A maximum principle related to volume growth and applications
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A maximum principle related to volume growth and applications Luis J. Alías1 · Antonio Caminha2 · F. Yure do Nascimento3 Received: 15 April 2020 / Accepted: 24 October 2020 © Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract In this paper, we derive a new form of maximum principle for smooth functions on a complete noncompact Riemannian manifold M for which there exists a bounded vector field X such that ⟨∇f , X⟩ ≥ 0 on M and divX ≥ af outside a suitable compact subset of M, for some constant a > 0 , under the assumption that M has either polynomial or exponential volume growth. We then use it to obtain some Bernstein-type results for hypersurfaces immersed into a Riemannian manifold endowed with a Killing vector field, as well as to some results on the existence and size of minimal submanifolds immersed into a Riemannian manifold endowed with a conformal vector field. Keywords Maximum principle · Riemannian manifolds · Volume growth · Bernstein-type results · Constant mean curvature hypersurfaces · Minimal submanifolds Mathematics Subject Classification Primary 53C42 · Secondary 53B30 · 53C50 · 53Z05 · 83C99
L. J. Alías: This research is a result of the activity developed within the framework of the Programme in Support of Excellence Groups of the Región de Murcia, Spain, by Fundación Séneca, Science and Technology Agency of the Región de Murcia. Luis J. Alías was partially supported by MICINN/FEDER project PGC2018-097046-B-I00 and Fundación Séneca project 19901/GERM/15, Spain. Antonio Caminha was partially supported by PRONEX/FUNCAP/CNPQ-PR2-0101-00089.01.00/15 and Estímulo à Cooperação Científica e Desenvolvimento da Pós-Graduação Funcap/CAPES Projeto no 88887.165862/2018-00, Brazil. * Luis J. Alías [email protected] Antonio Caminha [email protected] F. Yure do Nascimento [email protected] 1
Departamento de Matemáticas, Universidad de Murcia, E‑30100 Espinardo, Murcia, Spain
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Departamento de Matemática, Universidade Federal do Ceará, Campus do Pici, Fortaleza‑Ce 60455‑760, Brazil
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Universidade Federal do Ceará, BR 226, Km 4, Crateús, Ceará, Brazil
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1 Introduction Maximum principles appear naturally in differential geometry, due to the fact that many different geometric situations are analytically modeled by certain linear or quasilinear elliptic partial differential operators, for which several versions of maximum principles play a key role in the theory. See, for instance, [3] or [9] for two recent monographies on the topic corroborating this fact. In a recent paper of us [1], we derived a new form of maximum principle which is appropriate for controlling the behavior of a smooth vector field with nonnegative divergence on a complete noncompact Riemannian manifold, and which is the analogue of the simple fact that, on such a manifold, a nonnegative subharmonic function that vanishes at infinity actually vanishes identically (Theorem 2.2 in [1]). In this paper, we derive a maximum principl
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