Minimal Submanifolds and Related Topics

In the theory of minimal submanifolds, Bernstein's problem and Plateau's problem are central topics. This important book presents the Douglas–Rado solution to Plateau's problem, but the main emphasis is on Bernstein's problem and its new developments in various directions: the value distribution of the Gauss image of a minimal surface in Euclidean 3-space, Simons' work for minimal graphic hypersurfaces, and the author's own contributions to Bernstein type theorems for higher codimension. The author also introduces some related topics, such as submanifolds with parallel mean curvature, Weierstrass type representation for surfaces of mean curvature 1 in hyperbolic 3-space, and special Lagrangian submanifolds.

This new edition contains the author's recent work on the Lawson–Osserman's problem for higher codimension, and on Chern's problem for minimal hypersurfaces in the sphere. Both Chern's problem and Lawson–Osserman's problem are important problems in minimal surface theory which are still unsolved. In addition, some new techniques were developed to address those problems in detail, which are of interest in the field of geometric analysis.
Sample Chapter(s)
Introduction

Contents: IntroductionBernstein's Theorem and Its GeneralizationsWeistrass Type RepresentationsPlateau's Problem and Douglas–Rado SolutionIntrinsic Rigidity TheoremsStable Minimal HypersurfacesMinimal Submanifolds of Higher CodimensionBernstein Type Theorems for Higher CodimensionEntire Space-Like Submanifolds
Readership: Researchers and graduate students in differential geometry.
Minimal Submanifold;Bernstein Type Theorem;Plateau's Problem;Harmonic Gauss Map;Curvature Estimate;Grassman Manifold;Weierstrass Representation;Stable Minimal Hypersurface;Special Lagrangian Submanifold;Parallel Mean Curvature Submanifold;Space-Like Submanifold0Key Features:Contains new developments on the classical subjectsFeatures geometric analysis method on the subjectIt is a half textbook (Chapters 1 to 4) and a half monograph (Chapters 5 to 9)Features a unique treatment, among other books, on minimal surface theoryIncludes a comprehensive list of references