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arXiv · 2609.32001

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

Riccardo Caniato
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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 $Σ\subset M$ satisfies $b_1(Σ;\mathbb{Z}_2)\le C\bigl(\operatorname{Ind}(Σ)+\operatorname{Area}(Σ)\bigr)$. 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 $γ(Σ)\le 8\operatorname{Ind}(Σ)$.

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Riccardo Caniato. 2026-09-25. Genus, Morse Index, and Area of Minimal Surfaces in Three-Manifolds. https://arxiv.org/abs/2609.32001

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