Ricci curvature and the structure of manifolds

这个项目的目标是获得进一步了解的结构完全非紧流形。总的主题是理解一些偏微分方程的解的曲率可以用来获得更多的信息几何和拓扑的流形。在这种情况下，PI将研究拉普拉斯算子在Ricci曲率方面的谱底上的边界，并使用谱信息来研究流形的结构。PI还将研究完全梯度Ricci孤子的几何和分析性质，旨在更好地理解Ricci流的奇异性。拟议的项目将导致更好地理解无限几何物体的形状和结构。这些物体，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，例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。偏微分方程是几何学中的一个基本工具，例如里奇流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数学和科学许多领域的新发现。

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.