This volume contains three expanded lecture notes from the program Scalar Curvature in Manifold Topology and Conformal Geometry that was held at the Institute for Mathematical Sciences from 1 November to 31 December 2014. The first chapter surveys the recent developments on the fourth-order equations with negative exponent from geometric points of view such as positive mass theorem and uniqueness results. The next chapter deals with the recent important progress on several conjectures such as the existence of non-flat smooth hyper-surfaces and Serrin's over-determined problem. And the final chapter induces a new technique to handle the equation with critical index and the sign change coefficient as well as the negative index term. These topics will be of interest to those studying conformal geometry and geometric partial differential equations.
Contents:
Lectures on the Fourth-Order Q Curvature Equation (Fengbo Hang and Paul C Yang)
An Introduction to the Finite and Infinite Dimensional Reduction Methods (Manuel del Pino and Juncheng Wei)
Einstein Constraint Equations on Riemannian Manifolds (Quôc Anh Ngô)
Readership: Advanced undergraduates, graduate students and researchers interested in the study of conformal geometry and geometric partial differential equations. Scalar Curvatures;Negative Gradient Flow;Positive Mass Theorem;Einstein Equation;Scalar Field Equation;LyapunovâSchmidt Reduction Methods;Integral Estimate;Blow-Up Analysis;Manifolds Key Features:
Includes recent developments and several new methods on geometric analysis of scalar curvatures
Excellent references for graduate students as well as for young researchers
Starting from the Einstein equation in general relativity, we carefully derive the Einstein constraint equations which specify initial data for the Cauchy problem for the Einstein equation. Then, we show how to use the conformal method to study these constraint equations.
0.Introduction
It is well-known that the Einstein theory of relativity (or commonly known as general relativity) is a geometric theory of gravitation. In this theory, gravity is considered as a geometric property of space and time. Because of the geometric property of space and time, general relativity partially includes both special relativity and the Newton law of universal gravitation as special cases.
Theoretically, general relativity describes objects in large scale, like the universe, as Lorentzian manifolds on which gravitation interacts and the universe evolves over time through a system of partial differential equations known as the Einstein equations. Being the central object of the theory, studying the Einstein equations becomes a significant subject in order to understand the whole theory.
In an effort to solve the general Einstein equations, physicists first try to tackle the equations in some simple cases. Fortunately, some remarkably solutions have been found in this direction. Although general relativity nearly coincides with the Newton law of universal gravitation, those known solutions for the Einstein equations in particular cases have led theoretical physicists to predict some new phenomena which deserve investigation carefully. However, much less is known about the solutions of the general Einstein equations. On the other hand, due to the geometric nature of the theory, solving the general Einstein equations turns out to be a wonderful research topic not only for physicists but also for mathematicians, pushing the development of the research rapidly.
However, along with the rapid development of the research, it poses many challenging problems to mathematicians, for...
