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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Heinz type estimates for graphs in Euclidean space


Author: Francisco Fontenele
Journal: Proc. Amer. Math. Soc. 138 (2010), 4469-4478
MSC (2010): Primary 53A07, 53C42; Secondary 53A05, 35B50
Published electronically: August 2, 2010
MathSciNet review: 2680071
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Abstract: Let $ M^n$ be an entire graph in the Euclidean $ (n+1)$-space $ \mathbb{R}^{n+1}$. Denote by $ H$, $ R$ and $ \vert A\vert$, respectively, the mean curvature, the scalar curvature and the length of the second fundamental form of $ M^n$. We prove that if the mean curvature $ H$ of $ M^n$ is bounded, then $ \inf_M\vert R\vert=0$, improving results of Elbert and Hasanis-Vlachos. We also prove that if the Ricci curvature of $ M^n$ is negative, then $ \inf_M\vert A\vert=0$. The latter improves a result of Chern as well as gives a partial answer to a question raised by Smith-Xavier. Our technique is to estimate $ \inf\vert H\vert,\;\inf\vert R\vert$ and $ \inf\vert A\vert$ for graphs in $ \mathbb{R}^{n+1}$ of $ C^2$ real-valued functions defined on closed balls in $ \mathbb{R}^n$.


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Additional Information

Francisco Fontenele
Affiliation: Departamento de Geometria, Universidade Federal Fluminense, 24020-140, Niterói, Rio de Janeiro, Brazil
Email: fontenele@mat.uff.br

DOI: http://dx.doi.org/10.1090/S0002-9939-2010-10590-7
PII: S 0002-9939(2010)10590-7
Keywords: Curvature estimates for graphs in Euclidean space, negative Ricci curvature, Efimov’s theorem
Received by editor(s): August 31, 2009
Published electronically: August 2, 2010
Additional Notes: This work was partially supported by CNPq (Brazil)
Dedicated: To my wife, Andrea
Communicated by: Jon G. Wolfson
Article copyright: © Copyright 2010 American Mathematical Society