勷勤数学•专家报告

题      目：Gehring-Hayman inequality for quasigeodesics in Banach spaces and its applications

报  告  人：王仙桃   教授  (邀请人：刘名生 )

                                     湖南师范大学

时      间：6月3日  09：00-10：00

地     点：数科院东楼401

报告人简介:

       王仙桃，湖南省二级教授，博士生导师，教育部新世纪优秀人才计划入选者，湖南省新世纪121第二层次人选，湖南省杰出青年基金获得者，湖南省学科带头人；湖南省十二五重点学科数学、湖南省首届普通高校科技创新团队、湖南省普通高校教学团队的带头人，国家级双语教学示范课程、国家级一流线上课程《数学分析》的主持人。研究领域为Klein群、拟共形映射以及调和映射，解决了发表在国际数学最权威刊物《Acta.Math.》上悬而未决达三十多年的问题，在《Adv.Math.》、《Math.Ann.》、《IMRN》等刊物发表论文80多篇。

摘      要：

       Suppose that E is a Banach space with dim E ≥ 2, and D and D_0 are proper subdomains in E. Let f : D → D_0 be a coarsely quasihyperbolic homeomorphism. The main purpose of this paper is to establish the following result: If D_0 is a uniform domain, then the quasigeodesic in D essentially minimizes the length among all arcs in D with the same end-points, up to a universal multiplicative constant. This result gives affirmative answers to the related open problems raised by Heinonen and Rohde from 1993 and by Vaisala from 2005. As the first application, we obtain that the length of the image of a quasigeodesic in D under f minimizes the length among the images of all arcs in D whose end-points are the same as the given quasigeodesic, up to a universal multiplicative constant. As the second application, we show that D being John implies D being inner uniform. This is a generalization of the related result obtained by Kim and Langmeyer in 1998 since this result implies that the assumption of “each quasihyperbolic geodesic in a John domain in R^n being a cone arc”is redundant.

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