Joint Seminar

10.01.2019, 16:15  –  Free University of Berlin, Arnimallee 6, SR 025/026
Seminar "Topics in Geometric Analysis"

Piotr Bizoń, Niels Martin Møller

16:15Piotr BizońResonant dynamics in spatially confined Hamiltonian systems

The long-time behavior of nonlinear dispersive waves subject
to spatial confinement can be very rich and complex because, in contrast
to unbounded domains, waves cannot escape to infinity and keep
self-interacting for all times. If, in addition, the linear spectrum
around the ground state is fully resonant, then the nonlinearity can
produce significant effects for arbitrarily small perturbations. The
weak field dynamics of such systems can be approximated by solutions of
the corresponding infinite-dimensional time-averaged Hamiltonian systems
which govern resonant interactions between the modes and a major
mathematical challenge  is to describe the energy transfer between the
modes. I will discuss some recent progress in understanding this
problem, emphasizing universal features of resonant dynamics arising in
very different physical contexts such as, for instance, dynamics of 
Bose-Einstein condensates (modeled by the nonlinear Schroedinger
equation with a trapping potential) or a weakly turbulent behavior of
small perturbations of the anti-de Sitter spacetime (modeled by the
Einstein equations with negative cosmological constant).
17:45Niels Martin MøllerTranslating Solitons and (Bi-)Halfspace Theorems for Minimal Surfaces

I will present new results on the classification problem for
complete self-translating hypersurfaces for the mean curvature flow.
Such surfaces show up as singularity models in the flow (along with
other types of solitons, e.g. the self-shrinkers), and have been
studied since the first examples were found by Mullins in 1956.

Examples from gluing constructions show that one cannot easily
classify such solitons - nor can one classify their projections to one
dimension lower, nor their convex hulls. But if one does both of these
"forgetful" operations, the list becomes very short, coinciding with
(and implying) the one given by Hoffman-Meeks in 1989 for minimal
submanifolds: All
of R^n , halfspaces, slabs, hyperplanes and convex compacts in R^n.
This also implies several of the known obstructions to existence,
e.g. for convex translating solitons.

This is joint work with Francesco Chini (U Copenhagen).
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