PhD Defense by Francesco Chini

Title: Some classification results for translating solitons and ancient mean curvature flows

Abstract: In this PhD thesis we make some contributions to the study of translating solitons and ancient mean curvature flows.

In Paper A, which is joint work with my supervisor Niels Martin Møller, we focus on properly immersed translating solitons of the mean curvature flow, with compact (possibly empty) boundary. We prove that they cannot be contained in the intersection of two transverse vertical halfspaces. As an application, we classify their convex hull, up to an orthogonal projection. The proofs are crucially based on an Omori-Yau maximum principle.

In Paper B, also joint work with Niels Martin Møller, we extend the ideas contained in Paper A to the more general setting of ancient flows. We prove a parabolic Omori-Yau maximum principle for ancient flows, which is of independent interest, and we use it to show that properly immersed ancient mean curvature flows cannot be contained in the intersection of two transverse half-spaces. In particular we classify the convex hulls of their sets of reach.

In Paper C, we prove that 2-dimensional embedded simply connected translating solitons with entropy < 3 and which are contained in a slab, must be mean convex. In order to achieve this result, we provide a curvature estimate for 2-dimensional, simply connected translaters with finite entropy.

Supervisors:   

TT Assistant Professor Niels Martin Møller, University of Copenhagen

Professor Bergfinnur Durhuus, University of Copenhagen

Assessment committee:

Professor Henrik Schlichtkrull (Chairman), University of Copenhagen

Professor Francisco Martin, University of Granada

Professor Carlo Mantegazza, University of Napoli