Optimal rigidity estimates for nearly umbilical surfaces (1)
Dienstag, 27.10.15, 16:00-17:00, Raum 125, Eckerstr. 1
For closed connected surfaces in the 3-dimensional Euclidean space the L^2 distance of the second fundamental form to the identity is estimated by the L^2 norm of the traceless second fundamental form.
Optimal rigidity estimates for nearly umbilical surfaces (2)
Dienstag, 3.11.15, 16:00-17:00, Raum 125, Eckerstr. 1
Mass in Kähler geometry
Dienstag, 10.11.15, 16:00-17:00, Raum 125, Eckerstr. 1
We prove a simple, explicit formula for the mass of any asymptotically locally Euclidean (ALE) Kähler manifold, assuming only the sort of weak fall-off conditions required for the mass to actually be well-defined. For ALE scalar-flat Kähler manifolds, the mass turns out to be a topological invariant, depending only on the underlying smooth manifold, the first Chern class of the complex structure, and the Kähler class of the metric. When the metric is actually AE (asymptotically Euclidean), our formula not only implies a positive mass theorem for Kähler metrics, but also yields a Penrose-type inequality for the mass.
Quantitative rigidity results for conformal immersions
Dienstag, 17.11.15, 16:00-17:00, Raum 125, Eckerstr. 1
A phase field model for Willmore´s energy with topological constraint
Dienstag, 24.11.15, 16:00-17:00, Raum 125, Eckerstr. 1
We consider the problem of minimizing Willmore’s energy on confined and connected surfaces with prescribed surface area. To this end, we approximate the surface by a level set function u admitting the value +1 on the inside of the surface and -1 on its outside. The confinement of the surface is now simply given by the domain of definition of u. A diffuse interface approximation for the area functional, as well as for Willmore’s energy are well known. We address the main difficulty, namely the topological constraint of connectedness by a penalization of a geodesic distance which is chosen to be sensitive to connected components of the phase field level sets and provide a proof of Gamma-convergence of our model to the sharp interface limit in case of a two-dimensional ambient space. Furthermore, we show some numerical results. This is joint work with Stephan Wojtowytsch (Durham University) and Antoine Lemenant (Universit Paris 7).
Geometric inequalities in almost positive Ricci curvature
Dienstag, 1.12.15, 16:00-17:00, Raum 125, Eckerstr. 1
Various geometric estimates for an n-dimensional Riemannian manifold are discussed under the assumption of lower L^p bounds of the Ricci curvature, p>n/2.
Stability of closed, immersed hypersurfaces under pinching of the first Laplace eigenvalue,
Dienstag, 8.12.15, 16:00-17:00, Raum 125, Eckerstr. 1
On (spinoral) Yamabe equations on noncompact manifolds
Dienstag, 12.1.16, 16:00-17:00, Raum 125, Eckerstr. 1
We study generalizations of the Yamabe problem and its\nspinorial sybling to noncompact manifolds. We examine general properties\nof (sybling) Yamabe equations and there link to the corresponding\nvariational problems. In particular, we investigate the (non-)existence\nof solutions for certain model spaces.
Harnack inequalities in curvature flows
Dienstag, 19.1.16, 16:15-17:15, Raum 125, Eckerstr. 1
We give an overview over the state of research in the theory of Harnack inequalities for extrinsic geometric flows, such as the mean curvature flow. We also discuss some applications to the classification of ancient solutions.
Limits of \balpha-harmonic maps
Dienstag, 26.1.16, 16:15-17:15, Raum 125, Eckerstr. 1
In a famous paper, Sacks and Uhlenbeck introduced a perturbation of the\nDirichlet energy, the so-called \balpha-energy E\balpha, \balpha>1, to construct non-trivial\nharmonic maps of the two-sphere in manifolds with a non-contractible\nuniversal cover. The Dirichlet energy corresponds to \balpha= 1 and, as \balpha\ndecreases to 1, critical points of E\balpha are known to converge to harmonic\nmaps in a suitable sense.\nHowever, in a joint work with Andrea Malchiodi and Mario Micallef,\nwe show that not every harmonic map can be approximated by critical\npoints of such perturbed energies. Indeed, we prove that constant maps\nand the rotations of S^2 are the only critical points of E_\balpha for maps from\nS^2 to S^2 whose \balpha-energy lies below some threshold, which is independent\nof \balpha (sufficiently close to 1). In particular, nontrivial dilations (which are\nharmonic) cannot arise as strong limits of \balpha-harmonic maps. We shall\nalso discuss similar results for other perturbations of the Dirichlet energy.
Hessian metrics and local Lagrangian immersions
Dienstag, 2.2.16, 16:15-17:15, Raum 125, Eckerstr. 1