## Venue:

**Raum 404, 4th floor, Eckerstr. 1, Freiburg University**

## Timetable (subject to change):

Tuesday (July 26) | Wednesday (July 27) | |

08.30-09.00 | Registration | |

09.00-10.00 | Belletini | Novaga |

10:30-11:30 | Kröner | Laux |

11:30-12:30 | Wojtowytsch | Mondino |

14:00-15:00 | Lamm | Cabezas-Rivas |

15:30-16:30 | Blatt | Schnürer |

16:30-17:30 | Moser | Röger |

19:30 | Conference Dinner |

## Abstracts:

### Giovanni Bellettini: Constrained BV functions on covering spaces and a solution to Plateau's type problems

We link covering spaces with the theory of functions of bounded variation, in order to study Plateau's type problem without fixing a priori the topology of solutions. Our solution is obtained solving a minimization problem in the class of BV functions defined on a covering space of the complement of an (n-2)-dimensional manifold S without boundary, and satisfying a suitable constraint on the fibers. The model avoids all issues concerning the presence of the boundary S, and allows a phase transition approximation. Various different Plateau's type problems can be faced choosing different numbers of sheets of the covering space.### Simon Blatt: Some Möbius invariant geometric evolution equations

While the Willmore energy is invariant under Möbius transformations, its negative L(2)- gradient flow is not = simply because the L(2)-scalar product used in its definition does not have this invariance. In this talk we present Möbius invariant versions of the Willmore flow and other evolution equations picking up ideas of Ruben Jakob and Oded Schramm. We will discuss its uses and limitations, prove well-posedness of the Cauchy problem and attractivity of local minimizers, and present some open questions.### Esther Cabezas-Rivas: The Ricci Flow on manifolds with almost non-negative curvature operator

We show that n-manifolds with a lower volume bound v and upper diameter bound D whose curvature operator is bounded below by -ε(n,v,D) also admit metrics with nonnegative curvature operator. The proof relies on heat kernel estimates for the Ricci flow and shows that various smoothing properties of the Ricci flow remain valid if an upper curvature bound is replaced by a lower volume bound.### Heiko Kröner: Rates of convergence for regularized level set power mean curvature flow

In a paper by Schulze the evolution of hypersurfaces with normal speed equal to a power k>1 of the mean curvature is considered. The level set solution of the flow is obtained as the C^0-limit of a sequence of smooth functions solving regularized level set equations. We prove a rate for this convergence which is of interest from a numerical point of view.### Tobias Lamm: Conformal Willmore Tori

In this talk I am going to present recent existence and non-existence results for conformal Willmore Tori in R^4 which were obtained in a collaboration with Reiner M. Schätzle (Tübingen).### Tim Laux: Convergence of thresholding schemes for multi-phase mean-curvature flow

The thresholding scheme, a time discretization for mean-curvature flow was introduced by Meriman, Bence and Osher in 1992. In the talk I present new convergence results for modern variants of this scheme, in particular in the multi-phase case with arbitrary surface tensions. The first result establishes convergence towards a weak formulation of mean-curvature flow in the BV-framework of sets of finite perimeter. The proof is based on the interpretation of the thresholding scheme as a minimizing movements scheme by Esedoglu and Otto in 2014. This interpretation means that the thresholding scheme preserves the structure of (multi-phase) mean-curvature flow as a gradient flow w. r. t. the total interfacial energy. More precisely, the thresholding scheme is a minimizing movements scheme for an energy that $\Gamma $ -converges to the total interfacial energy. In this sense, our proof is similar to the convergence results of Almgren, Taylor and Wang in 1993 and Luckhaus and Sturzenhecker in 1995, which establish convergence of a more academic minimizing movements scheme. Like the one of Luckhaus and Sturzenhecker, ours is a conditional convergence result, which means that we have to assume that the time-integrated energy of the approximation converges to the time-integrated energy of the limit. This is a natural assumption, which is however not ensured by the compactness coming from the basic estimates.--- Based on joint works with Felix Otto (MPI MIS Leipzig) and Drew Swartz (Booz Allen Hamilton)