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On Lawson-Osserman Constructions & The generalized Klingenberg's lemma in complete Riemannian manifolds
发布时间: 2017-12-18     16:03   【返回上一页】 发布人:张永胜, 胥世成



北京师范大学数学科学学院

 

几何专题报告


报告一:


报告题目:On Lawson-Osserman Constructions

报告人: 张永胜 教授 (东北师范大学)

 

时间地点:201712月20日周三下午3-4点, 后主楼1223

 

邀请人:彦文娇

报告摘要: We make systematic developments on Lawson-Osserman constructions concerning the Dirichlet problem (over unit disks) for minimal surfaces of high codimension in their 1977' Acta paper. In particular, we show the existence of boundary functions for which infinitely many analytic solutions and at least one nonsmooth Lipschitz solution exist simultaneously. This newly-discovered amusing phenomenon enriches the understanding on the Lawson-Osserman philosophy. This talk is based on joint works with Xiaowei XU and Ling YANG.





报告二:


报告题目:The generalized Klingenberg's lemma in complete Riemannian manifolds

报告人: 胥世成 教授(首都师范大学)

 

时间地点:201712月20日周三下午4-5点, 后主楼1223

 

邀请人:彦文娇

报告摘要: Let M be a complete manifold and suppose that a closed submanifold W lies in the normal bundle of another closed submanifold N in M. Then can one estimate effectively on the size of normal bundle of W? Based on the work of Innami-Shiohama-Soga (GAFA,2012), we developed a new tool that can be used to deal with such problems. It can be viewed as a generalization of well-known Klingenberg's lemma on the existence of geodesic loops. Recently we use it to derive a curvature-free and sharp estimate on the size of convexity neighborhood around a point, which improve the classical Whitehead's theorem on the existence of convexity neighborhood, as well as a recent result of Jiaqiang Mei and James Dibble. It also follows that an answer of the question above is, the injectivity radius of W, inj.rad(W) is no less than min{inj.rad(N),foc.rad(W)}-d_H(W,N), where foc.rad(W) is the focal radius of W and d_H(W,N) is the Hausdorff distance between W and N. Thus, for submanifolds that have no focal points, inj.rad is a 1-Lipschitz function with respect to the Hausdorff distance.


Reference: Local estimate on convexity radius and decay of injectivity radius in a Riemannian manifold, Commun. Contemp. Math., 2017. (arXiv:1704.03269)