Related papers: Asymptotics of the Teichm\"uller harmonic map flow

The Teichm\"uller harmonic map flow is a gradient flow for the harmonic map energy of maps from a closed surface to a general closed Riemannian target manifold of any dimension, where both the map and the domain metric are allowed to…

The Teichm\"uller harmonic map flow deforms both a map from an oriented closed surface $M$ into an arbitrary closed Riemannian manifold, and a constant curvature metric on $M$, so as to reduce the energy of the map as quickly as possible…

The harmonic map energy of a map from a closed, constant-curvature surface to a closed target manifold can be seen as a functional on the space of maps and domain metrics. We consider the gradient flow for this energy. In the absence of…

In this paper we study the Teichm\"uller harmonic map flow as introduced by Rupflin and Topping [15]. It evolves pairs of maps and metrics $(u,g)$ into branched minimal immersions, or equivalently into weakly conformal harmonic maps, where…

We analyse finite-time singularities of the Teichm\"uller harmonic map flow -- a natural gradient flow of the harmonic map energy -- and find a canonical way of flowing beyond them in order to construct global solutions in full generality.…

We define a geometric flow that is designed to change surfaces of cylindrical type spanning two disjoint boundary curves into solutions of the Douglas-Plateau problem of finding minimal surfaces with given boundary curves. We prove that…

We consider the 1-harmonic flow of maps from a bounded domain into a submanifold of a Euclidean space, i.e. the gradient flow of the total variation functional restricted to maps taking values in the manifold. We restrict ourselves to…

We introduce a flow of maps from a compact surface of arbitrary genus to an arbitrary Riemannian manifold which has elements in common with both the harmonic map flow and the mean curvature flow, but is more effective at finding minimal…

We consider the question of whether solutions of variants of Teichm\"uller harmonic map flow from surfaces $M$ to general targets can degenerate in finite time. For the original flow from closed surfaces of genus at least $2$, as well as…

Wolf gave a homeomorphism from the Teichm\"uller space to the space of quadratic differentials on a closed Riemann surface by using harmonic maps. Moreover, using harmonic maps rays, he gave a compactification of the Teichm\"uller space and…

Using a flow first introduced by J.P. Anderson, we obtain some existence theorems for harmonic maps from a noncompact complete Riemannian manifold into a complete Riemannian manifold. In particular, we prove as a corollary a recent result…

We use meromorphic quadratic differentials with higher order poles to parametrize the Teichm\"uller space of crowned hyperbolic surfaces. Such a surface is obtained on uniformizing a compact Riemann surface with marked points on its…

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 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…

The purpose of this note is to prove the existence of global weak solutions to the flow associated to integro-differential harmonic maps into spheres and Riemannian homogeneous manifolds.

We analyse the asymptotic behaviour of solutions of the Teichm\"uller harmonic map flow from cylinders, and more generally of `almost minimal cylinders', in situations where the maps satisfy a Plateau-boundary condition for which the…

In this paper we give a geometric interpretation of the renormalization algorithm and of the continued fraction map that we introduced in arxiv:0905.0871 to give a characterization of symbolic sequences for linear flows in the regular…

Finding surface mappings with least distortion arises from many applications in various fields. Extremal Teichm\"uller maps are surface mappings with least conformality distortion. The existence and uniqueness of the extremal…

We introduce a flow that is designed to flow maps $u:\Sigma\to \mathbb{R}^n$ which map the boundary of a general domain surface $\Sigma$ into a given (not necessarily connected) submanifold $N\hookrightarrow \mathbb{R}^n$ towards a free…

We consider harmonic maps on simply connected Riemann surfaces into the group $\mathrm{U}(n)$ of unitary matrices of order $n$. It is known that a harmonic map with an associated algebraic extended solution can be deformed into a new…