Abstract
Let X be a toric surface with Delzant polygon P and u(t) be a solution of the Calabi flow equation on P. Suppose the Calabi flow exists in [0, T). By studying local estimates of the Riemann curvature and the geodesic distance under the Calabi flow, we prove a uniform interior estimate of u(t) for \(t < T\).
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Notes
Roughly speaking, a point contains a nontrivial concentration energy if there is certain line segment (in Euclidean sense, passing through this point) where the integration of \(|Rm|^2\) is nontrivial over this line segment.
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Acknowledgments
The second and third named author would like to express their gratitude to Professor Paul Gauduchon and Frank Pacard for their support. The second named author would like to thank Professor Pengfei Guan and Vestislav Apostolov for stimulating discussions. The third named author would like to thank Professor An-Min Li for his support.
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Communicated by F. H. Lin.
X. X. Chen is partially supported by NSF grant DMS-1515795. H. Huang is financially supported by the FMJH (Fondation mathématique Jacques Hadamard). Sheng acknowledges the support of NSFC grant 11471225.
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Chen, X., Huang, H. & Sheng, L. The interior regularity of the Calabi flow on a toric surface. Calc. Var. 55, 106 (2016). https://doi.org/10.1007/s00526-016-1028-1
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DOI: https://doi.org/10.1007/s00526-016-1028-1