Mean Curvature Flow Solitons in the Presence of Conformal Vector Fields

  • Luis J. AlíasEmail author
  • Jorge H. de Lira
  • Marco Rigoli


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.


Mean curvature Conformal fields Mean curvature flow solitons Maximum principle Weak maximum principle Parabolicity 

Mathematics Subject Classification

Primary 53C44 Secondary 53C42 



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Copyright information

© Mathematica Josephina, Inc. 2019

Authors and Affiliations

  1. 1.Departamento de MatemáticasUniversidad de MurciaEspinardoSpain
  2. 2.Departamento de MatematicaUniversidade Federal do CearáFortalezaBrazil
  3. 3.Dipartimento di MatematicaUniversita degli Studi di MilanoMilanItaly

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