Nankai Tracts in Mathematics
Minimal Submanifolds and Related Topics
2nd Edition
极小子流形和相关主题
by Yuanlong Xin (Fudan University, China)
 
380pp                                 
978-981-3236-05-9                   US$128          £113
 
Release Date (Asia): Aug 2018
Release Date (Rest of the World): Oct 2018
 
http://www.worldscientific.com/worldscibooks/10.1142/10880
 

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.
 
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

 
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 Submanifold;
 
Readership: Researchers and graduate students in differential geometry.