Analysis and Numerics In Curvature Energies-Meeting

 

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)

Andrea Mondino: A frame energy for immersed tori

In the seminar I will present some joint work with T. Riviere where we study the Dirichlet energy of moving frames on 2-dimensional tori immersed in the euclidean $3\le m$ -dimensional space. This functional, called frame energy, is naturally linked to the Willmore energy of the immersion and on the conformal structure of the abstract underlying surface. I will discuss a ''Willmore Conjecture'' type lower bounds for such energy, then I will introduce basic tools for doing the calculus of variations and finally I will give applications to regular homotopy classes of immersed tori in ${ℝ}^3$ .

Roger Moser: Some singular perturbation problems involving curvature

The free energy functional of a crystal surface is often modelled by an anisotropic area functional. Sometimes, this is augmented (for regularisation purposes or because of physical considerations) by a Willmore term. Consider therefore a functional consisting of an anisotropic area term and a Willmore energy term, where the latter is weighted with a factor that tends to 0. This gives rise to a singular perturbation problem that formally resembles a Modica-Mortola or Ginzburg-Landau problem (depending on the anisotropy). Due to some geometric constraints, however, the behaviour can be quite different. I will discuss some results towards Gamma-convergence in two different situations.

Matteo Novaga: Evolution of planar networks

I will consider the motion by curvature of a network of curves in the plane, and discuss existence, uniqueness and asymptotic behavior of the evolution. I will also discuss possible extensions to other geometric flows of networks.

Matthias Röger: Variational analysis of a mesoscale model for bilayer membranes

We present an asymptotic analysis of a mesoscale energy for biological membranes. The energy is both non-local and non-convex. It combines a surface area and a Monge–Kantorovich-distance term, leading to a competition between preferences for maximally concentrated and maximally dispersed configurations. We extend previous work to the three-dimensional case and identify a curvature energy of Canham–Helfrich type in the zero-thickness limit. For a closely related problem of energies on surfaces equipped with a director field we show compactness and a lower bound property with respect to a suitably generalized curvature energy.

Oliver Schnürer: Mean curvature flow without singularities

We present a method to flow hypersurfaces by mean curvature flow without encountering singularities. We will illustrate this concept of a solution and discuss aspects like different weak interpretations, uniqueness, Dirichlet boundary data and obstacle problems.

Stephan Wojtowytsch: Diffuse Interfaces and Topology: A Phase-Field Model for Willmore's Energy

Motivated from a biological model, we consider the problem of minimising Willmore's energy in the class of connected surfaces with prescribed area embedded into a bounded domain. From a computational point of view, it may be favourable to approximate the curvature energy by a phase field model. Diffuse Interfaces, however, can easily separate into multiple components along a gradient flow evolution. This is overcome using a topological penalty term in the functional involving a geodesic distance function. We present here a proof of Gamma-convergence to the sharp interface limit in two and three dimensions and numerical evidence of the effectiveness of our method in two dimensions.