2020年北师大秋季几何分析与PDE研讨会
时间：2020年11月22日
地点：北京师范大学后主楼 1129
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报告安排：
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葛化彬
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13:00—13:30
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李 畅
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13:30—14:00
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休 息
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10分钟
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沈良明
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14:10—14:40
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沈伟明
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14:40—15:10
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茶 歇
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20分钟
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王 博
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15:30—16:00
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王 越
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16:00—16:30
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休 息
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10分钟
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郑 涛
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16:40—17:10
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组织者：熊金钢 Email: jx@bnu.edu.cn
报告题目和摘要
葛化彬，中国人民大学
Title: Combinatorial Ricci flows and the hyperbolization of a class of compact 3-manifolds
Abstract: Thurston conjectured that all compact hyperbolic 3-manifolds can be geometrically triangulated. Under suitable combinatorial assumptions, we confirm this conjecture for such manifolds with higher genus boundary components. This is joint work with Ke Feng and Bobo Hua.
李畅，中科院数学与系统科学研究院
Title: The complex Hessian equations on Hermitian manifolds
Abstract: We consider the complex Hessian equation involving additional gradient terms. We derive the second order estimates for χ-plurisubharmonic solutions to this equation on compact Hermitian manifolds. Such an estimate may be helpful to study the existence and regularity for the solution of the equation.
沈良明，北京航空航天大学
Title: The Calabi-Yau metric and complex Monge-Ampere equation in noncompact setting
Abstract: We first recall Yau's folklore solution to the Calabi conjecture. Then we briefly discuss how to deduce the canonical metric problem to the complex Monge-Ampere equation and derive a priori estimates. After that we introduce the corresponding noncompact setting by Tian-Yau. Finally we talk a bit about some progress in the generalization to Tian-Yau's work.
沈伟明，首都师范大学
Title: Blow up sets of Ricci curvatures of complete conformal metrics
Abstract: A version of the singular Yamabe problem in bounded domains yields complete conformal metrics with negative constant scalar curvatures. In this talk, we will talk about blow-up phenomena of Ricci curvatures of these metrics on domains whose boundary is close to certain limit set of a lower dimension. We will characterize the blow-up set according to the Yamabe invariant of the underlying manifold. In particular, we will prove that all points in the lower dimension part of the limit set belong to the blow-up set on manifolds not conformally equivalent to the standard sphere and that all but one point in the lower dimension part of the limit set belong to the blow-up set on manifolds conformally equivalent to the standard sphere. We will demonstrate by examples that these results are optimal.
王博，北京理工大学
Title: On the $ \sigma_{k}$-Nirenberg problem
Abstract: In this talk, I will present some recent results concerning the $\sigma_{k}$-Nirenberg problem in the study of conformal geometry, including existence and compactness. This is a joint work with Professor YanYan Li and Professor Luc Nguyen.
王越，首都师范大学
Title：On Global Regularity and Boundary Layer Separation of Steady Prandtl Equations
Abstract: For the 2-D steady Prandtl Equations, Oleinik proved the global-in-x existence of solutions with finite order regularity in the case of favorable pressure gradient and the local-in-x existence of solutions in the case of adverse pressure gradient. In this talk, I will first review some related results and then report our recent works for the 2-D steady Prandtl Equations where 1. we proved the global C^\infty regularity in the case of favorable pressure gradient; 2. we proved the boundary layer separation for a large class of Oleinik’s solutions and study the local behavior of the solutions near the separation in the case of adverse pressure gradient.
郑涛，北京理工大学
Title: Notions Related to Negativity on K\"ahler Manifolds and Geometric Applications
Abstract: A recent celebrated theorem of Diverio-Trapani and Wu-Yau states that a compact K\"ahler manifold admitting a K\"ahler metric of quasi-negative holomorphic sectional curvature has an ample canonical line bundle, confirming a conjecture of Yau. In this talk, we shall introduce a natural notion of almost quasi-negative holomorphic sectional curvature and extend this theorem to compact K\"ahler manifolds of almost quasi-negative holomorphic sectional curvature. This is a joint work with Yashan Zhang.
