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Melanie Rupflin

Publications and source records attributed to Melanie Rupflin.

At least 19 recordsLinked to original sources
arXiv2026

Flowing to free boundary minimal surfaces

We introduce a flow that is designed to flow maps $u:Σ\to \mathbb{R}^n$ which map the boundary of a general domain surface $Σ$ into a given (not necessarily connected) submanifold $N\hookrightarrow \mathbb{R}^n$ towards a free boundary (branched) minimal immersion supported by $N$. In the case when $Σ$ is the unit disc $D$, this task can be achieved by means of the Plateau-flow introduced in the work [15] of the second author. When $Σ\neq D$, however, also the conformal type of the domain metric plays a role and it no longer suffices to deform the trace of the given map into a half-harmonic map as in [15]. In order to overcome this issue, here we combine ideas of the Plateau-flow from [15] with ideas of the Teichmüller harmonic flow from [12], in order to flow both an initial map $u_0$ with trace $u_0\colon\partial Σ\to N$ and an initial domain metric $g_0$ in a way that produces, as time tends to infinity, a half-harmonic map from $\partial Σ$ into $N$ whose harmonic extension is conformal and hence is a (branched) minimal immersion.

math.AP↗
arXiv2026

A sharp quantitative stability result near infinitely concentrated minimisers

We consider the question of quantitative stability of minimisers for a well-known variational problem for which the infimum of the energy is not achieved in the classical sense, namely for the Dirichlet energy of degree $1$ maps from closed surfaces $(Σ,g_Σ)$ of positive genus into the unit sphere $S^2\subset \mathbb{R}^3$. For this variational problem it is natural to view configurations which consist of a constant map from the given domain and an infinitely concentrated rotation as generalised minimisers and to hence ask whether the distance of almost minimisers $v:Σ\to S^2$ to this set of infinitely concentrated minimisers can be controlled in terms of the energy defect $δ_v=E(v)-\inf E=E(v)-4π$. In this paper we develop a new dynamic approach that allows us to change the topology of the domain in a well controlled manner and to deform almost minimising maps from surfaces of general genus into harmonic maps from the sphere in a way that yields sharp quantitative estimates on all key features that characterise the distance to the set of infinitely concentrated minimisers, i.e. the scale of concentration, the $H^1$-distance to the nearest bubble on the concentration region and the $H^1$-distance to the nearest constant away from the concentration point.

math.AP↗
arXiv2025

Quantitative estimates for the relative isoperimetric problem and its gradient flow outside convex bodies in the plane

We prove three related quantitative results for the relative isoperimetric problem outside a convex body $Ω$ in the plane: (1) Łojasiewicz estimates and quantitative rigidity for critical points, (2) rates of convergence for the gradient flow, and (3) quantitative stability for minimizers. These results come with explicit constants and optimal exponents/rates, and hold whenever a simple two-dimensional auxiliary variational problem for circular arcs outside of $Ω$ is nondegenerate. The proofs are inter-related, and in particular, for the first time in the context of isoperimetric problems, a flow approach is used to prove quantitative stability for minimizers.

math.AP↗
arXiv2025

Lojasiewicz inequalities for almost harmonic maps near simple bubble trees

We prove Lojasiewicz inequalities for the harmonic map energy for maps from surfaces of positive genus into general analytic target manifolds which are close to simple bubble trees and as a consequence obtain new results on the convergence of harmonic map flow and on the energy spectrum of harmonic maps with small energy. Our results and techniques are not restricted to particular targets or to integrable settings and we are able to lift general Lojasiewicz-Simon inequalities valid near harmonic maps $\hat ω:S^2\to N$ to the singular setting whenever the bubble $\hat ω$ is attached at a point which is not a branch point.

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arXiv2023

Sharp quantitative rigidity results for maps from $S^2$ to $S^2$ of general degree

As the energy of any map $v$ from $S^2$ to $S^2$ is at least $4π\vert deg(v)\vert$ with equality if and only if $v$ is a rational map one might ask whether maps with small energy defect $δ_v=E(v)-4π\vert deg(v)\vert$ are necessarily close to a rational map. While such a rigidity statement turns out to be false for maps of general degree, we will prove that any map $v$ with small energy defect is essentially given by a collection of rational maps that describe the behaviour of $v$ at very different scales and that the corresponding distance is controlled by a quantitative rigidity estimate of the form $dist^2\leq C δ_v(1+\vert\logδ_v\vert)$ which is indeed sharp.

math.AP↗
arXiv2023

Łojasiewicz inequalities near simple bubble trees

In this paper we prove a gap phenomenon for critical points of the $H$-functional on closed non-spherical surfaces when $H$ is constant, and in this setting furthermore prove that sequences of almost critical points satisfy Łojasiewicz inequalities as they approach the first non-trivial bubble tree. To prove these results we derive sufficient conditions for Łojasiewicz inequalities to hold near a finite-dimensional submanifold of almost-critical points for suitable functionals on a Hilbert space.

math.AP↗
arXiv2023

Low energy levels of harmonic maps into analytic manifolds

We consider the energy spectrum $Ξ_E(N)$ of harmonic maps from the sphere into a closed Riemannian manifold $N$. While a well known conjecture asserts that $Ξ_E(N)$ is discrete whenever $N$ is analytic, for most analytic targets it is only known that any potential accumulation point of the energy spectrum must be given by the sum of the energies of at least two harmonic spheres. The lowest energy level that could hence potentially be an accumulation point of $Ξ_E$ is thus $2 E_{min}$. In the present paper we exclude this possibility for generic 3 manifolds and prove additional results that establish obstructions to the gluing of harmonic spheres and Lojasiewicz-estimates for almost harmonic maps.

math.AP↗
arXiv2019

Uniqueness and nonuniqueness of limits of Teichmueller harmonic map flow

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 singularities, previous theory established that the flow converges to a branched minimal immersion, but only at a sequence of times converging to infinity, and only after pulling back by a sequence of diffeomorphisms. In this paper we investigate whether it is necessary to pull back by these diffeomorphisms, and whether the convergence is uniform as time tends to infinity.

math.DG↗
arXiv2019

Sharp eigenvalue estimates on degenerating surfaces

We consider the first non-zero eigenvalue $λ_1$ of the Laplacian on hyperbolic surfaces for which one disconnecting collar degenerates and prove that $8π\nabla\log(λ_1)$ essentially agrees with the dual of the differential of the degenerating Fenchel-Nielsen length coordinate. As a consequence, we can improve previous results of Schoen, Wolpert, Yau and Burger to obtain estimates with optimal error rates and obtain new information on the leading order terms of the polyhomogeneous expansion of $λ_1$ of Albin, Rochon and Sher.

math.DG↗
arXiv2018

Holomorphic quadratic differentials dual to Fenchel-Nielsen coordinates

We discuss bases of the space of holomorphic quadratic differentials that are dual to the differentials of Fenchel-Nielsen coordinates and hence appear naturally when considering functions on the set of hyperbolic metrics which are invariant under pull-back by diffeomorphisms, such as eigenvalues of the Laplacian. The precise estimates derived in the current paper form the basis for the proof of the sharp eigenvalue estimates on degenerating surfaces obtained in arXiv:1701.08491.

math.DG↗
arXiv2018

Finite-time degeneration for variants of Teichmüller harmonic map flow

We consider the question of whether solutions of variants of Teichmüller 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 the flow from cylinders, we prove that such a finite-time degeneration must occur in situations where the image of thin collars is `stretching out' at a rate of at least $\text{inj}(M,g)^{-(\frac14+δ)}$, and we construct targets in which the flow from cylinders must indeed degenerate in finite time. For the rescaled Teichmüller harmonic map flow, the condition that the image stretches out is not only sufficient but also necessary and we prove the following sharp result: Solutions of the rescaled flow cannot degenerate in finite time if the image stretches out at a rate of no more than $\lvert\log(\text{inj}(M,g))\rvert^{\frac12}$, but must degenerate in finite time if it stretches out at a rate of at least $\lvert\log(\text{inj}(M,g))\rvert^{\frac12+δ}$ for some $δ>0$.

math.DG↗
arXiv2018

Hyperbolic metrics on surfaces with boundary

We discuss an alternative approach to the uniformisation problem on surfaces with boundary by representing conformal structures on surfaces $M$ of general type by hyperbolic metrics with boundary curves of constant positive geodesic curvature. In contrast to existing approaches to this problem, the boundary curves of our surfaces $(M,g)$ cannot collapse as the conformal structure degenerates which is important in applications in which $(M,g)$ serves as domain of a PDE with boundary conditions.

math.DG↗
arXiv2018

Horizontal curves of hyperbolic metrics

We analyse the fine convergence properties of one parameter families of hyperbolic metrics, on a fixed underlying surface, that move always in a horizontal direction, i.e. orthogonal to the action of diffeomorphisms.

math.DG↗
arXiv2017

Global weak solutions of the Teichmüller harmonic map flow into general targets

We analyse finite-time singularities of the Teichmüller 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. Moreover, we prove a no-loss-of-topology result at finite time, which completes the proof that this flow decomposes an arbitrary map into a collection of branched minimal immersions connected by curves.

math.DG↗
arXiv2017

Analysis of boundary bubbles for almost minimal cylinders

We analyse the asymptotic behaviour of solutions of the Teichmüller 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 three-point condition degenerates. We prove that such a degenerating boundary condition forces the domain to stretch out as a boundary bubble forms. Our main result then establishes that for prescribed boundary curves that satisfy Douglas' separation condition, these boundary bubbles will not only be harmonic but will themselves be branched minimal immersions. Together with earlier work, this in particular completes the proof that the Teichmüller harmonic map flow changes every initial surface in $\mathbb{R}^n$ spanning such boundary curves into a solution of the corresponding Douglas-Plateau problem.

math.AP↗
arXiv2016

Smooth long-time existence of Harmonic Ricci Flow on surfaces

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.

math.DG↗
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↗