# Geometric variational problems (13w5070)

Arriving in Banff, Alberta Sunday, December 15 and departing Friday December 20, 2013

## Organizers

Jingyi Chen (University of British Columbia)

Ailana Fraser (University of British Columbia)

Tobias Lamm (Karlsruhe Institute of Technology (KIT))

## Objectives

In the proposed workshop we want to focus on three important geometric variational problems: Minimal surfaces, harmonic maps and Willmore surfaces. We plan to invite experts from all of the above mentioned fields and our hope is that this will lead to new collaborations and the solution of open problems in this currently very active and exciting research area.

In recent years there has been striking progress in understanding the blow-up sets and limit laminations of sequences of embedded minimal surfaces in 3-manifolds. In particular, this has led to new developments in the classical theory of minimal surfaces in $mathbb{R}^3$, such as the classification of topologically simple minimal surfaces. Important applications of minimal surfaces to other fields of mathematics have been discovered. In fact minimal surfaces play a crucial role in the proof of the Poincare conjecture. In relativity there is interest in dynamical horizons which are 3-dimensional spacelike hypersurfaces foliated by apparent horizons in a slicing of a spacetime. Apparent horizons are minimal 2-spheres in some cases, but usually solutions of a prescribed mean curvature equation of a particular type. There is also interest in higher dimensional black holes related to string theory. One focus of the workshop will be on existence theory. Constructing special submanifolds such as minimal surfaces and constant mean curvature surfaces is an important and classical topic in differential geometry with applications in topology and physics (e.g. the theory of general relativity). A further important topic is calibrated submanifolds , i.e. submanifolds which minimize the volume functional in their homology classes. Typical examples are complex submanifolds in K"ahler manifolds and special Lagrangian submanifolds in Calabi-Yau manifolds, and they are the building blocks in various duality phenomena in mirror symmetry. Minimal surfaces/submanifolds are characterized by nonlinear partial differential equations which are difficult to solve in general. Douglas-Morrey's solution to the Plateau problem is perhaps the first major advance. One may use the gluing method (a delicate application of the implicit function theorem), variational method (such as min-max principle or minimizing certain functionals, e.g. Sacks-Uhlenbeck's perturbed energy functional) and the heat flow method (e.g. the mean curvature flow).

Another focus of the workshop will be on Willmore surfaces. These are solutions of a variational problem for which minimal surfaces are minimizers. Willmore surfaces arise naturally in a large number of different fields such as elasticity theory, general relativity and conformal geometry. Moreover, since the system of PDE's solved by Willmore surfaces is of fourth order, they can also be seen as an interesting model for nonlinear higher order systems of PDE's. Recently there have been a number of new analytic and geometric results on the existence, regularity and energy quantization for Willmore surfaces in Euclidean space and in Riemannian manifolds. One of the difficulties in the study of Willmore surfaces is the fact that the system of PDE's is not elliptic since it is invariant under diffeomorphisms. This problem can be overcome by choosing a good gauge, e.g. by considering graphical solutions or by working with a conformal parametrization of the surface. This last approach recently led to the introduction of $W^{2,2}$-conformal immersions and some interesting applications on the existence of minimizing Willmore surfaces in a given conformal class have been obtained. The recent work of Rivi'ere led to a major breakthrough in the regularity theory, of two-dimensional conformally invariant variational problems such as Willmore surfaces, harmonic maps, CMC surfaces or more generally surfaces of prescribed mean curvature in Riemannian manifolds.

The workshop will bring together people working on applications of these new methods to related problems including geometric flows.

In recent years there has been striking progress in understanding the blow-up sets and limit laminations of sequences of embedded minimal surfaces in 3-manifolds. In particular, this has led to new developments in the classical theory of minimal surfaces in $mathbb{R}^3$, such as the classification of topologically simple minimal surfaces. Important applications of minimal surfaces to other fields of mathematics have been discovered. In fact minimal surfaces play a crucial role in the proof of the Poincare conjecture. In relativity there is interest in dynamical horizons which are 3-dimensional spacelike hypersurfaces foliated by apparent horizons in a slicing of a spacetime. Apparent horizons are minimal 2-spheres in some cases, but usually solutions of a prescribed mean curvature equation of a particular type. There is also interest in higher dimensional black holes related to string theory. One focus of the workshop will be on existence theory. Constructing special submanifolds such as minimal surfaces and constant mean curvature surfaces is an important and classical topic in differential geometry with applications in topology and physics (e.g. the theory of general relativity). A further important topic is calibrated submanifolds , i.e. submanifolds which minimize the volume functional in their homology classes. Typical examples are complex submanifolds in K"ahler manifolds and special Lagrangian submanifolds in Calabi-Yau manifolds, and they are the building blocks in various duality phenomena in mirror symmetry. Minimal surfaces/submanifolds are characterized by nonlinear partial differential equations which are difficult to solve in general. Douglas-Morrey's solution to the Plateau problem is perhaps the first major advance. One may use the gluing method (a delicate application of the implicit function theorem), variational method (such as min-max principle or minimizing certain functionals, e.g. Sacks-Uhlenbeck's perturbed energy functional) and the heat flow method (e.g. the mean curvature flow).

Another focus of the workshop will be on Willmore surfaces. These are solutions of a variational problem for which minimal surfaces are minimizers. Willmore surfaces arise naturally in a large number of different fields such as elasticity theory, general relativity and conformal geometry. Moreover, since the system of PDE's solved by Willmore surfaces is of fourth order, they can also be seen as an interesting model for nonlinear higher order systems of PDE's. Recently there have been a number of new analytic and geometric results on the existence, regularity and energy quantization for Willmore surfaces in Euclidean space and in Riemannian manifolds. One of the difficulties in the study of Willmore surfaces is the fact that the system of PDE's is not elliptic since it is invariant under diffeomorphisms. This problem can be overcome by choosing a good gauge, e.g. by considering graphical solutions or by working with a conformal parametrization of the surface. This last approach recently led to the introduction of $W^{2,2}$-conformal immersions and some interesting applications on the existence of minimizing Willmore surfaces in a given conformal class have been obtained. The recent work of Rivi'ere led to a major breakthrough in the regularity theory, of two-dimensional conformally invariant variational problems such as Willmore surfaces, harmonic maps, CMC surfaces or more generally surfaces of prescribed mean curvature in Riemannian manifolds.

The workshop will bring together people working on applications of these new methods to related problems including geometric flows.