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