A characterization of revolution quadrics by a system of partial differential equations
Author:
Vladimir I. Oliker
Journal:
Proc. Amer. Math. Soc. 138 (2010), 40754080
MSC (2010):
Primary 53A05, 53C40
Published electronically:
June 29, 2010
MathSciNet review:
2679628
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Abstract: It is shown that existence of a global solution to a particular nonlinear system of second order partial differential equations on a complete connected Riemannian manifold has topological and geometric implications and that in the domain of positivity of such a solution, its reciprocal is the radial function of only one of the following rotationally symmetric hypersurfaces in : paraboloid, ellipsoid, one sheet of a twosheeted hyperboloid, and a hyperplane.
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A characterization of quadrics by the principal curvature functions. Arch. Math. (Basel), 81:342347, 2003. MR 2013266 (2004h:53085)
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Private communication.
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Prescribing symmetric functions of the eigenvalues of the Ricci tensor. Ann. of Math. (2), 166(2):475531, 2007. MR 2373147 (2008k:53068)
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Differential Geometry, Curves  Surfaces  Manifolds, 2nd edition. Translated from the German by B. Hunt. AMS Student Mathematical Library Series, vol. 16, 2006. MR 1882174 (2002k:53001)
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On closed Weingarten surfaces. Monatshefte Math., 146:113126, 2005. MR 2176338 (2006g:53005)
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 F.J. Lange and U. Simon.
Eigenvalues and eigenfunctions of Riemannian manifolds. Proc. of the AMS, 7(2):237242, 1979. MR 542091 (80h:58053)
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 A.M. Li, U. Simon, and G. Zhao.
Global Affine Differential Geometry of Hypersurfaces. W. de Gruyter, BerlinNew York, 1993. MR 1257186 (95e:53016)
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 E. Lutwak.
On some affine isoperimetric inequalities. J. of Diff. Geometry, 23:113, 1986. MR 840399 (87k:52030)
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 M. Obata.
Certain conditions for a Riemannian manifold to be isometric with a sphere. J. Math. Soc. Japan, 14(3):333340, 1962. MR 0142086 (25:5479)
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 M. Obata.
The conjectures on conformal transformations of Riemannian manifolds. J. Differential Geometry, 6(3):247258, 1971/72. MR 0303464 (46:2601)
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Kummer configurations and reflector problems: Hypersurfaces in with given mean intensity, Result. Math. 56:519535, 2009. MR 2575876
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Additional Information
Vladimir I. Oliker
Affiliation:
Department of Mathematics and Computer Science, Emory University, Atlanta, Georgia 30322
Email:
oliker@mathcs.emory.edu
DOI:
http://dx.doi.org/10.1090/S000299392010104392
Keywords:
Hypersurfaces of revolution,
Obata’s theorem
Received by editor(s):
June 6, 2008
Received by editor(s) in revised form:
February 20, 2009
Published electronically:
June 29, 2010
Additional Notes:
The research of the author was partially supported by National Science Foundation grant DMS0405622.
Communicated by:
Matthew J. Gursky
Article copyright:
© Copyright 2010
American Mathematical Society
