GEOMETRIC ANALYSIS (draft)

Описание

4 PETER LI
§0 Introduction
The main goal of this book is to present the basic tools that are necessary for research in geometric
analysis. Though the main theme centers around linear theory, i.e., the Laplace equation, the heat equation,
and eigenvalues for the Laplacian, the methods in dealing with these problems are quite often useful to the
study of nonlinear partial differential equations that arise in geometry.
The first part of this book originated from a series of lectures given by the author at a Geometry Summer
Program in 1990 at the Mathematical Sciences Research Institute in Berkeley. The lecture notes were revised
and expanded when the author taught a regular course in geometric analysis. During the author’s visit with
the Global Analysis Research Institute at Seoul National University, he was encouraged to submit these
notes, though still in a rather crude form, for publication in their lecture notes series [L6].
Another part of this book on harmonic functions originated from the lecture notes of the author while
he gave a sequence of courses on this topic at the University of California, Irvine. A certain part of this
material was also used in a series of lectures the author gave in the XIV Escola de Geometria Diferencial in
Brazil during the summer of 2006. These notes [L9] were printed for distribution to the participants of the
program.
After updating the Korea lecture notes with more recent developments and combining with them the
harmonic function notes, the author also inserted a few sections on the heat equation. The resulting product
now takes the form of an introduction to the subject of geometric analysis on the one hand, with some
application to geometric problems via linear theory on the other. Due to the vast literature in geometric
analysis, it is prudent not to make any attempt to try to discuss the nonlinear theory. The interested reader
is encouraged to consult the excellent book of Schoen and Yau [SY2] in this direction.
The readers should be aware that the aim of this book is to address entry level geometric analysts by
introducing the basic techniques in geometric analysis in the most economical way. The theorems discussed
are chosen sometimes for their fundamental usefulness and sometimes for purpose of demonstrating various
techniques. In many cases, they often do not represent the best possible results that are available.
The author would like to express his gratitude to Ovidiu Munteanu, Lei Ni and Jiaping Wang for their
suggestions to improve this book. He is particular in debt to Munteanu for his detail proof-reading of the
draft. Acknowledgement is also due to his graduate students Lihan Wang and Fei He, who were extremely
helpful in pointing out necessary corrections to the manuscript.

Детали