Abstract:
We state various conjectures on the behavior of minimal
surfaces and Ricci flow, explaining heuristic motivations for the conjectures
and proofs in some very special cases. We also prove a version of Kneser-Haken
finiteness for stable minimal surfaces (joint with Hass), i.e. given a
closed riemannian 3-manifold, there is a number n such that if one has
n disjoint stable minimal surfaces, then two must be parallel. We use this
to show finiteness of components of stable minimal surfaces in bumpy metrics,
and conjecture finiteness in the case of analytic metrics which are not
foliated by compact minimal surfaces.