Minimal Hypersurfaces with Finite Index

Abstract

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