A gap theorem for Willmore tori and an application to the Willmore flow

摘要

In 1965, Willmore conjectured that the integral of the square of the mean curvature of a torus immersed in R~3 is at least 2π~2 and attains this minimal value if and only if the torus is a M?bius transform of the Clifford torus. This was recently proved by Marques and Neves (2012). In this paper, we show for tori that there is a gap to the next critical point of the Willmore energy and we discuss an application to the Willmore flow. We also prove an energy gap from the Clifford torus to surfaces of higher genus.