发表机构
University of Warwick（华威大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明闭三维流形中极小曲面的第一Betti数受指标与面积之和控制，并给出正曲率下亏格与指标的不等式。
AI 中文摘要
我们证明，在每个闭黎曼三维流形$(M^3,\bar g)$中，存在常数$C>0$，使得每个闭的光滑嵌入极小曲面$\Sigma\subset M$满足$b_1(\Sigma;\mathbb{Z}_2)\le C\bigl(\operatorname{Ind}(\Sigma)+\operatorname{Area}(\Sigma)\bigr)$。该估计在无定向性或双侧性假设下成立，并建立了Song所猜想的下加性亏格-指标-面积估计。对于闭的连通定向双侧极小浸入，在环境截面曲率有下界的条件下，我们还获得了显式的亏格界。在正环境Ricci曲率下，我们证明了普适不等式$\gamma(\Sigma)\le 8\operatorname{Ind}(\Sigma)$。
英文摘要
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 $γ(Σ)\le 8\operatorname{Ind}(Σ)$.
Comments24 pages. Any comments are welcome
