奇性空间上的几何分析

The aim of the project is to systematically investigate the analysis on singular spaces, i.e. nonsmooth metric spaces. The analysis on singular spaces becomes more and more important than ever before due to the wide spectrum of theoretical (nonsmooth geometry) and practical applications. We adopt the geometric analysis methods to study singular spaces in order to figure out the analogues and the discrepancies between smooth Riemannian manifolds and singular spaces. Two classes of singular spaces are of particular interest: metric measure spaces with generalized curvature and discrete spaces (i.e. graphs). We propose to develop the theory of elliptic PDEs on singular spaces, such as the W^{1,p} estiamtes of Poisson equations and Alexandrov-Bakelman-Pucci estimates, and to prove Jost’s conjecture, i.e. harmonic maps from metric measure spaces with generalized Ricci curvature into CAT(0) spaces (i.e. metric spaces with nonpositive generalized sectional curvature) are Lipschitz continuous, by using the new Bochner formula. On graphs, we want to show the volume doubling property and the Poincare inequality under various curvature conditions, and hence to prove Yau’s conjecture on polynomial growth harmonic functions. In addition, we try to find more applications of the new Davies-Gaffney estimate for the heat kernel on graphs and Lp spectral theory. The purpose of this project is to deeply understand nonsmooth analysis techniques in singular spaces.

本项目的内容是研究奇性空间（即不光滑度量空间）上的分析。由于非光滑几何理论和实际应用的需要，奇性空间上的分析日趋重要。我们用几何分析的方法研究奇性空间，试图找出光滑黎曼流形和奇性空间上分析性质的相似和区别。我们关注两类奇性空间：有广义曲率的测度度量空间以及离散空间（也叫做图）。我们试图在奇性空间上发展椭圆型方程理论，如泊松方程的W^{1,p}估计、ABP估计等；应用新证明的Bochner公式，尝试证明Jost猜想：从广义Ricci曲率空间出发到CAT(0)空间（即广义截面曲率非正的空间）的调和映照是Lipschitz连续的。在图上，我们研究各种曲率条件下的体积倍增条件和Poincare不等式，从而证明多项式增长调和函数的Yau猜想；考察图上新的热核Davies估计的应用和Lp谱论。本项目的目的是进一步深入理解奇性空间上的非光滑分析。

本项目主要研究奇性空间(即不光滑度量空间)上的分析，包括测度度量空间和离散空间上的几何分析相关问题，主要涉及Laplace算子的相关性质，包括调和函数、热核、算子谱、调和映照、随机完备性等。取得的研究成果主要包括：1.2017年Math. Ann.的文章中我们证明了图上连续时间热核最优的Davies估计。2.2017年Adv. Math.的文章中证明了Bakry-Emery曲率有一致下界的图的随机完备性。3.2017年Calc.Var. PDE的文章中，定义了图上的Steklov特征值问题，并利用等周型常数给出了第一非零特征值的估计。通过该项目的研究，我们将几何分析的方法应用到不光滑的空间上，加深了对奇性空间的理解。