2024/02/12 by Manuel Schlierf, Schlierf, Manuel · 1 citation
Mathematics · #35B40 #35J35 (secondary) #35K41 #49Q10 (primary) #53E40 #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Navier-Stokes equation solutions #Nonlinear Partial Differential Equations
paper · pdf · doi:10.48550/arxiv.2402.07755
openalex publication_date 2024/02/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 8π - 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 CLY below which surfaces with this boundary are embedded. By a slight modification, one obtains a threshold CLYrot 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 CLYrot. 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 CLYrot which improves previous results for the stationary problem.