An estimate for the sectional curvature of cylindrically bounded submanifolds

Alías, Luis; Bessa, G.; Montenegro, J.

doi:10.1090/S0002-9947-2012-05439-0

An estimate for the sectional curvature of cylindrically bounded submanifolds
HTML articles powered by AMS MathViewer

by Luis J. Alías, G. Pacelli Bessa and J. Fabio Montenegro PDF

Trans. Amer. Math. Soc. 364 (2012), 3513-3528 Request permission

Abstract:

We give sharp sectional curvature estimates for complete immersed cylindrically bounded $m$-submanifolds $\varphi \colon M^m \to N^{n-\ell }\times \mathbb {R}^{\ell }$, $n+\ell \leq 2m-1$, provided that either $\varphi$ is proper with the norm of the second fundamental form with certain controlled growth or $M$ has scalar curvature with strong quadratic decay. The latter gives a non-trivial extension of the Jorge-Koutrofiotis Theorem. In the particular case of hypersurfaces, that is, $m=n-1$, the growth rate of the norm of the second fundamental form is improved. Our results will be an application of a generalized Omori-Yau Maximum Principle for the Hessian of a Riemannian manifold, in its newest elaboration given by Pigola, Rigoli and Setti (2005).

References

Luis J. Alías, G. Pacelli Bessa, and Marcos Dajczer, The mean curvature of cylindrically bounded submanifolds, Math. Ann. 345 (2009), no. 2, 367–376. MR 2529479, DOI 10.1007/s00208-009-0357-1
G. Pacelli Bessa and J. Fabio Montenegro, On compact $H$-hypersurfaces of $N\times \Bbb R$, Geom. Dedicata 127 (2007), 1–5. MR 2338510, DOI 10.1007/s10711-007-9145-9
E. Calabi, Problems in Differential Geometry (S. Kobayashi and J. Eells, Jr., eds.) Proc. of the United States-Japan Seminar in Differential Geometry, Kyoto, Japan, 1965, Nippon Hyoronsha Co. Ltd., Tokyo (1966) 170.
Q. Chen and Y. L. Xin, A generalized maximum principle and its applications in geometry, Amer. J. Math. 114 (1992), no. 2, 355–366. MR 1156569, DOI 10.2307/2374707
S. Y. Cheng and S. T. Yau, Differential equations on Riemannian manifolds and their geometric applications, Comm. Pure Appl. Math. 28 (1975), no. 3, 333–354. MR 385749, DOI 10.1002/cpa.3160280303
S. S. Chern, The geometry of $G$-structures, Bull. Amer. Math. Soc. 72 (1966), 167–219. MR 192436, DOI 10.1090/S0002-9904-1966-11473-8
Shiing-shen Chern and Nicolaas H. Kuiper, Some theorems on the isometric imbedding of compact Riemann manifolds in euclidean space, Ann. of Math. (2) 56 (1952), 422–430. MR 50962, DOI 10.2307/1969650
Cláudio Carlos Dias, Isometric immersions with slow growth of curvature, An. Acad. Brasil. Ciênc. 54 (1982), no. 2, 293–295. MR 680268
N. V. Efimov, Hyperbolic problems in the theory of surfaces, Proc. Internat. Congr. Math. (Moscow, 1966) Izdat. “Mir”, Moscow, 1968, pp. 177–188 (Russian). MR 0238232
Fernando Giménez, Estimates for the curvature of a submanifold which lies inside a tube, J. Geom. 58 (1997), no. 1-2, 95–105. MR 1434182, DOI 10.1007/BF01222930
Th. Hasanis and D. Koutroufiotis, Immersions of Riemannian manifolds into cylinders, Arch. Math. (Basel) 40 (1983), no. 1, 82–85. MR 720897, DOI 10.1007/BF01192755
David Hilbert, Ueber Flächen von constanter Gaussscher Krümmung, Trans. Amer. Math. Soc. 2 (1901), no. 1, 87–99 (German). MR 1500557, DOI 10.1090/S0002-9947-1901-1500557-5
L. Jorge and D. Koutroufiotis, An estimate for the curvature of bounded submanifolds, Amer. J. Math. 103 (1981), no. 4, 711–725. MR 623135, DOI 10.2307/2374048
Luquésio P. de M. Jorge and Frederico V. Xavier, An inequality between the exterior diameter and the mean curvature of bounded immersions, Math. Z. 178 (1981), no. 1, 77–82. MR 627095, DOI 10.1007/BF01218372
Shoshichi Kobayashi and Katsumi Nomizu, Foundations of differential geometry. Vol. II, Interscience Tracts in Pure and Applied Mathematics, No. 15 Vol. II, Interscience Publishers John Wiley & Sons, Inc., New York-London-Sydney, 1969. MR 0238225
John Douglas Moore, An application of second variation to submanifold theory, Duke Math. J. 42 (1975), 191–193. MR 377776
John Nash, The imbedding problem for Riemannian manifolds, Ann. of Math. (2) 63 (1956), 20–63. MR 75639, DOI 10.2307/1969989
Hideki Omori, Isometric immersions of Riemannian manifolds, J. Math. Soc. Japan 19 (1967), 205–214. MR 215259, DOI 10.2969/jmsj/01920205
Barrett O’Neill, Immersion of manifolds of nonpositive curvature, Proc. Amer. Math. Soc. 11 (1960), 132–134. MR 117683, DOI 10.1090/S0002-9939-1960-0117683-0
Tominosuke Ôtsuki, Isometric imbedding of Riemann manifolds in a Riemann manifold, J. Math. Soc. Japan 6 (1954), 221–234. MR 67550, DOI 10.2969/jmsj/00630221
Stefano Pigola, Marco Rigoli, and Alberto G. Setti, Maximum principles on Riemannian manifolds and applications, Mem. Amer. Math. Soc. 174 (2005), no. 822, x+99. MR 2116555, DOI 10.1090/memo/0822
R. Schoen and S.-T. Yau, Lectures on differential geometry, Conference Proceedings and Lecture Notes in Geometry and Topology, I, International Press, Cambridge, MA, 1994. Lecture notes prepared by Wei Yue Ding, Kung Ching Chang [Gong Qing Zhang], Jia Qing Zhong and Yi Chao Xu; Translated from the Chinese by Ding and S. Y. Cheng; With a preface translated from the Chinese by Kaising Tso. MR 1333601
Edsel F. Stiel, Immersions into manifolds of constant negative curvature, Proc. Amer. Math. Soc. 18 (1967), 713–715. MR 212743, DOI 10.1090/S0002-9939-1967-0212743-4
C. Tompkins, Isometric embedding of flat manifolds in Euclidean space, Duke Math. J. 5 (1939), no. 1, 58–61. MR 1546106, DOI 10.1215/S0012-7094-39-00507-7
Shing Tung Yau, Harmonic functions on complete Riemannian manifolds, Comm. Pure Appl. Math. 28 (1975), 201–228. MR 431040, DOI 10.1002/cpa.3160280203