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Lagrangian Homothetic Solitons for the Inverse Mean Curvature Flow

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Abstract

We study solutions to the inverse mean curvature flow which evolve by homotheties of a given submanifold with arbitrary dimension and codimension. We first show that the closed ones are necessarily spherical minimal immersions and so we reveal the strong rigidity of the Clifford torus in this setting. Mainly we focus on the Lagrangian case, obtaining numerous examples and uniqueness results for some products of circles and spheres that generalize the Clifford torus to arbitrary dimension. We also characterize the pseudoumbilical ones in terms of soliton curves for the inverse curve shortening flow and minimal Legendrian immersions in odd-dimensional spheres. As a consequence, we classify the rotationally invariant Lagrangian homothetic solitons for the inverse mean curvature flow.

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Authors and Affiliations

  1. Departamento de Matemáticas, Universidad de Jaén, 23071, Jaén, Spain

    Ildefonso Castro

  2. Departamento de Didáctica de las Ciencias, Universidad de Jaén, 23071, Jaén, Spain

    Ana M. Lerma

Authors
  1. Ildefonso Castro
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  2. Ana M. Lerma
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Correspondence to Ildefonso Castro.

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Research partially supported by a MEC-Feder grant MTM2014-52368-P.

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Castro, I., Lerma, A.M. Lagrangian Homothetic Solitons for the Inverse Mean Curvature Flow. Results Math 71, 1109–1125 (2017). https://doi.org/10.1007/s00025-016-0574-3

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