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Legendrian Mean Curvature Flow in \(\eta \)-Einstein Sasakian Manifolds

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Abstract

Recently, there are a great deal of work done which connects the Legendrian isotopic problem with contact invariants. The isotopic problem of Legendre curve in a contact 3-manifold was studied via the Legendrian curve shortening flow which was introduced and studied by K. Smoczyk. On the other hand, in the SYZ Conjecture, one can model a special Lagrangian singularity locally as the special Lagrangian cones in \({\mathbb {C}}^{3}\). This can be characterized by its link which is a minimal Legendrian surface in the 5-sphere. Then in these points of view, we will focus on the existence of the long-time solution and asymptotic convergnce along the Legendrian mean curvature flow in the \((2n+1)\)-dimensional \(\eta \)-Einstein Sasakian manifolds under the suitable stability condition due to the Thomas-Yau conjecture.

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Acknowledgements

The authors wish to thank the referees for helpful comments, suggestions and remarks which make the present form of the paper possible.

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Authors and Affiliations

  1. Mathematical Science Research Center, Chongqing University of Technology, 400054, Chongqing, People’s Republic of China

    Shu-Cheng Chang

  2. School of Mathematics and Statistics, Xinyang Normal University, Xinyang, 464000, Henan, People’s Republic of China

    Yingbo Han

  3. Department of Applied Mathematics, National Pingtung University, Pingtung, 90003, Taiwan

    Chin-Tung Wu

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  1. Shu-Cheng Chang
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  2. Yingbo Han
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  3. Chin-Tung Wu
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Correspondence to Yingbo Han.

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Shu-Cheng Chang, Chin-Tung Wu: Research supported in part by NSC.

Yingbo Han is partially supported by an NSFC 11971415 and Nanhu Scholars Program for Young Scholars of Xinyang Normal University.

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Chang, SC., Han, Y. & Wu, CT. Legendrian Mean Curvature Flow in \(\eta \)-Einstein Sasakian Manifolds. J Geom Anal 34, 89 (2024). https://doi.org/10.1007/s12220-023-01537-x

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