"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 $(M,g_b)$ 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.