Related papers: Short-time existence of the $\alpha$-Dirac-harmoni…

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…

For a homotopy class $[u]$ of maps between a closed Riemannian manifold $M$ and a general manifold $N$, we want to find a Dirac-harmonic map with the map component in the given homotopy class. Most known results require the index to be…

$\alpha$-Dirac-harmonic maps are variations of Dirac-harmonic maps, analogous to $\alpha$-harmonic maps that were introduced by Sacks-Uhlenbeck to attack the existence problem for harmonic maps from surfaces. For $\alpha >1$, the latter are…

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…

We study the existence of harmonic maps and Dirac-harmonic maps from degenerating surfaces to non-positive curved manifold via the scheme of Sacks and Uhlenbeck. By choosing a suitable sequence of $\alpha$-(Dirac-)harmonic maps from a…

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 study a new set of coupled field equations motivated by the non-linear supersymmetric sigma model of quantum field theory. These equations couple a map into a Riemannian manifold controlled by a harmonic map like action with a spinor…

The conformal heat flow of harmonic maps is a system of evolution equations combined with harmonic map flow with metric evolution in conformal direction. It is known that global weak solution of the flow exists and smooth except at mostly…

In this article, we study the the harmonic map heat flow from a manifold with conic singularities to a closed manifold. In particular, we have proved the short time existence and uniqueness of solutions as well as the existence of global…

Using mixed analytical and numerical methods we investigate the development of singularities in the heat flow for corotational harmonic maps from the $d$-dimensional sphere to itself for $3\leq d\leq 6$. By gluing together shrinking and…

We study the heat flow of p-harmonic maps between complete Riemannian manifolds. We prove the global existence of the flow when the initial datum has values in a generalised regular ball. In particular, if the target manifold has…

In this paper, we consider the heat flow for p-pseudoharmonic maps from a closed Sasakian manifold M into a compact Riemannian manifold N. We prove global existence and asymptotic convergence of the solution for the p-pseudoharmonic map…

As a commutative version of the supersymmetric nonlinear sigma model, Dirac-harmonic maps from Riemann surfaces were introduced fifteen years ago. They are critical points of an unbounded conformally invariant functional involving two…

In this paper, we discuss the heat flow of a pseudo-harmonic map from a closed pseudo-Hermitian manifold to a Riemannian manifold with non-positive sectional curvature, and prove the existence of the pseudo-harmonic map which is a…

Let $B_1$ be the unit open disk in $\Real^2$ and $M$ be a closed Riemannian manifold. In this note, we first prove the uniqueness for weak solutions of the harmonic map heat flow in $H^1([0,T]\times B_1,M)$ whose energy is non-increasing in…

Motivated by emerging applications from imaging processing, the heat flow of a generalized $p$-harmonic map into spheres is studied for the whole spectrum, $1\leq p<\infty$, in a unified framework. The existence of global weak solutions is…

We study the new geometric flow that was introduced in [11] that evolves a pair of map and (domain) metric in such a way that it changes appropriate initial data into branched minimal immersions. In the present paper we focus on the…

In this paper, we consider the pseudoharmonic heat flow with small initial horizontal energy and give the existence of pseudoharmonic maps from closed pseudo-Hermitian manifolds to closed Riemannian manifolds.

Let $(M,g)$ be a four dimensional compact Riemannian manifold with boundary and $(N,h)$ be a compact Riemannian manifold without boundary. We show the existence of a unique, global weak solution of the heat flow of extrinsic biharmonic maps…

In this paper, we study an $\alpha$-flow for the Sack-Uhlenbeck functional on Riemannian surfaces and prove that the limiting map by the $\alpha$-flows is a weak solution to the harmonic map flow. By an application of the $\alpha$-flow, we…