Geometry Learning Seminar: N.M. Møller (U Copenhagen)

GeoTop Geometry Learning Seminar

Speaker: Niels Martin Møller (U Copenhagen)

Title: Talk #1 on (Non)uniqueness of tangent planes and removable singularities at infinite time for the translating soliton equation

Abstract: Translating solitons for mean curvature flow are, in certain collapsed cases, known to have local subconvergence to “tangent planes at infinity”, as time t → ±∞. This raises the natural question whether L^∞ solutions to the quasilinear equation with a non-trivial drift term may at large scales "wobble" off towards infinity, or if there hold removable singularities theorems at infinite times?

The question turns out to yield a delicate "yes and no": "Yes", as we will in this series of talks prove, for complete solitons, such tangent planes are indeed uniquely defined (with geometric consequences in the classification program). "No", as we construct solitons complete with boundary which, despite decay of all derivatives, admit a continuum of vertical planes as subsequential limits.

This series of three talks will be chiefly about PDEs, starting (almost) from first principles (learning seminar style),
and can be followed independently of most of the geometry aspects,
which were already covered in depth in F. Martin's Minicourse on Mean Curvature Flow Translators.

Based on joint work with E.S. Gama and F. Martín, in: https://arxiv.org/abs/2509.11473.

Figure 1: The pitchfork translating soliton for mean curvature flow, which was proven to exist by Martín-Hoffman-White.
But what are the finer asymptotics of such surfaces?