arXiv subjects
Publications and source records attributed to Jan-Henrik Metsch.
We establish the existence and smoothness of minimizers of the Willmore energy among axially symmetric surfaces of spherical type with prescribed isoperimetric ratio. Afterwards, we study the behavior of these minimizers as the isoperimetric ratio tends to zero.
These notes provide a self-contained introduction to Schauder theory on manifolds. First, we derive Schauder estimates for a fourth-order parabolic linear problem with a first- and third-order boundary condition on a smooth compact manifold M. Then, applying these, we prove the existence of solutions and study the associated elliptic problem.
We study half-spheres with small radii sitting on the boundary of a smooth bounded domain while meeting it orthogonally. Even though it is known that there exist families of CMC and Willmore type half-spheres near a nondegenerate critical point p of the domains boundaries mean curvature, it is unknown in both cases whether these provide a foliation of any deleted neighborhood of p. We prove that this is not guaranteed and establish a criterion in terms of the boundaries geometry that ensures or prevents the respective surfaces from providing such a foliation. This perhaps surprising phenomenon of conditional foliations is absent in the closely related Riemannian setting, where a foliation is guaranteed. We show how this unconditional foliation arises from symmetry considerations and how these fail to apply to the 'domain-setting'.
In this paper, we propose the concept of accelerated convergence that has originally been developed to speed up the convergence of numerical methods for extremum seeking (ES) loops. We demonstrate how the dynamics of ES loops may be analyzed to extract structural information about the generated output of the loop. This information is then used to distil the limit of the loop without having to wait for the system to converge to it.
We consider the area-preserving Willmore evolution of surfaces that are close to a half-sphere with a small radius, sliding on the boundary S of a domain while meeting it orthogonally. We prove that the flow exists for all times and keeps a 'half-spherical' shape. Additionally, we investigate the asymptotic behaviour of the flow and prove that for large times the barycenter of the surfaces approximately follows an explicit ordinary differential equation. Imposing additional conditions on the mean curvature of S, we then establish convergence of the flow.