Applications of geometric identities to rigidity problems

Abstract

In this thesis, we explore the applications of integral identities, such as the Reilly-type and Pohozaev-type identities, in various geometric contexts, highlighting their roles in obtaining inequalities and rigidity results for specific classes of Riemannian manifolds. First, we consider the context of V -static manifolds, which are Riemannian manifolds with boundary, constant scalar curvature, and a metric that is a critical point of the volume functional with a fixed boundary metric. In this context, we employ our Reilly-type identity to establish Heintze-Karcher and Minkowski inequalities for bounded domains. Furthermore, we examine the rigidity phenomena associated with these inequalities, especially in cases where equality is achieved, shedding light on the geometric structure of these manifolds. Additionally, we obtain an inequality for domains in m-quasi Einstein manifolds along with a rigidity characterization. This inequality is motivated by the stability of the Wang-Yau energy. Finally, we turn our attention to weighted overdetermined problems on Riemannian manifolds with density. By studying a Poisson problem associated with the weighted Laplacian, we derive a Heintze-Karcher inequality and a Soap Bubble-type theorem that characterize geodesic balls in these weighted spaces. By imposing Dirichlet and Neumann boundary conditions, we also establish a Serrin-type result in generalized cones and convex cones of Euclidean space, identifying metric balls as the unique solutions to the underlying overdetermined problem.