Isometric embeddings, isoperimetric inequalities and geometric nonlinear PDE

等距嵌入、等周不等式和几何非线性 PDE

• 批准号: RGPIN-2018-04443
• 负责人: Guan, Pengfei
• 金额: $ 8.3万
• 依托单位: McGill University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=758982
• 关键词: Isometric; embeddings; isoperimetric; inequalities; geometric

The proposed research program is centred on fundamental problems in differential geometry and nonlinear PDE: the isometric embedding problem, the isoperimetric type inequalities on general manifolds, and regularity of solutions to nonlinear geometric partial differential equations. The first topic is isometric embedding problem for compact surfaces to three dimensional Riemannian manifold with horizons. When the ambient space is Euclidean space, it is the classical Weyl problem. It is of importance in geometry to consider general ambient space, this also is related to the notions of quasi local masses in general relativity. The most interesting case is that when the ambient space is a anti de Sitter-Schwarzchilds space. The second topic concerns various global geometric quantities on manifolds, like volume, surface area, quermassintegrals etc. We would like to establish optimal isoperimetric type inequalities for these geometric quantities. Our approach will be based on nonlinear partial differential equations of parabolic type. For each pair of geometric quantities, we would like to design a curvature flow such that: along the flow, one quantity is preserved and another is monotone. The key is to prove the longtime existence and convergence of the flow. The last topic addresses some longstanding regularity problems of curvature type equations. Pogorelov type counter-examples indicate that interior regularity fails for Monge-Amp\`ere equation when dimension is larger or equal to three. One longstanding open problem is that, if interior estimate holds for scalar curvature equation and $\sigma_2$ Hessian equation. These geometric equations are of fundamental importance, for example, scalar curvature equation naturally arising from the isometric embedding problems. A breakthrough will have great impact in geometric analysis. A common thread linking our program is the analysis of the geometric fully nonlinear equations. These equations are the main subjects of the research program. Besides the regularity and existence of solutions of these equations (which are still important subjects of the study), there emerge some new directions of research from the proposed problems. One main challenge is for the isometric embedding problem discussed is the existence of homotopic paths, we propose a novel approach using geometric flows in combination with elliptic method. The flow approach will also be devised to establish isoperimetric type inequalities: explore the variational properties of the associated functionals to design a flow with appropriate monotonicity properties. For the regularity problems of solutions to geometric nonlinear PDE, we propose new ideas to deal with the issue. Our objective is to develop various analytic tools for geometric nonlinear partial differential equations, investigate structures of solutions and derive geometric consequences.

拟议的研究计划以微分几何和非线性偏微分方程的基本问题为中心：等距嵌入问题、一般流形上的等周型不等式以及非线性几何偏微分方程解的正则性。 第一个主题是紧致曲面到具有视界的三维黎曼流形的等距嵌入问题。当环境空间是欧几里得空间时，就是经典的Weyl问题。在几何学中考虑一般环境空间非常重要，这也与广义相对论中的准局部质量的概念有关。最有趣的情况是当周围空间是反德西特-史瓦西空间时。 第二个主题涉及流形上的各种全局几何量，例如体积、表面积、质量积分等。我们希望为这些几何量建立最佳等周型不等式。我们的方法将基于抛物型非线性偏微分方程。对于每一对几何量，我们希望设计一个曲率流，使得：沿着流，一个量被保留，另一个量是单调的。关键是要证明流的长期存在性和收敛性。最后一个主题解决了曲率类型方程的一些长期存在的正则性问题。 Pogorelov 型反例表明，当维度大于或等于 3 时，Monge-Amp\'ere 方程的内部正则性失效。一个长期悬而未决的问题是，内部估计是否适用于标量曲率方程和 $\sigma_2$ Hessian 方程。这些几何方程具有根本重要性，例如，等距嵌入问题自然产生的标量曲率方程。一个突破将对几何分析产生巨大影响。连接我们程序的一个共同线索是几何完全非线性方程的分析。这些方程是该研究计划的主要主题。除了这些方程的规律性和解的存在性（这仍然是研究的重要课题）之外，所提出的问题还出现了一些新的研究方向。对于所讨论的等距嵌入问题的一个主要挑战是同伦路径的存在，我们提出了一种使用几何流与椭圆方法相结合的新方法。还将设计流方法来建立等周类型不等式：探索相关泛函的变分属性，以设计具有适当单调性属性的流。针对几何非线性偏微分方程解的正则性问题，我们提出了处理该问题的新思路。 我们的目标是开发用于几何非线性偏微分方程的各种分析工具，研究解的结构并导出几何结果。