Abstract
In this paper, we recall some well known Berger’s formulas. As their applications, we prove that if the local holomorphic pinching constant is \(\lambda <2\), then there exists a positive constant \(\delta >\frac{29(\lambda -1)}{\sqrt{(48-24\lambda )^{2}+(29\lambda -29)^{2}}}\) such that \(\cos \alpha \ge \delta \) is preserved along the mean curvature flow, improving Li–Yang’s main theorem in Li and Yang (Geom. Dedicata 170 (2014) 63–69). We also prove that when \(\cos \alpha \) is close enough to 1, then the symplectic mean curvature flow exists globally and converges to a holomorphic curve.
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Acknowledgements
The author is partially supported by NSFC No. 11301017, Research Fund for the Doctoral Program of Higher Education of China and the Fundamental Research Funds for the Central Universities. He thanks Professor Jiayu Li, Xiaoli Han, Jun Sun and Liuqing Yang for many useful discussions. He also thanks the referees for useful suggestions.
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Zhang, S. Berger’s formulas and their applications in symplectic mean curvature flow. Proc Math Sci 130, 29 (2020). https://doi.org/10.1007/s12044-020-0552-2
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DOI: https://doi.org/10.1007/s12044-020-0552-2