Some Nonlinear Problems in Riemannian Geometry

A manifold M _n , of dimension n, is a Haussdorff topological space such that each point of M _n , has a neighborhood homeomorphic to ℝ ⁿ . Thus a manifold is locally compact and locally connected. Hence a connected manifold is pathwise connected.

We are going to define Sobolev spaces of integer order on a Riemannian manifold. First we shall be concerned with density problems. Then we shall prove the Sobolev imbedding theorem and the Kondrakov theorem. After that we shall introduce the notion of best constant in the Sobolev imbedding theorem. Finally, we shall study the exceptional case of this theorem (i.e., H ₁ ⁿ on n-dimensional manifolds).

A normed space is a vector space F(over ℂ or ℝ), which is provided with a norm. A norm, denoted by ∥ ∥, is a real-valued functional on F, which satisfies:

The main aim of this book is to present some methods for solving nonlinear elliptic (or parabolic) problems and to use them concretely in Riemannian Geometry. The present chapter 4 consists in six sections. In the first two, we prove the existence of Green’s function for compact Riemannian manifolds. In §3 and 4, we present some material concerning Riemannian Geometry and Partial Differential Equations, the two main fields of this book. This material, which completes the previous one (in Chapter 3), is crucially used in the sequel of this volume. Many theorems will be quoted without proof, except if they are not available in other books. Then we describe the methods and we mention the sections of the book where one finds concrete applications of them to some problems concerning curvature and also harmonic maps. For instance, to illustrate the steepest descent method, the pioneering article of Eells-Sampson is the best example. We end this chapter with a new result on the best constant in the Sobolev inequality. Its proof shows the power of the method of points of concentration.

Yamabe wanted to solve the Poincaré conjecture (see 9.14). For this he thought, as a first step, to exhibit a metric with constant scalar curvature. He considered conformal metrics (the simplest change of metric is a conformal one), and gave a proof of the following statement “On a compact Riemannian manifold (M, g), there exists a metric g′ conformal to g, such that the corresponding scalar curvature R′ is constant”. The Yamabe problem was born, since there is a gap in Yamabe’s proof. Now there are many proofs of this statement. We will consider some of them, but if the reader wants to see one proof, he has to read only sections 5.11, 5.21, 5.29 and 5.30.

Let (M _n , g) be a C ^∞ Riemannian manifold of dimension n ≥ 2. Given f a smooth function on M _n , the Problem is:

Does there exist a metric g′ on M such that the scalar curvature R′ of g′ is equal to f ?

In this chapter we deal with problems concerning Ricci Curvature mainly:

Let (M, g) and (\(\tilde M,\tilde g\)) be two C ^∞ riemannian manifolds, M of dimension n and \(\tilde M\) of dimension m. M will be compact with boundary or without and {x ⁱ }(1 ≤ i ≤ n) will denote local coordinates of x in a neighbourhood of a point P ∈ M and y ^α (1 ≤ α ≤ m) local coordinates of y in a neighbourhood of f(P) ∈ \(\tilde M\).