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Mean Curvature Flow Solitons in the Presence of Conformal Vector Fields

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Abstract

In this paper we introduce and study a notion of mean curvature flow soliton in Riemannian ambient spaces general enough to encompass target spaces of constant sectional curvature, Riemannian products or, in increasing generality, warped product spaces. As expected, our definition is motivated by the self-similarity of certain special solutions of the mean curvature flow with respect to the flow generated by a distinguished vector field on the target manifold. Our approach allows us to identify some natural geometric quantities that satisfy elliptic equations or differential inequalities in a simple and manageable form for which the machinery of weak maximum principles is valid. The latter is one of the main tools we apply to derive several new characterizations and rigidity results for mean curvature flow solitons that extend to our much more general setting known properties, for instance, in Euclidean space.

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Authors and Affiliations

  1. Departamento de Matemáticas, Universidad de Murcia, 30100, Espinardo, Murcia, Spain

    Luis J. Alías

  2. Departamento de Matematica, Universidade Federal do Ceará, Bloco 914, Campus do Pici, Fortaleza, Ceará, 60455-760, Brazil

    Jorge H. de Lira

  3. Dipartimento di Matematica, Universita degli Studi di Milano, Via Saldini 50, 20133, Milan, Italy

    Marco Rigoli

Authors
  1. Luis J. Alías
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  2. Jorge H. de Lira
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  3. Marco Rigoli
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Correspondence to Luis J. Alías.

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This research of L.J. Alías 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. Partially supported by MINECO/FEDER Project Reference MTM2015-65430-P and Fundación Séneca Project Reference 19901/GERM/15, Spain. J.H. de Lira is partially supported by CNPq Produtividade em Pesquisa Grant \(\#\) 302067/2014-0 and FUNCAP/CNPq/PRONEX Grant “Núcleo de Análise Geométrica e Aplicações” \(\#\) 09.01.00/11.

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Alías, L.J., de Lira, J.H. & Rigoli, M. Mean Curvature Flow Solitons in the Presence of Conformal Vector Fields. J Geom Anal 30, 1466–1529 (2020). https://doi.org/10.1007/s12220-019-00186-3

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