# STRUCTURE OF STABLE MINIMAL HYPERSURFACES IN A RIEMANNIAN MANIFOLD OF NONNEGATIVE RICCI CURVATURE

• Published : 2013.07.31
• 33 6

#### Abstract

Let N be a complete Riemannian manifold with nonnegative Ricci curvature and let M be a complete noncompact oriented stable minimal hypersurface in N. We prove that if M has at least two ends and ${\int}_M{\mid}A{\mid}^2\;dv={\infty}$, then M admits a nonconstant harmonic function with finite Dirichlet integral, where A is the second fundamental form of M. We also show that the space of $L^2$ harmonic 1-forms on such a stable minimal hypersurface is not trivial. Our result is a generalization of one of main results in [12] because if N has nonnegative sectional curvature, then M admits no nonconstant harmonic functions with finite Dirichlet integral. And our result recovers a main theorem in [3] as a corollary.

#### Keywords

stable minimal hypersurface;end;$L^2$ harmonic form;parabolicity;non-parabolicity

#### References

1. F. Almgren, Jr., Some interior regularity theorems for minimal surfaces and an extension of Bernstein's Theorem, Ann. of Math. (2) 84 (1966), 277-292. https://doi.org/10.2307/1970520
2. E. Bombieri, E. DeGiorgi, and E. Guisti, Minimal cones and the Bernstein problem, Invent. Math. 7 (1969), 243-268. https://doi.org/10.1007/BF01404309
3. H.-D. Cao, Y. Shen, and S. Zhu, The structure of stable minimal hypersurfaces in $R^{n+1}$, Math. Res. Lett. 4 (1997), no. 5, 637-644. https://doi.org/10.4310/MRL.1997.v4.n5.a2
4. S. S. Chern, Minimal Submanifolds in a Riemannian Manifolds, University of Kansas, 1968.
5. E. DeGiorgi, Una estensione del teorema di Bernstein, Ann. Scuola Norm. Sup. Pisa 19 (1965), 79-85.
6. M. do Carmo and C. K. Peng, Stable complete minimal surfaces in $\mathbb{R}^3$ are planes, Bull. Amer. Math. Soc. (N.S.) 1 (1979), no. 6, 903-906. https://doi.org/10.1090/S0273-0979-1979-14689-5
7. M. do Carmo and C. K. Peng, Stable minimal hypersufaces, Proceedings of the 1980 Beijing Symposium on Differential Geometry and Differential Equations, Vol. 1, 2, 3 (Beijing, 1980), 1349-1358, Sci. Press Beijing, Beijing, 1982.
8. D. Fischer-Colbrie and R. Schoen, The structure of complete stable minimal surfaces in 3-manifolds of nonnegative scalar curvature, Comm. Pure Appl. Math. 33 (1980), no. 2, 199-211. https://doi.org/10.1002/cpa.3160330206
9. W. Fleming, On the oriented Plateau problem, Rend. Circ. Mat. Palerimo 11 (1962), 69-90. https://doi.org/10.1007/BF02849427
10. A. Grigor'yon, On the existence of positive fundamental solution of the Laplacian equation on Riemannian manifolds, Math. USSR Sbornik 56 (1987), 349-358. https://doi.org/10.1070/SM1987v056n02ABEH003040
11. P. Li, Curvature and Function Theory on Riemannian Manifolds, Surveys in differential geometry, 375-32, Surv. Differ. Geom., VII, Int. Press, Somerville, MA, 2000.
12. P. Li and J. Wang, Stable minimal hypersurfaces in a nonnegatively curved manifold, J. Reine Angew. Math. 566 (2004), 215-230.
13. R. Miyaoka, Harmonic 1-forms on a complete stable minimal hypersurfaces, Geometry and global analysis (Sendai, 1993), 289-293, Tohoku Univ., Sendai, 1993.
14. R. Schoen, L. Simon, and S.-T. Yau, Curvature estimates for minimal hypersurfaces, Acta Math. 134 (1975), no. 3-4, 275-288. https://doi.org/10.1007/BF02392104
15. K. Seo, Rigidity of minimal submanifolds in hyperbolic space, Arch. Math. 94 (2010), no. 2, 173-181. https://doi.org/10.1007/s00013-009-0096-2
16. Y. Shen and S. Zhu, Rigidity of stable minimal hypersurfaces, Math. Ann. 309 (1997), no. 1, 107-116. https://doi.org/10.1007/s002080050105
17. J. Simons, Minimal varieties in Riemannian manifolds, Ann. of Math. (2) 88 (1968), 62-05. https://doi.org/10.2307/1970556
18. N. Th. Varopoulos, Potential theory and diffusion on Riemannian manifolds, Conference on harmonic analysis in honor of Antoni Zygmund, Vol. I, II (Chicago, Ill., 1981), 821-837, Wadsworth Math. Ser., Wadsworth, Belmont, CA, 1983.
19. G. Yun, Stable minimal hypersurfaces in locally symmetric spaces, Math. Nachr. 280 (2007), no. 15, 1744-1751. https://doi.org/10.1002/mana.200410575
20. G. Yun, Total scalar curvature and $L^2$ harmonic 1-forms on a minimal hypersurface in Euclidean space, Geom. Dedicata 89 (2002), 135-141.

#### Acknowledgement

Supported by : National Research Foundation of Korea(NRF)