Mathematics

Dr. Atreyee Bhattacharya
IISER Bhopal

Curvature functionals are important variational tools in Riemannian geometry for understanding the rigidity of standard Riemannian structures. 
Given a compact smooth manifold, these functionals are defined on the space of Riemmanian metrics on that manifold by integrating (suitable powers of) norms of different curvature quantities (e.g. scalar curvature, Ricci curvature, full curvature tensor, Weyl curvature, etc). One can make sense of differentiability of such functionals and also talk about their critical points. Irreducible symmetric spaces turn out to be critical points of all such functionals. One further describes the stability of  a critical point by analyzing the second order behaviour of a functional at that point. While classification of the irreducible symmetric spaces as stable critical points for certain functionals is available, the moduli space of critical points or the stability of generic critical points are far from being well understood. In this talk we will focus on some of the widely studied curvature functionals and their critical points with an aim of analyzing their stability. Starting with an overview of the existing literature, we will mention some of our own results in this direction where we construct some unstable critical points of curvature functionals and also provide a stability criterion for critical points that are symmetric spaces but not necessarily Einstein. This is a joint work with Soma Maity.