

We introduce a rectified $n$-harmonic map flow from an n-dimensional closed Riemannian manifold to another closed Riemannian manifold. We prove existence of a global solution, which is regular except for a finite number of points, of the…

In this paper, we investigate $V$-harmonic heat flows from complete Riemannian manifolds with nonnegative Bakry-Emery Ricci curvature to complete Riemannian manifolds with sectional curvature bounded above. We give a gradient estimate of…

We consider maps between Riemannian manifolds in which the map is a stationary point of the nonlinear Hodge energy. The variational equations of this functional form a quasilinear, nondiagonal, nonuniformly elliptic system which models…

In this note we establish estimates for the harmonic map heat flow from $S^1$ into a closed manifold, and use it to construct sweepouts with the following good property: each curve in the tightened sweepout, whose energy is close to the…

The heat flow for Dirac-harmonic maps on Riemannian spin manifolds is a modification of the classical heat flow for harmonic maps by coupling it to a spinor. It was introduced by Chen, Jost, Sun, and Zhu as a tool to get a general existence…

We calculate heat invariants of arbitrary Riemannian manifolds without boundary. Every heat invariant is expressed in terms of powers of the Laplacian and the distance function. Our approach is based on a multi-dimensional generalization of…

We prove that the Benjamin-Ono equation is globally well-posed in $ H^s(\T) $ for $ s\ge 0 $. Moreover we show that the associated flow-map is Lipschitz on every bounded set of $ {\dot H}^s(\T) $, $s\ge 0$, and even real-analytic in this…

We establish global existence, uniqueness, regularity and long-time asymptotics of strong solutions to the half-harmonic heat flow and dissipative Landau-Lifshitz equation, valid for initial data that is small in the homogeneous Sobolev…

For a sequence of coupled fields $\{(\phi_n,\psi_n)\}$ from a compact Riemann surface $M$ with smooth boundary to a general compact Riemannian manifold with uniformly bounded energy and satisfying the Dirac-harmonic system up to some…

We introduce and study a conformal heat flow of harmonic maps defined by an evolution equation for a pair consisting of a map and a conformal factor of metric on the two-dimensional domain. This flow is designed to postpone finite time…

We study the heat flow from an open, bounded set $D$ in $\R^2$ with a polygonal boundary $\partial D$. The initial condition is the indicator function of $D$. A Dirichlet $0$ boundary condition has been imposed on some but not all of the…

The paper considers the Ricci flow, coupled with the harmonic map flow between two manifolds. We derive estimates for the fundamental solution of the corresponding conjugate heat equation and we prove an analog of Perelman's differential…