Minimal Hypersurfaces with Finite Index
Minimal Hypersurfaces with Finite Index
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DOI：
10.4310/mrl.2002.v9.n1.a7
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发表时间：
2002
影响因子：
1
通讯作者：
Peter Li;Jiaping Wang
中科院分区：
文献类型：
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作者：
Peter Li;Jiaping Wang
在曹沈朱[C-S-Z]的一篇文章中，他们证明了R中n ≥ 3的完备、浸入、稳定的极小超曲面M必只有一端。当n = 2时，Do Carmo-Peng [dC-P]和FischerColbrie-Schoen [FC-S]分别独立地证明了R中完备的、浸入的、定向的稳定极小曲面必为平面。后来Gulliver [G]和Fischer-Colbrie [FC]证明了如果R中的一个完备的浸入极小曲面有有限指标，那么它一定共形等价于一个有1000个穿刺的紧致Riemann曲面。Fischer-Colbrie实际上证明了这一点的极小曲面在一个完整的流形与非负标量曲率。在任何情况下，一个推论是，如果R中的一个完全的、浸入的、定向的极小曲面有有限的指数，那么它一定有多个端点。本文的目的是将这一结果推广到欧氏空间中的高维极小超曲面（定理5）。事实上，我们还将证明这样的流形的第一个L-Betti数必须是有限的。曹沈朱的策略是利用Schoen-Yau [S-Y]的一个结果，即R的完备的、稳定的极小超曲面不允许具有有限Dirichlet积分的非常数调和函数。在M有多个端点的假设下，曹沈朱构造了一个具有有限Dirichlet积分的非常数调和函数。这种方法非常适合第一作者和Tam在[L-T]中研究的方案。事实上，作者证明了任何完备黎曼流形的非抛物端的个数都是由具有有限Dirichlet积分的有界调和函数空间的维数上有界的。Cao-Shen-Zhu的证明可以修改，以证明一个完全的浸入极小子流形的每个端点都必须是非抛物的。由于这种联系
In an article of Cao-Shen-Zhu [C-S-Z], they proved that a complete, immersed, stable minimal hypersurface M of R with n ≥ 3 must have only one end. When n = 2, it was proved independently by do Carmo-Peng [dC-P] and FischerColbrie-Schoen [FC-S] that a complete, immersed, oriented stable minimal surface in R must be a plane. Later Gulliver [G] and Fischer-Colbrie [FC] proved that if a complete, immersed, minimal surface in R has finite index, then it must be conformally equivalent to a compact Riemann surface with finitely many punctures. Fischer-Colbrie actually proved this for minimal surfaces in a complete manifold with non-negative scalar curvature. In any event, a corollary is that if a complete, immersed, oriented minimal surface in R has finite index then it must have finitely many ends. The purpose of this paper is to generalize this result for finitely many ends to higher dimensional minimal hypersurfaces in Euclidean space (see Theorem 5). In fact, we will also show that the first L-Betti number of such a manifold must be finite. The strategy of Cao-Shen-Zhu was to utilize a result a Schoen-Yau [S-Y] asserting that a complete, stable minimal hypersurface of R cannot admit a non-constant harmonic function with finite Dirichlet integral. Assuming that M has more than one end, Cao-Shen-Zhu constructed a non-constant harmonic function with finite Dirichlet integral. This approach very much fits into the scheme studied by the first author and Tam in [L-T]. In fact, the authors showed that the number of non-parabolic ends of any complete Riemannian manifold is bounded above by the dimension of the space of bounded harmonic functions with finite Dirichlet integral. The proof of Cao-Shen-Zhu can be modified to show that each end of a complete, immersed, minimal submanifold must be non-parabolic. Due to this connection