Category Archives: Curvature dimension inequalities

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 →

In this short Lecture, as in the previous one, we consider a complete and -dimensional Riemannian manifold with non negative Ricci curvature. The volume doubling property that was proved is closely related to sharp lower and upper Gaussian bounds that … Continue reading →

In this Lecture we consider a complete and -dimensional Riemannian manifold with non negative Ricci curvature. Our goal is to prove the following fundamental result, which is known as the volume doubling property. Theorem: There exists a constant such that … Continue reading →

Let be a complete -dimensional Riemannian manifold and, as usual, denote by its Laplace-Beltrami operator. As in the previous lecture, we will assume that the Ricci curvature of is bounded from below by with . Our purpose in this lecture … Continue reading →

Let be a complete -dimensional Riemannian manifold and, as usual, denote by its Laplace-Beltrami operator. Throughout the Lecture, we will assume that the Ricci curvature of is bounded from below by with . Our purpose is to prove a first … Continue reading →

Let be a complete -dimensional Riemannian manifold and, as usual, denote by its Laplace-Beltrami operator. Throughout the Lecture, we will assume again that the Ricci curvature of is bounded from below by . The Lecture is devoted to the proof … Continue reading →