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Melanie Rupflin

Publications and source records attributed to Melanie Rupflin.

27 records · Page 2Linked to original sources
arXiv2016

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 0 case, our flow is just the harmonic map flow, and it tries to find branched minimal 2-spheres as in Sacks-Uhlenbeck and Struwe etc. In the genus 1 case, we show that our flow is exactly equivalent to that considered by Ding-Li-Lui. In general, we recover the result of Schoen-Yau and Sacks-Uhlenbeck that an incompressible map from a surface can be adjusted to a branched minimal immersion with the same action on $π_1$, and this minimal immersion will be homotopic to the original map in the case that $π_2=0$.

math.DG↗
arXiv2016

Teichmüller harmonic map flow into nonpositively curved targets

The Teichmüller 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 [16]. The flow then tries to converge to a branched minimal immersion when it can [16,18]. The only thing that can stop the flow is a finite-time degeneration of the metric on $M$ where one or more collars are pinched. In this paper we show that finite-time degeneration cannot happen in the case that the target has nonpositive sectional curvature, and indeed more generally in the case that the target supports no bubbles. In particular, when combined with [16,18,9], this shows that the flow will decompose an arbitrary such map into a collection of branched minimal immersions.

math.DG↗
arXiv2015

Refined asymptotics of the Teichmüller harmonic map flow into general targets

The Teichmüller 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 evolve. Given a weak solution of the flow that exists for all time $t\geq 0$, we find a sequence of times $t_i\to\infty$ at which the flow at different scales converges to a collection of branched minimal immersions with no loss of energy. We do this by developing a compactness theory, establishing no loss of energy, for sequences of almost-minimal maps. Moreover, we construct an example of a smooth flow for which the image of the limit branched minimal immersions is disconnected. In general, we show that the necks connecting the images of the branched minimal immersions become arbitrarily thin as $i\to\infty$.

math.DG↗
arXiv2015

Teichmüller harmonic map flow from cylinders

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 also in this new setting and for arbitrary initial data, solutions of the Teichmüller harmonic map flow exist for all times. Furthermore, for solutions for which a three-point-condition does not degenerate as $t\to\infty$, we show convergence along a sequence $t_i\to\infty$ to a critical point of the area given either by a minimal cylinder or by two minimal discs spanning the given boundary curves.

math.AP↗
arXiv2014

Weak solutions to mean curvature flow respecting obstacles I: the graphical case

We consider the problem of evolving hypersurfaces by mean curvature flow in the presence of obstacles, that is domains which the flow is not allowed to enter. In this paper, we treat the case of complete graphs and explain how the approach of M. Saez and the second author yields a global weak solution to the original problem for general initial data and onesided obstacles.

math.DG↗
arXiv2012

Asymptotics of the Teichmüller harmonic map flow

The Teichmüller harmonic map flow, introduced in [9], evolves both a map from a closed Riemann surface to an arbitrary compact Riemannian manifold, and a constant curvature metric on the domain, in order to reduce its harmonic map energy as quickly as possible. In this paper, we develop the geometric analysis of holomorphic quadratic differentials in order to explain what happens in the case that the domain metric of the flow degenerates at infinite time. We obtain a branched minimal immersion from the degenerate domain.

math.DG↗
arXiv2012

Flowing maps to minimal surfaces: Existence and uniqueness of solutions

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 existence theory as well as the issue of uniqueness of solutions. We establish that a (weak) solution exists for as long as the metrics remain in a bounded region of moduli space, i.e. as long as the flow does not collapse a closed geodesic in the domain manifold to a point. Furthermore, we prove that this solution is unique in the class of all weak solutions with non-increasing energy. This work complements the paper [11] of Topping and the author where the flow was introduced and its asymptotic convergence to branched minimal immersions is discussed.

math.DG↗
arXiv2010

Selfsimilar expanders of the harmonic map flow

We study the existence, uniqueness, and stability of self-similar expanders of the harmonic map heat flow in equivariant settings. We show that there exist selfsimilar solutions to any admissiable initial data and that their uniqueness and stability properties are essentially determined by the energy-minimising properties of the so called equator maps.

math.AP↗