Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Some characterizations of space-forms

Authors: Stefano Pigola, Marco Rigoli and Alberto G. Setti
Journal: Trans. Amer. Math. Soc. 359 (2007), 1817-1828
MSC (2000): Primary 53C21; Secondary 35J60, 35B05
Published electronically: November 3, 2006
Erratum: Tran. Amer. Math. Soc. 360 (2008), 3943-3944.
MathSciNet review: 2272150
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Integral conditions on the traceless Ricci tensor are used to characterize Euclidean and hyperbolic spaces among complete, locally conformally flat manifolds of constant scalar curvature. The main tools are vanishing-type results for $ L^{p}$-solutions of a large class of differential inequalities. Further applications of the technique are also given.

References [Enhancements On Off] (What's this?)

  • [A] M. Anderson, The compactification of a minimal submanifold in Euclidean space by the Gauss map. Preprint.
  • [B] P. Bérard, Remarques sur l'équation de J. Simons. Differential geometry, 47-57, Pitman Monogr. Surveys Pure Appl. Math., 52, Longman Sci. Tech., Harlow, 1991. MR 1173032 (93g:53082)
  • [Bo] J.P. Bourguignon,The magic of Weitzenböck formulas, Variational Methods Paris (1998), 251-271. Progess in Nonlinear Differential Equations and Applications IV, Birkauser, 1990. MR 1205158 (94a:58181)
  • [Ca1] G. Carron, Une suite exacte en $ L^{2}$ -cohomologie.Duke Math. J. 95 (1998), 343-372. MR 1652017 (99g:58115)
  • [Ca2] G. Carron, $ L^2$-Cohomologie et inégalités de Sobolev, Math. Ann. 314 (1999), 614-639. MR 1709104 (2000f:53045)
  • [CGS] L. Caffarelli, B. Gidas, J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth. Comm. Pure Appl. Math. 42 (1989), 271-297. MR 0982351 (90c:35075)
  • [CSZ] H.-D. Cao, Y. Shen, S. Zhu, The structure of stable minimal hypersurfaces in $ \mathbb{R}^{m+1}$. Math. Res. Lett. 4 (1997), 637-644. MR 1484695 (99a:53037)
  • [CW] Q.M. Cheng, B.Q. Wu, The generalized maximum principle and conformally flat spaces, Northeastern Math. J. 8 (1992), 54-56. MR 1164748 (93g:53052)
  • [F] A. Friedman, Partial Differential Equations, Holt, Rinehart and Winston, Inc., New York 1970. MR 0445088 (56:3433)
  • [G] S.I. Goldberg, An application of Yau's maximum principle to conformally flat spaces. Proc. Amer. Math. Soc. 79 (1980), 268-270. MR 0565352 (81j:53043)
  • [H] T. Hasanis, Conformally flat spaces and a pinching problem for the Ricci tensor. Proc. Amer. Math. Soc. 86 (1982), 312-315. MR 0667296 (84k:53043)
  • [HS] D. Hoffman, J. Spruck, Sobolev and isoperimetric inequalities for Riemannian submanifolds. Comm. Pure Appl. Math. 27 (1974), 715-727. MR 0365424 (51:1676)
  • [K] N. Kuiper, On conformally flat spaces in the large, Annals of Math. 50 (1949), 916-924. MR 0031310 (11:133b)
  • [LW] P. Li and J. Wang, Complete manifolds with positive spectrum, J. Diff. Geom. 58 (2001), 501-534. MR 1906784 (2003e:58046)
  • [N] L. Ni, Gap theorems for minimal submanifolds in $ \mathbb{R}^{n+1}$. Comm. Anal. Geom. 9 (2001), 641-656. MR 1895136 (2002m:53097)
  • [O1] M. Okumura, Hypersurfaces and a pinching problem on the second fundamental tensor. Amer. J. Math. 96 (1974), 207-213. MR 0353216 (50:5701)
  • [O2] M. Okumura, Submanifolds and a pinching problem on the second fundamental tensors. Trans. Amer. Math. Soc. 178 (1973), 285-291. MR 0317246 (47:5793)
  • [PRS] S. Pigola, M. Rigoli, A.G. Setti, Vanishing theorems on Riemannian manifolds and geometric applications. J. Funct. Anal. 229 (2005), 424-461. MR 2182595
  • [SY] R. Schoen, S.T. Yau, Lectures on differential geometry. Conference Proceedings and Lecture Notes in Geometry and Topology, I. International Press, Cambridge, MA, 1994. MR 1333601 (97d:53001)
  • [T] M. Tani, On a conformally flat Riemannian space with positive Ricci curvature. Tôhoku Math. J. (2) 19 (1967), 227-231. MR 0220213 (36:3279)
  • [Z] S. Zhu, The classification of complete conformally flat manifolds of nonnegative Ricci curvature, Pacific J. Math. 163 (1994), 189-199. MR 1256184 (95d:53045)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 53C21, 35J60, 35B05

Retrieve articles in all journals with MSC (2000): 53C21, 35J60, 35B05

Additional Information

Stefano Pigola
Affiliation: Dipartimento di Fisica e Matematica, Università dell’Insubria - Como, via Valleggio 11, I-22100 Como, Italy

Marco Rigoli
Affiliation: Dipartimento di Matematica, Università di Milano, via Saldini 50, I-20133 Milano, Italy

Alberto G. Setti
Affiliation: Dipartimento di Fisica e Matematica, Università dell’Insubria - Como, via Valleggio 11, I-22100 Como, Italy

Keywords: Space forms, vanishing theorems, isolation phenomena
Received by editor(s): January 29, 2005
Published electronically: November 3, 2006
Dedicated: Dedicated to the memory of Franca Burrone Rigoli
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society