Author Archives: Fabrice Baudoin

It is a well-known result that if is a complete -dimensional Riemannian manifold with , for some , then has to be compact with diameter less than . The proof of this fact can be found in any graduate book … Continue reading →

In this Lecture, we are interested in sharp Sobolev inequalities in positive curvature. Let be a complete and -dimensional Riemannian manifold such that where . We assume . As we already know from Lecture 15 , we have , but … Continue reading →

In this Lecture, we study in further details the connection between volume growth of metric balls, heat kernel upper bounds and the Sobolev inequality. As we shall see, on a manifold with non negative Ricci curvature, all these properties are … Continue reading →

In this Lecture, we show how Sobolev inequalities on a Riemannian manifold are related to the volume growth of metric balls. The link between the Hardy-Littlewood-Sobolev theory and heat kernel upper bounds is due to Varopoulos, but the proof I … Continue reading →

Let be a complete Riemannian manifold and be a non empty bounded set. Let be the set of smooth functions such that for every , It is easy to see that is essentially self-adjoint on . Its Friedrichs extension, still … Continue reading →