Seminar Analysis: Alessandro Carlotto (ETH-ITS Zürich)

The finiteness problem for minimal surfaces of bounded index in a 3-manifold

Given a closed, Riemannian 3-manifold (N, g) without symmetries (more precisely: generic) and a non-negative integer p, can we say something about the number of minimal surfaces it contains whose Morse index is bounded by p? More realistically, can we prove that such number is necessarily finite? This is the classical ”generic finiteness” problem, which has a rich history and exhibits interesting subtleties even in its basic counterpart concerning closed geodesics on surfaces. We settle such question when g is a bumpy metric of positive scalar curvature by proving that either finiteness holds or N does contain a copy of RP³ in its prime decomposition and we discuss the obstructions to any further generalisation of such result. When g is assumed to be strongly bumpy (meaning that all closed, immersed minimal surfaces do not have Jacobi fields, a notion recently proved to be generic by White) then the finiteness conclusion is true for any compact 3-manifold without boundary.

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