Abstract
In this paper, we investigate the positive solutions of \(\mathfrak {L}u=0\) on a self-shrinker. First, we prove a global gradient estimate for the positive solutions, and obtain a strong Liouville theorem. Then by the generalized Laplacian comparison theorem for the distance function on a self-shrinker, we derive a local gradient estimate for the positive solutions. At last, we collect some applications of the local gradient estimate for the positive solutions on self-shrinkers.
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The authors express their sincere thanks to the referees and editors for their careful reading of the original manuscript and for their comments which improved the paper.
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Zhu, Y., Chen, Q. Gradient Estimates for the Positive Solutions of \(\mathfrak {L}u=0\) on Self-Shrinkers. Mediterr. J. Math. 15, 28 (2018). https://doi.org/10.1007/s00009-018-1072-5
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DOI: https://doi.org/10.1007/s00009-018-1072-5