Theodora Bourni

We develop an Aleksandrov reflection framework for a large class of expanding curvature flows in hyperbolic space, with inverse mean curvature flow serving as a model case. The method applies to the level-set formulation of the flow. As a consequence, we obtain graphical and Lipschitz estimates. Using these estimates, we show that solutions become starshaped and therefore converge exponentially fast to an umbilic hypersurface at infinity. We also extend our results to the non-compact setting, assuming that the solution has a unique point at infinity. In this case, we prove that the flow becomes a graph over a horosphere with uniform gradient bounds and converges to a limiting horosphere.

We establish rigidity results for ancient solutions to the free boundary mean curvature flow in manifolds with convex boundary. In particular, we show that any free boundary minimal hypersurface of Morse index I admits an I-parameter family of ancient solutions that emanate from it. Moreover, among ancient solutions that backward converge exponentially fast to the minimal hypersurface, these exhaust all possibilities. Additionally, we construct a smooth free boundary mean convex foliation around an unstable free boundary minimal hypersurface that enables us to provide a more detailed geometric description of mean-convex ancient solutions that backward converge to that minimal surface.

This tandem workshop with MATRIX in Creswick, Australia, brought together leading experts from the fields of geometric partial differential equations and geometric analysis in general. The focus of the workshop was on recent developments and directions in non-linear geometric diffusion equations. The main flows considered were mean curvature flow, inverse mean curvature flow, Ricci flow, Willmore flow, as well as related flows. A number of the results and methods were in the setting of general relativity, where flows have been very successful in helping solve major problems (for example the resolution of the Penrose conjecture by Huisken/Ilmanen using the inverse mean curvature flow). For four days of the workshop there were combined (with MATRIX) morning talks and extensive discussions.

We construct a translating solution to anisotropic curve shortening flow and show that for a given anisotropic factor g\colon S^{1}\to\mathbb{R}_{+} , and a given direction and speed, this translator is unique. We then construct an ancient compact solution to anisotropic curve shortening flow, and show that this solution, along with the appropriate translating solution, are the unique solutions to anisotropic curve shortening flow that lie in a slab of a given width, and in no smaller slab.

We classify convex ancient curve shortening flows with free boundary on general bounded convex domains.

Using a combination of direct geometric methods and an analysis of the linearization of the flow about the horizontal bisector, we prove that there exists a unique (modulo rotations about the origin) convex ancient curve-shortening flow in the disc with free boundary on the circle. This appears to be the first result of its kind in the free-boundary setting.

We construct a slingshot, that is a compact, embedded solution to curve shortening flow that comes out of a non compact curve and exists for a finite time.