Minimal Submanifolds and Related Topics

Yuanlong Xin

• Publisher: WSPC
• Year: 2018
• ISBN:
  9789813236073
• Edition: 2
• Topic: Matematica
• Subtopic: Geometria differenziale

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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:

• Introduction
• Bernstein's Theorem and Its Generalizations
• Weistrass Type Representations
• Plateau's Problem and Douglas–Rado Solution
• Intrinsic Rigidity Theorems
• Stable Minimal Hypersurfaces
• Minimal Submanifolds of Higher Codimension
• Bernstein Type Theorems for Higher Codimension
• Entire Space-Like Submanifolds

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--> 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 Submanifold0 Key Features:

• Contains new developments on the classical subjects
• Features geometric analysis method on the subject
• It 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 theory
• Includes a comprehensive list of references

Chapter 1

Introduction

This chapter is an introductory material to the theory of minimal submanifolds. We begin with the notion of the second fundamental form, from which the minimal submanifolds can be neatly defined. Then they are characterized by a variational property.

In this chapter we also give important properties for minimal submanifolds in Euclidean space and specify the equation for minimal graphs of codimension one. This is a famous equation in mathematics. Some basic properties for minimal submanifolds in the sphere are described which enable us to give examples of minimal submanifolds in the sphere.

In order to prove Bernstein type theorems for minimal submanifolds in Euclidean space we study the Gauss maps whose images lie in Grassmannian manifolds. It is natural to study the geometry of the Grassmannian manifolds. It is interesting in its own right, but is not available in the literature. We write it down in detail in the last section of the chapter. In fact, the Grassmannian manifolds are minimal submanifolds in the unit sphere in a suitable sense.

1.1The Second Fundamental Form

In the classical surface theory in

³ there are first and second fundamental forms. We know that a plane and a circular cylinder in

³ are locally isometric. But their shapes are different, since they have different second fundamental forms. The invariants determined only by the first fundamental form are intrinsic; others are extrinsic invariants which are dependent not only on the first fundamental form, but also on t...