报告题目:Bounded differentials on unit disk and the associated geometry
报告人:戴嵩,天津大学应用数学中心
时间: 2022年10月7日(周五)14:30-15:30
腾讯会议:647-4448-8711
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摘要:Let D be the unit disk with hyperbolic metric. Given a holomorphic quadratic differential q, there is a harmonic map f from D to itself such that q is the Hopf differential. Under certain completeness condition, Wan showed that q is bounded (with respect to the hyperbolic metric) is equivalent to either the energy density is bounded or f is quasi-conformal. In this talk, we study more holomorphic differentials and the associated geometries. For holomorphic cubic, quartic and sextic differentials, the associated geometries are hyperbolic affine spheres in R^3, maximal surfaces in H^{2,n} and J-holomorphic curves in H^{4,2}. Combining Wan's PDE approach and Higgs bundle techniques, we show that under the completeness condition, the holomorphic differential is bounded is equivalent to either the induced metric is mutually bounded with the hyperbolic metric or the curvature of the induced metric is bounded above by a negative constant. We also generalize Wan’s result from the single equation to the equation system, which relates to the Toda system and the Hitchin equation in the non-Abelian Hodge theory. This is a joint work with Qiongling Li.
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