局部极小极大曲面与强不可约极小Heegaard分裂
Local min-max surfaces and strongly irreducible minimal Heegaard splittings
摘要：
设$(M,g)$是一个闭的定向黎曼$3$ -流形，并假设存在一个强不可约Heegaard分裂$H$ .证明了$H$与指数至多为1的极小曲面是同构的，或者与不可定向的极小曲面的稳定定向复盖是同构的.特别地，这证明了Rubinstein证明的一个结果。一些后果包括存在于任何$\mathbb{R}P^3$的一个最小的环面或一个最小的射影平面稳定的普遍覆盖。在正数量曲率的情况下，证明了对于不同构于$S^3$或$\mathbb{R}P^3$的球面空间形式，任何强不可约的Heegaard分裂在它的合痕类中都有一个极小的代表，并且如果$R\geq 6$，则存在一个面积小于$4π$的极小Heegaard分裂.
英文摘要：
Let $(M,g)$ be a closed oriented Riemannian $3$-manifold and suppose that there is a strongly irreducible Heegaard splitting $H$. We prove that $H$ is either isotopic to a minimal surface of index at most one or isotopic to the stable oriented double cover of a non-orientable minimal surface with a vertical handle attached. In particular, this proves a result conjectured by Rubinstein. Some consequences include the existence in any $\mathbb{R}P^3$ of either a minimal torus or a minimal projective plane with stable universal cover. In the case of positive scalar curvature, it is shown for spherical space forms not diffeomorphic to $S^3$ or $\mathbb{R}P^3$ that any strongly irreducible Heegaard splitting admits a minimal representative in its isotopy class, and that there is a minimal Heegaard splitting of area less than $4π$ if $R\geq 6$.
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