Generic finiteness of minimal surfaces with bounded Morse index
Generic finiteness of minimal surfaces with bounded Morse index
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发表时间：
2015-09
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通讯作者：
A. Carlotto
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作者：
A. Carlotto
给定一个无边界的紧致3-流形N，我们证明了对于正数量曲率的凹凸度量，除非流形本身包含嵌入的极小RP，否则具有一致的莫尔斯指数上界的极小曲面空间总是有限的^[2]。特别地，当N在其素分解中不包含RP ^3的拷贝时，我们得到了一个通用的有限性结果。我们讨论了任何进一步推广这样的结果的障碍。当要求度量g是（纯量正的）强凹凸（意味着所有封闭的浸入极小曲面不具有雅可比场，这一概念最近被B证明是通用的。白色）相同的结论对任何闭的3-流形都成立。
Given a compact 3-manifold N without boundary, we prove that for a bumpy metric of positive scalar curvature the space of minimal surfaces having a uniform upper bound on the Morse index is always finite unless the manifold itself contains an embedded minimal RP^2. In particular, we derive a generic finiteness result whenever N does not contain a copy of RP^3 in its prime decomposition. We discuss the obstructions to any further generalization of such a result. When the metric g is required to be (scalar positive and) strongly bumpy (meaning that all closed, immersed minimal surfaces do not have Jacobi fields, a notion recently proved to be generic by B. White) the same conclusion holds true for any closed 3-manifold.