Ricci curvature and the structure of manifolds

本课题的目标是进一步了解完全非紧流形的结构。总的主题是，理解一些偏微分方程的曲率解可以用来获得更多关于流形的几何和拓扑的信息。在这种情况下，PI将根据里奇曲率研究拉普拉斯函数的谱底界，并利用谱信息来研究流形的结构。PI还将研究完全梯度Ricci孤子的几何和解析性质，旨在更好地理解Ricci流的奇点。提出的项目将导致对无限几何物体的形状和结构的更好理解。这些物体被称为流形，经常出现在科学中，因为物理定律可以通过流形上的某些偏微分方程来建模。了解流形的结构在许多领域都很重要，如物理学中的黑洞和虫洞，化学中的分子结构或工程中的液晶运动。偏微分方程已经成为几何中的一个基本工具，例如，里奇流已经被用来对所有三维流形的形状进行分类。流形上几何偏微分方程研究的进一步进展将导致适用于许多数学和科学领域的新发现。

The goal of this project is to obtain further understanding of the structure of complete noncompact manifolds. The general theme is that understanding the solutions of some partial differential equations in terms of curvature can be used to obtain more information about the geometry and topology of the manifold. In this context, the PI will study bounds on the bottom of spectrum of the Laplacian in terms of the Ricci curvature and use the spectral information to study the structure of the manifold. The PI will also study geometric and analytic properties of complete gradient Ricci solitons, aiming at a better understanding of the singularities of the Ricci flow.The proposed project will lead to a better understanding of the shape and structure of infinite geometric objects. These objects, called manifolds, appear frequently in science because physical laws can be modeled by certain partial differential equations on manifolds. Understanding the structure of manifolds is important in many fields, such as black holes and worm holes in physics, the structure of molecules in chemistry or the motion of liquid crystal in engineering. Partial differential equations have been a fundamental tool in geometry, for example the Ricci flow has been used to classify the shape of all three dimensional manifolds. Further advances in the study of geometric partial differential equations on manifolds would lead to new discoveries applicable to many areas of mathematics and science.