Topology, Morse Index, and Area of Minimal Surfaces in Three-Manifolds

Abstract

We prove that, in every closed Riemannian three-manifold $(M^3,\bar{g})$, there exists a constant $C=C(M,\bar{g})>0$ such that every closed smoothly embedded minimal surface $\Sigma \subset M$ satisfies $b_1(\Sigma ;{\mathbb{Z}}_2)\le C(\mathrm{Ind}(\Sigma )+\mathrm{Area}(\Sigma ))$, establishing Song's additive genus-index-area conjecture. For closed connected orientable two-sided minimal immersions, we prove the universal inequality $\gamma (\Sigma )\le 8\mathrm{Ind}(\Sigma )$ in positive Ricci curvature and the estimate $\gamma (\Sigma )\le C(M,\bar{g})\mathrm{Ind}(\Sigma )$ in positive scalar curvature. In the latter case, we also bound both area and total curvature by $C(M,\bar{g})(1+\mathrm{Ind}(\Sigma ))$.

arXiv comment24 pages. Any comments are welcome