Some geometric and analytical results on weighted riemannian manifolds

Abstract

Let (M, x , y, f) be a weighted Riemannian manifold. In this thesis we obtain some geometric and analytical results in (M, x , y, f) assuming that Bakry-Emery Ricci tensor is non-negative in some results in other results we assuming that the weighted mean curvature is bounded from below. Moreover, assuming that the radial generalized sectional curvature is bounded from below we obtain a comparison theorem for the Hessian of the distance function and some consequences of it. Let Σ be a closed surface in M, assuming that the Perelman scalar curvature is bounded from below, we obtain an upper bound for the first non-zero eigenvalue of the weighted Jacobi operator for surfaces Σ Ă M and we generalize a result of Shoen and Yau about stable minimal surfaces, see [45]. We also obtained, for surfaces with boundary, a sharp estimate from below for the first non-zero Stekloff’s eigenvalue. For surfaces we also obtain an upper bound for the first non-zero eigenvalue of the weighted Jacobi operator and some consequences of it, for instance, we show that in R 3 there exist no closed stable selfshrinker. In higher dimension we obtain upper bound and lower bound for the first non-zero Stekloff’s eigenvalue on suitable hypotheses. We conclude our work with a weighted splitting theorem.