Pure Maths Colloquium: Translating solitons from Riemannian foliations.
6 March 2017 16:00 in CM221
The mean curvature flow is one of the most important geometric flows on Riemannian submanifolds. Translating solitons are special solutions, where the flow is along straight lines, and have particular importance since they exist for all time and are related to certain singularities (Type II) of the flow. We study translating solitons in a more general context than Eulidean space: the product of a (semi-)Riemannian manifold with the real line, and develop new tools to study them. Considering Riemannian submersions where the total space is a translating soliton, we show that under certain hypotheses involving the mean curvature of the fiber, this data is equivalent to a
certain type of object on the base manifold. In the case where the submanifold is a leaf of a codimension-one foliation by orbits of a Lie group of symmetries (such as SO(n) or SO(p,q) acting on Euclidean or Minkowski space), we reduce the existence of a translating soliton to an ODE that we explicitly solve in many examples.
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