University of Newcastle geometric analysis seminar
2022
Friday, May 25, 11:00
Stephen Lynch (Universität Tübingen)
Title: Singularity formation in fully nonlinear flows of hypersurfaces.
Abstract: We consider curvature flows of hypersurfaces in a Riemannian manifold. An important example is the flow by mean curvature, but for many geometric problems flows by other speeds have proven to be a more effective tool. The key step for most applications is to understand the structure of singularities. We will discuss a new a priori 'convexity' estimate, which implies that singularities are positively curved, even if the initial datum is highly non-convex. As a consequence, we obtain a complete classification of Type I singularities under very general conditions, despite having no analogue of Huisken's monotonicity formula for mean curvature flow.
2021
Wednesday, March 17, 14:00
Timothy Buttsworth (University of Queensland)
Title: The Prescribed Cross Curvature Equation on the Three-Sphere
Abstract: For a given Riemannian manifold, the cross curvature tensor is a symmetric (0,2)-tensor field which describes how close the underlying geometry is to being hyperbolic. The cross curvature was introduced by Chow and Hamilton in 2004; they hoped that the corresponding cross curvature flow could be used to continuously deform an arbitrary Riemannian metric of negative sectional curvature into one of constant negative sectional curvature. In this talk, I will discuss the 'prescribed cross curvature equation', which is the underlying inhomogeneous steady-state version of the cross curvature flow. About this problem, Hamilton conjectured that any positive symmetric tensor on the three-sphere was the cross curvature of exactly one Riemannian metric. I will discuss some recent results which support the existence component of this conjecture, and refute the uniqueness component.
Wednesday, March 17, 15:00
Jonathan Zhu (Australian National University)
Title: Explicit Łojasiewicz inequalities for mean curvature flow shrinkers
Abstract: Łojasiewicz inequalities are a popular tool for studying the stability of geometric structures. For mean curvature flow, Schulze used Simon’s reduction to the classical Łojasiewicz inequality to study compact tangent flows. For round cylinders, Colding and Minicozzi instead used a direct method to prove Łojasiewicz inequalities. We’ll discuss similarly explicit Łojasiewicz inequalities and applications for other shrinking cylinders and Clifford shrinkers.