Author Archives: Fabrice Baudoin

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 →

Let be a complete -dimensional Riemannian manifold and denote by its Laplace-Beltrami operator. As usual, we denote by the heat semigroup generated by . Throughout the Lecture, we will assume that the Ricci curvature of is bounded from below by … Continue reading →

In this Lecture, we will prove a first interesting consequence of the Bochner’s identity: We will prove that if, on a complete Riemannian manifold , the Ricci curvature is bounded from below, then the heat semigroup is stochastically complete, that … Continue reading →

The goal of this lecture is to prove the Bochner formula: A fundamental formula that relates the so-called Ricci curvature of the underlying Riemannian structure to the analysis of the Laplace–Beltrami operator. The Bochner’s formula is a local formula, we … Continue reading →