The question of estimating the number of minimal surfaces that bound a prescribed contour has been open since Douglas's solution of the Plateau problem in 1931. In this book, the authors formulate and prove an index theorem for minimal surfaces of higher topological type spanning one boundary contour. The Index Theorem for Minimal Surfaces of Higher Genus describes, in terms of Fredholm Index, a rough measure on the set of curves bounding minimal surfaces of prescribed branching type and genus.

Readership

Mathematicians working in global analysis and/or minimal surface theory.

Table of Contents

• Introduction
• The differential geometric approach to Teichmüller theory
• Minimal surfaces of higher genus as critical points of Dirichlet's functional
• Review of some basic results in Riemann surface theory
• Vector bundles over Teichmüller space
• Minimal surfaces of higher genus as the zeros of a vector field and the conformality operators
• The corank of the partial conformality operators
• The corank of the complete conformality operators
• Manifolds of harmonic surfaces of prescribed branching type
• The index theorem
• Appendix I. A supplement to the boundary regularity theorems for minimal surfaces
• Appendix II. Maximal ideals in Sobolev algebras of holomorphic functions
• References