Genus, 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 $C>0$ such that every closed smoothly embedded minimal surface $\Sigma\subset M$ satisfies $b_1(\Sigma;\mathbb{Z}_2)\le C\bigl(\operatorname{Ind}(\Sigma)+\operatorname{Area}(\Sigma)\bigr)$. The estimate holds without orientability or two-sidedness assumptions and establishes the additive genus-index-area estimate conjectured by Song. For closed connected orientable two-sided minimal immersions, we also obtain explicit genus bounds under a lower bound on the ambient sectional curvature. Under positive ambient Ricci curvature, we prove the universal inequality $\gamma(\Sigma)\le 8\operatorname{Ind}(\Sigma)$.