Regularity of Minimal Surfaces

Abstract

With contributions by (2010): 49Q05, 53A05, 53A07, 53B20, 35J20, 35J47, 35J50, 35J75, 49Q20, 30C20 c Springer-Verlag Berlin Heidelberg 1992, 2010 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: VTEX, VilniusPrinted on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com) PrefaceThis book is the second volume of a treatise on minimal surfaces consisting of altogether three volumes which can be read and studied independently of each other. The central theme is boundary value problems for minimal surfaces such as Plateau's problem. The present treatise forms a greatly extended version of the monograph Minimal Surfaces I, II by U. Dierkes, S. Hildebrandt, A. Küster, and O. Wohlrab, published in 1992, which is often cited in the literature as [DHKW]. New coauthors are Friedrich Sauvigny for the first volume and Anthony J. Tromba for the second and third volume.The four main topics of this second volume are free boundary value problems, regularity of minimal surfaces and their geometric properties, and finally a new method is introduced to show that minimizers of area are immersed. Since minimal surfaces in R 3 are understood as harmonic, conformally parametrized mappings X : Ω → R3 of an open domain Ω in R 2 , they are real analytic in Ω, and so the problem of smoothness for X is the question how smooth X is at the boundary ∂Ω if X is subject to certain boundary conditions. However, even if X is "analytically regular", it might not be "geometrically regular" since it could have branch points. We investigate how X behaves in the neighbourhood of branch points, and secondly whether such points actually exist. In addition we describe geometric properties of minimal surfaces in R 3 or, more generally, of H-surfaces in an n-dimensional Riemannian manifold. This book can be read independently from the preceding volume of this treatise although we use some terminology and results from the previous material.We thank E. Kuwert, F. Müller, D. Schwab, H. von der Mosel, D. Wienholtz, and S. Winklmann for pointing out errors and misprints in [DHKW] which are corrected here. Particularly we are indebted to Frank Müller for some penetrating contributions to Chapter 3, and to Albrecht Küster who...

A General Existence Theorem for Embedded Minimal Surfaces with Free Boundary

“…Boundary regularity results were obtained in a variety of settings. Interested readers are encouraged to consult the recent treatise [8] [9] on boundary value problems of minimal surfaces.…”

Mixed Boundary Value Problems for Minimal Graphs on a Riemann Surface and Winding Staircases

“…Then we immerse this configuration < Γ, Z > into a soap liquid, and a soap pellicle is installed which minimizes the area and is bounded by the thin wire Γ. Moreover, this minimal surface possesses a trace on the cylinder In particular for the well-established theory of minimal graphs with mixed boundary conditions of Dirichlet and Neumann, we refer our readers to the book [1] by U. Dierkes, S. Hildebrandt, and F. Sauvigny as well as to the following books by U. Dierkes, S. Hildebrandt, and A. Tromba [2] and [3] on the theory of minimal surfaces, which appeared within the Springer-Grundlehren in September 2010. In the geometric situation of our winding staircase, a minimal graph over a Riemannian surface R appears.…”

On the nonexistence of boundary branch points for minimal surfaces spanning smooth contours II

“…In [2], Dierkes, Hildebrandt and Tromba introduced the method of taking higher-order derivatives of Dirichlet's energy in the direction of forced Jacobi fields, discovered by both Böhme and Tromba, to treat the question of interior branch points. The calculation of higher-order derivatives was achieved by applying the theory of residues.…”

“…The cases 1 + m = 2(n + 1) and 1 + m = 3(n + 1) are called exceptional. The notion of exceptional is the infinitesimal notion of analytically false [2].…”

“…The cases 1 + m = 2(n + 1) and 1 + m = 3(n + 1) are called exceptional. The notion of exceptional is the infinitesimal notion of analytically false [2].If a minimal surfaceX has a boundary branch point of order n and index m satisfying 2m − 2 < 3n, then Wienholtz [18] showed thatX cannot be a minimizer in any C r topology. Wienholtz's condition implies that the branch point is nonexceptional.…”