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Stability properties and topology at infinity of \(f\)-minimal hypersurfaces

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Abstract

We study stability properties of \(f\)-minimal hypersurfaces isometrically immersed in weighted manifolds with non-negative Bakry–Émery Ricci curvature under volume growth conditions. Moreover, exploiting a weighted version of a finiteness result and the adaptation to this setting of Li–Tam theory, we investigate the topology at infinity of \(f\)-minimal hypersurfaces. On the way, we prove a new comparison result in weighted geometry and we provide a general weighted \(L^1\)-Sobolev inequality for hypersurfaces in Cartan–Hadamard weighted manifolds, satisfying suitable restrictions on the weight function.

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Acknowledgments

Part of this work was done while we were visiting the Institut Henri Poincaré, Paris. We would like to thank the institute for the warm hospitality. Moreover we are deeply grateful to Stefano Pigola for useful conversations during the preparation of the manuscript. We would also like to thank Marcio Batista and Heudson Mirandola and the anonymous referee for useful comments.

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  1. Dipartimento di Matematica e Applicazioni, Università degli Studi di Milano Bicocca, Via Cozzi 55, 20125, Milan, Italy

    Debora Impera & Michele Rimoldi

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  1. Debora Impera
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  2. Michele Rimoldi
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Correspondence to Debora Impera.

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Impera, D., Rimoldi, M. Stability properties and topology at infinity of \(f\)-minimal hypersurfaces. Geom Dedicata 178, 21–47 (2015). https://doi.org/10.1007/s10711-014-9999-6

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