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Martin Ulrich Schmidt

Publications and source records attributed to Martin Ulrich Schmidt.

12 recordsLinked to original sources
arXiv2025

Universal Deformations of a Curve and a Differential

We construct universal local deformations (Kuranishi families) for pairs consisting of a compact complex curve and a meromorphic 1-form. Each pair is assumed to be locally planar, a condition which in particular forces the periods of the meromorphic differential to be preserved by local deformations. The hyperelliptic case yields universal local deformations for the spectral data of integrable systems such as simply-periodic solutions of the KdV equation or of the sinh-Gordon equation (cylinders of constant mean curvature). This is the first of two papers in which we shall develop a deformation theory of the spectral curve data of an integrable system.

math.AG↗
arXiv2025

Quaternionic Analysis of Conformal Maps and the Willmore Functional

Quaternionic analysis, which describes conformal maps from Riemann surfaces into $\mathbb{R}^3$ or $\mathbb{R}^4$, is extended to weakly conformal maps. As a consequence we present a new proof that on any compact Riemann surface $X$ the Willmore functional, the integral of the square of the mean curvature, attains a minimum on the space of smooth conformal maps from $X$ to $\mathbb{R}^3$ or $\mathbb{R}^4$. This was first proven by Kuwert and Schätzle under the assumption that the infimum of the Willmore functional is less than $8π$. In this case all conformal maps are unbranched, due to an estimate of Li and Yau. Rivière removed this restriction by allowing as limits conformal maps with ramification points. Our approach admits these weakly conformal maps from the very beginning, by extending the quaternionic function theory as developed by Pedit and Pinkall to square-integrable potentials. In Part~I we carry over the most important properties of holomorphic functions to quaternionic analysis with square-integrable potentials. We develop a differential operator, the Darboux transformation, that generalizes the $\partial$-derivative and with respect to which these holomorphic functions are infinitely many times differentiable. It transforms the Kodaira representation of a weakly conformal map into its Weierstraß representation. Parts~II and~III treat respectively the existence of a minima on the space of weakly conformal maps with ramification points and the regularity of this minima.

math.DG↗
arXiv2025

Blowing up sequences of constant mean curvature tori in $\mathbb{R}^3$ to minimal surfaces

This paper is motivated by the question of whether a sequence of solutions of a given integrable system can be blown up to obtain a solution of a different integrable system in the limit. We study a specific example of this phenomenon. Namely, we describe a blow-up for immersed constant mean curvature (cmc) planes of finite type with unbounded principal curvatures and derive sufficient conditions under which this blow-up converges to a minimal surface immersion. Passing to the respective Gauss-Codazzi equations, we are blowing up a sequence of solutions to the sinh-Gordon integrable system to obtain a solution to Liouville's equation, whose integrable system will turn out to be closely related to the Korteweg-de Vries integrable system. Our most important tool for this investigation is the algebraic-geometric correspondence that was established by Pinkall/Sterling and by Hitchin for cmc planes of finite type, which include all cmc tori.

math.DG↗
arXiv2023

The space of genus two spectral curves of constant mean curvature tori in $\mathbb{R}^3$

We use Whitham deformations to give a complete account of spectral data of real solutions of the sinh--Gordon equation of spectral genus 2. We parameterise the closure of spectral data of constant mean curvature tori in $\mathbb{R}^3$ by an isosceles right triangle and analyse its boundary. We prove that the Wente family, which is described by spectral data with real coefficients, is parameterised by the bisector of the right angle. Our methods combine blowups of Whitham deformations and spectral data in an innovative way that changes the underlying integrable system.

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arXiv2020

Singular curves and Baker-Akhiezer functions

We present the concept of Baker-Akhiezer functions on singular complex curves. For this purpose, we translate the algebraic presentation of such curves in [Se, Chapter~IV] into the analytic setting. Generalised divisors and their interplay with partial desingularisations are the fundament of the construction of Baker-Akhiezer functions.

math.AG↗
arXiv2020

Burchnall-Chaundy Theory

The Burchnall-Chaundy theory concerns the classification of all pairs of commuting ordinary differential operators. We phrase this theory in the language of spectral data for integrable systems. In particular, we define spectral data for rank 1 commutative algebras $A$ of ordinary differential operators. We solve the inverse problem for such data, i.e. we prove that the algebra $A$ is (essentially) uniquely determined by its spectral data. The isomorphy type of $A$ is uniquely determined by the underlying spectral curve.

math.SP↗
arXiv2014

The Prevalence of Tori amongst Constant Mean Curvature Planes in $\mathbb{R}^3$

Constant mean curvature (CMC) tori in Euclidean 3-space are described by an algebraic curve, called the spectral curve, together with a line bundle on this curve and a point on $ S ^ 1 $, called the Sym point. For a given spectral curve the possible choices of line bundle and Sym point are easily described. The space of spectral curves of tori is totally disconnected. Hence to characterise the "moduli space" of CMC tori one should, for each genus $g$, determine the closure $\overline{\mathcal{P}^g}$ of spectral curves of CMC tori within the spectral curves of CMC planes having spectral genus $g$. We identify a real subvariety $\mathcal{R}^g$ and a subset $\mathcal{S}^g\subseteq\mathcal{R}^g $ such that $\mathcal{R}^g_{\text{max}}\subseteq\overline{\mathcal{P}^g}\subseteq\mathcal{S}^g$, where $\mathcal{R}^g_{\text{max}}$ denotes the points of $\mathcal{R}^g$ having maximal dimension. The lowest spectral genus for which tori exist is $g=2$ and in this case $\mathcal{R}^2=\mathcal{R}^2_{\text{max}}=\overline{\mathcal{P}^2}=\mathcal{S}^2$. For $g>2 $, we conjecture that $\mathcal{R}^g\supsetneq\mathcal{R}^g_{\text{max}}=\mathcal{S}^g$. We give a number of alternative characterisations of $\mathcal{R}^g_{\text{max}}$ and in particular introduce a new integer invariant of a CMC plane of finite type, called its winding number.

math.DG↗
arXiv2013

Flows of constant mean curvature tori in the 3-sphere: The equivariant case

We present a deformation for constant mean curvature tori in the 3-sphere. We show that the moduli space of equivariant constant mean curvature tori in the 3-sphere is connected, and we classify the minimal, the embedded, and the Alexandrov embedded tori therein. We conclude with an instability result.

math.DG↗
arXiv2012

The Closure of Spectral Data for Constant Mean Curvature Tori in $ S ^ 3 $

The spectral curve correspondence for finite-type solutions of the sinh-Gordon equation describes how they arise from and give rise to hyperelliptic curves with a real structure. Constant mean curvature (CMC) 2-tori in $ S ^ 3 $ result when these spectral curves satisfy periodicity conditions. We prove that the spectral curves of CMC tori are dense in the space of smooth spectral curves of finite-type solutions of the sinh-Gordon equation. One consequence of this is the existence of countably many real $ n $-dimensional families of CMC tori in $ S ^ 3 $ for each positive integer $ n $.

math.DG↗
arXiv2004

Existence of minimizing Willmore surfaces of prescribed conformal class

We consider the class of all conformal mappings from a compact Riemann surface into the threedimensional or fourdimensional Euclidean space. A sequence in this class with bounded Willmore functional is shown to have a sequence of conformal transformations of the target space, such that a subsequence of the transformed sequence converges. This implies that there exists a smooth conformal mapping, which minimizes the Willmore functional in this class. For this purpose we extend the quaternionic function theory of Pedit and Pinkall to square integrable Hopf fields. In particular, we proof the Pluecker formula for such Hopf fields.

math.DG↗
arXiv2002

A proof of the Willmore conjecture

A proof of the Willmore conjecture is presented. With the help of the global Weierstrass representation the variational problem of the Willmore functional is transformed into a constrained variational problem on the moduli space of all spectral curves corresponding to periodic solutions of the Davey-Stewartson equation. The subsets of this moduli space, which correspond to bounded first integrals, are shown to be compact. With respect to another topology the moduli space is shown to be a Banach manifold. The subset of all periodic solutions of the Davey-Stewartson equation, which correspond to immersion of tori into the three-dimensional Euclidean space, are characterized by a singularity condition on the corresponding spectral curves. This yields a proof of the existence of minimizers for all conformal classes and the determination of the absolute minimum, which is realized by the Clifford torus.

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