Published online by Cambridge University Press: 06 August 2010
Notation concerning Riemannian manifolds
We want now to replace the Euclidean space ℝn by a Riemannian manifold M and consider the possibility of having some kind of Sobolev inequalities. This brings in a whole new point of view. On Euclidean space, we could only discuss whether inequalities were true or not. In the more general setting of Riemannian manifolds, we can investigate the relations between various functional inequalities and the relations between these functional inequalities and the geometry of the manifold. We can search for necessary and/or sufficient conditions for a given Sobolev-type inequality to hold true. This leads to a better understanding of what information about M is encoded in various Sobolev-type inequalities.
Sobolev inequalities are useful when developing analysis on Riemannian manifolds, even more so than on Euclidean space, because other tools such as Fourier analysis are not available any more. This is particularly true when one studies large scale behavior of solutions of partial differential equations such as the Laplace and heat equations.
In the sequel, we will focus on complete, non-compact Riemannian manifolds. For compact manifolds, local Euclidean-type Sobolev inequalities are always satisfied and the interesting questions have to do with controlling the constants arising in these inequalities in geometric terms. We refer the interested reader to where this is discussed at length.