The description for this book, Seminar On Minimal Submanifolds. (AM-103), Volume 103, will be forthcoming.

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.

This proceedings on pure and applied differential geometry, discusses several subjects in submanifold theory, such as the Willmore problem, minimal surfaces, submanifolds of finite type, affine differential geometry, indefinite Riemannian geometry, and applications of differential geometry in human and artificial vision.

This book contains the papers presented at the ICM2002 Satellite Conference on Nonlinear Evolution Equations and Dynamical Systems. About 50 mathematicians and scientists attended the meeting ? including E Witten (IAS), C Nappi (Princeton), K Khanin (Cambridge), D Phong (Columbia), d'Hoker (UCLA) and Peng Chiakuei (CAS). The book covers several fields, such as nonlinear evolution equations and integrable systems, infinite-dimensional algebra, conformal field theory and geometry.The proceedings have been selected for coverage in: ? Index to Scientific & Technical Proceedings (ISTP CDROM version / ISI Proceedings)

This volume of selected academic papers demonstrates the significance of the contribution to mathematics made by Manfredo P. do Carmo. Twice a Guggenheim Fellow and the winner of many prestigious national and international awards, the professor at the institute of Pure and Applied Mathematics in Rio de Janeiro is well known as the author of influential textbooks such as Differential Geometry of Curves and Surfaces. The area of differential geometry is the main focus of this selection, though it also contains do Carmo's own commentaries on his life as a scientist as well as assessment of the impact of his researches and a complete list of his publications. Aspects covered in the featured papers include relations between curvature and topology, convexity and rigidity, minimal surfaces, and conformal immersions, among others. Offering more than just a retrospective focus, the volume deals with subjects of current interest to researchers, including a paper co-authored with Frank Warner on the convexity of hypersurfaces in space forms. It also presents the basic stability results for minimal surfaces in the Euclidean space obtained by the author and his collaborators. Edited by do Carmo's first student, now a celebrated academic in her own right, this collection pays tribute to one of the most distinguished mathematicians.

Geometric Measure Theory: A Beginner's Guide, Fifth Edition provides the framework readers need to understand the structure of a crystal, a soap bubble cluster, or a universe. The book is essential to any student who wants to learn geometric measure theory, and will appeal to researchers and mathematicians working in the field. Brevity, clarity, and scope make this classic book an excellent introduction to more complex ideas from geometric measure theory and the calculus of variations for beginning graduate students and researchers. Morgan emphasizes geometry over proofs and technicalities, providing a fast and efficient insight into many aspects of the subject, with new coverage to this edition including topical coverage of the Log Convex Density Conjecture, a major new theorem at the center of an area of mathematics that has exploded since its appearance in Perelman's proof of the Poincaré conjecture, and new topical coverage of manifolds taking into account all recent research advances in theory and applications. - Focuses on core geometry rather than proofs, paving the way to fast and efficient insight into an extremely complex topic in geometric structures - Enables further study of more advanced topics and texts - Demonstrates in the simplest possible way how to relate concepts of geometric analysis by way of algebraic or topological techniques - Contains full topical coverage of The Log-Convex Density Conjecture - Comprehensively updated throughout

The third of three volumes on partial differential equations, this is devoted to nonlinear PDE. It treats a number of equations of classical continuum mechanics, including relativistic versions, as well as various equations arising in differential geometry, such as in the study of minimal surfaces, isometric imbedding, conformal deformation, harmonic maps, and prescribed Gauss curvature. In addition, some nonlinear diffusion problems are studied. It also introduces such analytical tools as the theory of L^p Sobolev spaces, Holder spaces, Hardy spaces, and Morrey spaces, and also a development of Calderon-Zygmund theory and paradifferential operator calculus. The book is targeted at graduate students in mathematics and at professional mathematicians with an interest in partial differential equations, mathematical physics, differential geometry, harmonic analysis, and complex analysis. The third edition further expands the material by incorporating new theorems and applications throughout the book, and by deepening connections and relating concepts across chapters. It includes new sections on rigid body motion, on probabilistic results related to random walks, on aspects of operator theory related to quantum mechanics, on overdetermined systems, and on the Euler equation for incompressible fluids. The appendices have also been updated with additional results, ranging from weak convergence of measures to the curvature of Kahler manifolds. Michael E. Taylor is a Professor of Mathematics at the University of North Carolina, Chapel Hill, NC. Review of first edition: “These volumes will be read by several generations of readers eager to learn the modern theory of partial differential equations of mathematical physics and the analysis in which this theory is rooted.” (Peter Lax, SIAM review, June 1998)