Simons’ equation and minimal hypersurfaces in space forms

Wang, Biao

doi:10.1090/proc/13781

Simons’ equation and minimal hypersurfaces in space forms
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Proc. Amer. Math. Soc. 146 (2018), 369-383 Request permission

Abstract:

Let $n\geq {}3$ be an integer, and let $\Sigma ^n$ be a non-totally geodesic complete minimal hypersurface immersed in the $(n+1)$-dimensional space form $\overline {M}^{n+1}(c)$, where the constant $c$ denotes the sectional curvature of the space form. If $\Sigma ^n$ satisfies the Simons’ equation (3.9), then either (1) $\Sigma ^n$ is a catenoid if $c\leq {}0$, or (2) $\Sigma ^n$ is a Clifford minimal hypersurface or a compact Ostuki minimal hypersurface if $c>0$. This paper is motivated by a 2009 work of Tam and Zhou.

References

Fabiano Brito and Maria Luiza Leite, A remark on rotational hypersurfaces of $S^n$, Bull. Soc. Math. Belg. Sér. B 42 (1990), no. 3, 303–318. MR 1081607
S. S. Chern, Minimal submanifolds in a Riemannian manifold, University of Kansas, Department of Mathematics Technical Report 19 (New Series), University of Kansas, Lawrence, Kan., 1968. MR 0248648
S. S. Chern, M. do Carmo, and S. Kobayashi, Minimal submanifolds of a sphere with second fundamental form of constant length, Functional Analysis and Related Fields (Proc. Conf. for M. Stone, Univ. Chicago, Chicago, Ill., 1968) Springer, New York, 1970, pp. 59–75. MR 0273546
M. do Carmo and M. Dajczer, Rotation hypersurfaces in spaces of constant curvature, Trans. Amer. Math. Soc. 277 (1983), no. 2, 685–709. MR 694383, DOI 10.1090/S0002-9947-1983-0694383-X
Manfredo P. do Carmo and Nolan R. Wallach, Representations of compact groups and minimal immersions into spheres, J. Differential Geometry 4 (1970), 91–104. MR 266104
Wu-yi Hsiang, On generalization of theorems of A. D. Alexandrov and C. Delaunay on hypersurfaces of constant mean curvature, Duke Math. J. 49 (1982), no. 3, 485–496. MR 672494
Wu-Yi Hsiang, On rotational $W$-hypersurfaces in spaces of constant curvature and generalized laws of sine and cosine, Bull. Inst. Math. Acad. Sinica 11 (1983), no. 3, 349–373. MR 726983
Yongqiang Ji, The goemetry of submanifolds (in Chinese), Science Press Ltd., Beijing, China, 2004.
H. Blaine Lawson Jr., Local rigidity theorems for minimal hypersurfaces, Ann. of Math. (2) 89 (1969), 187–197. MR 238229, DOI 10.2307/1970816
Haizhong Li and Guoxin Wei, Compact embedded rotation hypersurfaces of $S^{n+1}$, Bull. Braz. Math. Soc. (N.S.) 38 (2007), no. 1, 81–99. MR 2302750, DOI 10.1007/s00574-007-0037-2
Tominosuke Ôtsuki, Minimal hypersurfaces in a Riemannian manifold of constant curvature, Amer. J. Math. 92 (1970), 145–173. MR 264565, DOI 10.2307/2373502
Tominosuke Ôtsuki, On integral inequalities related with a certain nonlinear differential equation, Proc. Japan Acad. 48 (1972), 9–12. MR 308521
R. Schoen, L. Simon, and S. T. Yau, Curvature estimates for minimal hypersurfaces, Acta Math. 134 (1975), no. 3-4, 275–288. MR 423263, DOI 10.1007/BF02392104
James Simons, Minimal varieties in riemannian manifolds, Ann. of Math. (2) 88 (1968), 62–105. MR 233295, DOI 10.2307/1970556
Michael Spivak, A comprehensive introduction to differential geometry. Vol. V, Publish or Perish, Inc., Boston, Mass., 1975. MR 0394453
Tsunero Takahashi, Minimal immersions of Riemannian manifolds, J. Math. Soc. Japan 18 (1966), 380–385. MR 198393, DOI 10.2969/jmsj/01840380
Luen-Fai Tam and Detang Zhou, Stability properties for the higher dimensional catenoid in $\Bbb R^{n+1}$, Proc. Amer. Math. Soc. 137 (2009), no. 10, 3451–3461. MR 2515414, DOI 10.1090/S0002-9939-09-09962-6
Yuanlong Xin, Minimal submanifolds and related topics, Nankai Tracts in Mathematics, vol. 8, World Scientific Publishing Co., Inc., River Edge, NJ, 2003. MR 2035469, DOI 10.1142/9789812564382