Geometric Inequalities and Fully Nonlinear Elliptic Equations

首席调查员(PI)将在保形几何和子流形几何的背景下研究各种几何不等式及其相应的非线性偏微分方程式。其中一个主题计划研究高阶曲率对等周不等式有效性的影响。特别是，她打算定量地分析q-曲率和等周常数之间的相互作用。PI还建议研究嵌入子流形的不同阶曲率不等式。这些不等式最初是在凸几何的背景下考虑的，最近已知在非常大的一类非凸域上是有效的。PI的调查努力寻找这种不平等的全部普遍性。同时，她还致力于发展新的技巧来理解问题中自然产生的相应的完全非线性椭圆型偏微分方程。在所提出的研究中，PI的研究兴趣在于共形几何、子流形几何和偏微分方程(PDE)的交集。几何不等式和几何偏微分方程组的研究集中在共形不变量上，从物理学家的角度来看，共形不变量是一种重要的机制，并在数学物理的基本原理中找到了应用。推广凸几何经典结果的研究项目将提高我们对已建立理论的刚性的理解，并将有助于揭示其更大的应用范围。

The Principal Investigator (PI) will study various geometric inequalities and their corresponding nonlinear partial differential equations in the context of conformal geometry and the geometry of submanifolds. One theme plans to investigate the effect of the higher order curvatures on the validity of the isoperimetric inequality. In particular, she intends to quantitatively analyze the interaction between the Q-curvature and the isoperimetric constant. The PI also proposes to study curvature inequalities of different orders for embedded submanifolds. These inequalities, originally considered in the context of convex geometry, have recently known to be valid on a very large class of non-convex domains. The PI's investigation strives to look for the full generality of such inequalities. In the meanwhile, she also aims to develop new skills to understand the corresponding fully nonlinear elliptic partial differential equations that arise naturally in the problem.In the proposed study, the PI's research interest lies at the intersection of conformal geometry, the geometry of submanifolds and partial differential equations (PDEs). The study of geometric inequalities and geometric PDEs focuses on conformal invariants, which form an important machinery from physicists' point of view and have found applications to fundamental principles in mathematical physics. The research project to generalize classical results of convex geometry will improve our understanding on the rigidity of the established theory and will shed light on a greatly larger scope of its application.