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SOME REMARKS ON THE PIGOLA–RIGOLI–SETTI VERSION OF THE OMORI–YAU MAXIMUM PRINCIPLE

Part of: Partial differential equations Global differential geometry

Published online by Cambridge University Press:  19 August 2013

ALEXANDRE PAIVA BARRETO  and
FRANCISCO FONTENELE
ALEXANDRE PAIVA BARRETO*
Affiliation:
Departamento de Matemática, Universidade Federal de São Carlos, São Carlos, SP, Brazil
FRANCISCO FONTENELE
Affiliation:
Departamento de Geometria, Universidade Federal Fluminense, Niterói, RJ, Brazil email fontenele@mat.uff.br
*
✉alexandre@dm.ufscar.br
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Abstract

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We prove that the hypotheses in the Pigola–Rigoli–Setti version of the Omori–Yau maximum principle are logically equivalent to the assumption that the manifold carries a ${C}^{2} $ proper function whose gradient and Hessian (Laplacian) are bounded. In particular, this result extends the scope of the original Omori–Yau principle, formulated in terms of lower bounds for curvature.

Keywords

maximum principlesOmori–Yau maximum principle

MSC classification

Secondary: 53C21: Methods of Riemannian geometry, including PDE methods; curvature restrictions 35B50: Maximum principles
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 89 , Issue 2 , April 2014 , pp. 337 - 342
Copyright
Copyright ©2013 Australian Mathematical Publishing Association Inc. 

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