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Cédric Tarquini 《Comptes Rendus Mathematique》2004,339(3):209-214
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Philippe G. Ciarlet Liliana Gratie Cristinel Mardare 《Comptes Rendus Mathematique》2005,341(3):201-206
The main purpose of this Note is to show how a ‘nonlinear Korn's inequality on a surface’ can be established. This inequality implies in particular the following interesting per se sequential continuity property for a sequence of surfaces. Let ω be a domain in , let be a smooth immersion, and let , , be mappings with the following properties: They belong to the space ; the vector fields normal to the surfaces , , are well defined a.e. in ω and they also belong to the space ; the principal radii of curvature of the surfaces stay uniformly away from zero; and finally, the three fundamental forms of the surfaces converge in toward the three fundamental forms of the surface as . Then, up to proper isometries of , the surfaces converge in toward the surface as . To cite this article: P.G. Ciarlet et al., C. R. Acad. Sci. Paris, Ser. I 341 (2005). 相似文献
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《Journal de Mathématiques Pures et Appliquées》2005,84(10):1295-1361
Let be a symmetric diffusion operator with an invariant measure on a complete non-compact Riemannian manifold M. We give the optimal conditions on “the m-dimensional Ricci curvature associated with L” so that various Liouville theorems hold for L-harmonic functions, and that the heat semigroup has the -diffusion property and is unique in . As applications, we give the optimal conditions for the uniqueness of the positive L-invariant measure and the -uniqueness of the intrinsic Schrödinger operators on complete non-compact Riemannian manifolds. We also give a criterion for the finiteness of the total mass of the L-invariant measure and establish the Calabi–Yau volume growth theorem for the L-invariant measure on complete Riemannian manifolds on which “the m-dimensional Ricci curvature associated with L” is non-negative. This leads us to prove that if M is a complete Riemannian manifold with a finite L-invariant measure for which the associated m-dimensional Ricci curvature is non-negative, then M is compact. Moreover, we obtain an upper bound diameter estimate of such Riemannian manifolds by using the dimension of L, the total μ-volume of M and the upper bound of the μ-volume of geodesic balls of a fixed radius. Finally, using the variational formulae in Riemannian geometry, we give a new proof of the Bakry–Qian generalized Laplacian comparison theorem. 相似文献
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Alexandru Dimca 《Journal of Algebra》2009,321(11):3145-3157