On the Heinze-Karcher inequality

The aim of this thesis is to prove the Heintze-Karcher inequality in two different geometric settings: Euclidean and Riemannian. In the first case we consider a compact smooth hypersurface in R^(n+1) whose mean curvature is everywhere positive. Then we move to the Riemannian setting. Here we consider a (n+1)-dimensional Riemannian manifold M with non negative Ricci curvature, and a n-dimensional embedded compact submanifold with positive mean curvature. In particular in the Riemannian context we prove the Heintze-Karcher inequality with two different approach. The first one is based on a more general theorem, due to Heintze and Karcher, in which we prove an inequality in the case in which the Ricci curvature of M is bounded from below by a constant, and then we recover our case when the constant is zero. The second one is based on Reilly's formula and the Laplace operator. In conclusion we discuss, both in the Euclidean and Riemannian settings, the equality case in the Heintze-Karcher inequality. In particular in the Euclidean case we use it to prove the famous Alexandrov's theorem for hypersurfaces with constant mean curvature.