Melanie Rupflin

Flowing maps to minimal surfaces

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 surfaces. In the genus

Horizontal curves of hyperbolic metrics

0

Smooth long‐time existence of Harmonic Ricci Flow on surfaces

case, our flow is just the harmonic map flow, and it tries to find branched minimal 2-spheres as in Sacks-Uhlenbeck (1981) and Struwe (1985), etc. In the genus 1 case, we show that our flow is exactly equivalent to that considered by Ding-Li-Liu (2006). In general, we recover the result of Schoen-Yau (1979) and Sacks-Uhlenbeck (1982) that an incompressible map from a surface can be adjusted to a branched minimal immersion with the same action on

Supercritical elliptic equations

\pi_1

Teichmüller harmonic map flow from cylinders

, and this minimal immersion will be homotopic to the original map in the case that

Refined asymptotics of the Teichmüller harmonic map flow into general targets

\pi_2=0

Analysis of boundary bubbles for almost minimal cylinders

.

Addendum to: Supercritical Elliptic Equations

We analyse the fine convergence properties of one parameter families of hyperbolic metrics that move always in a horizontal direction, i.e. orthogonal to the action of diffeomorphisms. Such families arise naturally in the study of general curves of metrics on surfaces, and in one of the gradients flows for the harmonic map energy.

An improved uniqueness result for the harmonic map flow in two dimensions

We prove that at a finite singular time for the Harmonic Ricci Flow on a surface of positive genus both the energy density of the map component and the curvature of the domain manifold have to blow up simultaneously. As an immediate consequence, we obtain smooth long-time existence for the Harmonic Ricci Flow with large coupling constant.

Flowing maps to minimal surfaces: Existence and uniqueness of solutions

Abstract For an elliptic model equation with supercritical power non-linearity we give a complete description of radial solutions and discuss self-similar blow-up solutions.