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Minimal hypersurfaces with finite index. (English) Zbl 1019.53025

The main result of this paper is the following: Let \(M=M^n\) be a complete immersed oriented minimal hypersurface in \({\mathbb R}^{n+1}\) with \(n \geq 3\). Suppose \(M\) has finite index. Then \(M\) must have finite first \(L^2\)-Betti number, i.e. \(\dim H^1(L^2(M)) < \infty\). In particular, \(M\) must have finitely many ends.
As former works in this direction, one can refer to R. Gulliver [Proc. Symp. Pure Math. 44, 207-211 (1986; Zbl 0592.53006)] and D. Fischer-Colbrie [Invent. Math. 82, 121-132 (1985; Zbl 0573.53038)]. In fact, they obtained more stringent results for \(n=2\). Gulliver proved that if a complete immersed minimal surface in \({\mathbb R}^3\) has finite index, then it must be conformally equivalent to a compact Riemann surface with finitely many punctures. Fischer-Colbrie proved the same result for minimal surfaces in any complete three-dimensional manifold with non-negative scalar curvature.
The essence of the proof of the main result is as follows: By assumption, there exists a compact subset \(\Omega \subset M\) such that \(M \setminus \Omega\) is stable. Now, by using a refinement of the argument of R. Schoen and S.-T. Yau [Comment. Math. Helv. 51, 333-341 (1976; Zbl 0361.53040)], the authors prove \(\dim H^1(L^2(M)) < \infty\), which implies that the dimension of the space \({\mathcal H}^o_D(M)\) of bounded h armonic functions with finite Dirichlet integral is finite. On the other hand, a modification of the method of H.-D. Cao, Y. Shen and S. Zhu [Math. Res. Lett. 4, 637-644 (1997; Zbl 0906.53004)] shows that each end of \(M\) must be non-parabolic. Now, the fact that the number of non-parabolic ends of any complete Riemannian manifold is bound from above by the dimension of \({\mathcal H}^o_D(M)\) [P. Li and L. F. Tam, J. Differ. Geom. 35, 359-383 (1992; Zbl 0768.53018)] implies the main result.
The authors pose a problem of estimating the number of ends by the index directly.

MSC:

53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
53A07 Higher-dimensional and -codimensional surfaces in Euclidean and related \(n\)-spaces
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