"Bubbling" of the prescribed curvature flow on the torus

Abstract

By a classical result of Kazdan-Warner, for any smooth sign-changing function f with negative mean on the torus (M, gb) there exists a conformal metric g = e(2u) g(b) with Gauss curvature K-g = f, which can be obtained from a minimizer u of Dirichlet's integral in a suitably chosen class of functions. As shown by Galimberti, these minimizers exhibit "bubbling" in a certain limit regime. Here we sharpen Galimberti's result by showing that all resulting "bubbles" are spherical. Moreover, we prove that analogous "bubbling" occurs in the prescribed curvature flow.