Yamabe flow on noncompact manifolds

Dr Mario Schulz (QMUL). Part of the Analysis and Applications Seminar Series.

Yamabe flows are solutions to a geometric evolution equation which tends to evolve a given Riemannian metric towards a conformal metric of constant scalar curvature. On any closed Riemannian manifold the existence and uniqueness of a Yamabe flow is known. The question of well-posedness is more delicate if the initial manifold is noncompact and uniqueness fails in general. We examine the existence and uniqueness of instantaneously complete Yamabe flows starting from noncompact, possibly incomplete Riemannian manifolds of arbitrary dimension. In this context, we find a sharp condition on the dimension of a submanifold which implies that its removability as a singularity is necessarily preserved along the Yamabe flow.