Skip to main content

Niels Martin Møller: Rigidity of the grim reaper cylinder as a collapsed self-translating soliton

Time: Thu 2023-06-01 09.15 - 10.15

Location: 3721, Lindstedtsvägen 25

Language: English

Participating: Niels Martin Møller, University of Copenhagen

Export to calendar

Mean curvature flow self-translating solitons are minimal hypersurfaces for a certain incomplete conformal background metric, and are among the possible singularity models for the flow. In the collapsed case, they are confined to slabs in space. The simplest non-trivial such example, the grim reaper curve $\Gamma$ in $\mathbb{R}^2$, has been known since 1956, as an explicit ODE-solution, which also easily gave its uniqueness.

We consider here the case of surfaces, where the rigidity result for $\Gamma\times\mathbb{R}$ that we'll show is:
The grim reaper cylinder is the unique (up to rigid motions) finite entropy unit speed self-translating surface which has width equal to $\pi$ and is bounded from below. (Joint with D. Impera and M. Rimoldi.)

Time permitting, we'll also discuss recent uniqueness results in the collapsed simply-connected low entropy case (joint with E. Gama & F. Martín), using Morse theory and nodal set techniques, which extend Chini's classification.