Perelman has recently given a description of Ricci flow as a gradient flow on the space of Riemannian geometries. While this has received attention because of its application to the Poincar\'e and Thurston conjectures, his discovery clearly has other applications as well. In this talk, I will describe its application to a nonlinear sigma model (NLSM) that arises from string theory. It has long been recognized that Ricci flow is an approximation to a purely gravitational NLSM. In this talk, I will explain the necessary part of Perelman's formalism, present a gradient flow for an NLSM with B-field (and dilaton) as well as gravity, and use the Hessian to discuss the stability and rigidity of certain fixed points.