Global existence for the Willmore flow with boundary via Simon's Li-Yau inequality

Abstract

It is well-known that the Willmore flow of closed spherical immersions exists globally in time and converges if the initial datum has Willmore energy below $8\pi$ - exactly the Li-Yau energy threshold below which all closed immersions are embedded. Extending the Li-Yau inequality for closed surfaces via Simon's monotonicity formula also for surfaces with boundary, given Dirichlet boundary conditions, one obtains an energy threshold $C_{\mathrm{LY}}$ below which surfaces with this boundary are embedded. By a slight modification, one obtains a threshold $C_{\mathrm{LY}}^{\mathrm{rot}}$ below which surfaces of revolution satisfying the boundary data have no self-intersections on the rotation axis. With a new argument, using this modified Li-Yau inequality and tools from geometric measure theory, we show that the Willmore flow with Dirichlet boundary data starting in cylindrical surfaces of revolution exists globally in time if the energy of the initial datum is below $C_{\mathrm{LY}}^{\mathrm{rot}}$. Moreover, given Dirichlet boundary data, we also obtain the existence of a Willmore minimizer in the class of cylindrical surfaces of revolution if the corresponding infimum lies below $C_{\mathrm{LY}}^{\mathrm{rot}}$ which improves previous results for the stationary problem.