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Graphical mean curvature flow with bounded bi-Ricci curvature

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Abstract

We consider the graphical mean curvature flow of strictly area decreasing maps \(f:M\rightarrow N\), where M is a compact Riemannian manifold of dimension \(m> 1\) and N a complete Riemannian surface of bounded geometry. We prove long-time existence of the flow and that the strictly area decreasing property is preserved, when the bi-Ricci curvature \({\text {BRic}}_M\) of M is bounded from below by the sectional curvature \(\sigma _N\) of N. In addition, we obtain smooth convergence to a minimal map if \({\text {Ric}}_M\ge \sup \{0,{\sup }_N\sigma _N\}\). These results significantly improve known results on the graphical mean curvature flow in codimension 2.

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Notes

  1. Norms are with respect to the metrics induced by the corresponding immersions.

  2. In this article we assume manifolds are connected.

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Authors and Affiliations

  1. Institute of Differential Geometry, Leibniz University Hannover, Welfengarten 1, 30167, Hannover, Germany

    Renan Assimos

  2. Department of Mathematics, Section of Algebra and Geometry, University of Ioannina, 45110, Ioannina, Greece

    Andreas Savas-Halilaj

  3. Riemann Center for Geometry and Physics, Institute of Differential Geometry, Leibniz University Hannover, Welfengarten 1, 30167, Hannover, Germany

    Knut Smoczyk

Authors
  1. Renan Assimos
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  2. Andreas Savas-Halilaj
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  3. Knut Smoczyk
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Correspondence to Andreas Savas-Halilaj.

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Communicated by A. Mondino.

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The second author is supported by HFRI: Grant 133, and the third by DFG SM 78/7-1.

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Assimos, R., Savas-Halilaj, A. & Smoczyk, K. Graphical mean curvature flow with bounded bi-Ricci curvature. Calc. Var. 62, 12 (2023). https://doi.org/10.1007/s00526-022-02369-3

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