A maximum principle related to volume growth and applications

DOI10.1007/S10231-020-01051-9zbMATH Open1469.53104arXiv2001.07079OpenAlexW3101641036MaRDI QIDQ2035960

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Publication date: 2 July 2021

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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 $langleablaf,Xanglegeq0$ on $M$ and $mathrmdivXgeqaf$ 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 straightforward applications to smooth functions and, more interestingly, to 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.

Full work available at URL: https://arxiv.org/abs/2001.07079

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