非负标量曲率的三维流形上的单调量

Monotone Volume Formulas for Geometric Flows

We consider a closed manifold M with a Riemannian metric g(t) evolving in direction -2S(t) where S(t) is a symmetric two-tensor on (M,g(t)). We prove that if S satisfies a certain tensor inequality, then one can construct a forwards and a backwards reduced volume quantity, the former being non-increasing, the latter being non-decreasing along the flow. In the case where S=Ric is the Ricci curvature of M, the result corresponds to Perelman's well-known reduced volume monotonicity for the Ricci flow. Some other examples are given in the second section of this article, the main examples and motiv…

Ricci curvature and monotonicity for harmonic functions

In this paper we generalize the monotonicity formulas of [C] for manifolds with nonnegative Ricci curvature. Monotone quantities play a key role in analysis and geometry; see, e.g., [A], [CM1] and [GL] for applications of monotonicity to uniqueness. Among the applications here is that level sets of Green's function on open manifolds with nonnegative Ricci curvature are asymptotically umbilic.

三维流形上调和函数的单调性

Monotonicity of harmonic functions on $3$-manifolds with an asymptotically flat end

研究了三维流形上具有渐近平坦端的正调和函数的单调性，并探讨了具有两个端的Schwarzschild流形的刚性特征。

具有非负数量曲率的3-流形中p-容度函数的新单调性

New monotonicity for $p$-capacitary functions in $3$-manifolds with nonnegative scalar curvature

本文在具有非负数量曲率的渐近平坦3-流形中，导出了与p-容度函数相关的新单调量和几何不等式，将Miao的结果从p=2推广至p∈(1, 3)，并恢复了质量与p-容度及p-容度与面积不等式。

渐近平坦半空间中的质量-p-容量不等式

Mass-$p$-Capacity Inequalities in Asymptotically Flat Half-Spaces

本文在三维简单拓扑、非负标量曲率和非负边界平均曲率的渐近平坦半空间中，建立了与p-容量函数相关的一般单调量及尖锐的质量-容量不等式，等号在旋转对称半球外的Schwarzschild半空间上成立。

Entropy and reduced distance for Ricci expanders

Perelman has discovered two integral quantities, the shrinker entropy $\cW$ and the (backward) reduced volume, that are monotone under the Ricci flow $\pa g_{ij}/\pa t=-2R_{ij}$ and constant on shrinking solitons. Tweaking some signs, we find similar formulae corresponding to the expanding case. The {\it expanding entropy} $\ctW$ is monotone on any compact Ricci flow and constant precisely on expanders; as in Perelman, it follows from a differential inequality for a Harnack-like quantity for the conjugate heat equation, and leads to functionals $μ_+$ and $ν_+$. The {\it forward reduced volume}…

利用Colding单调性公式研究Ricci曲率的定量化刚性

Quantitative Rigidity Using Colding's Monotonicity Formulas for Ricci Curvature

本文基于Colding单调性公式，研究Ricci曲率的定量化刚性，通过构造格林函数和k-分割函数，定量控制正则性并控制到近锥体的距离。

非线性势的细特性及单调性公式的统一视角

Fine properties of nonlinear potentials and a unified perspective on monotonicity formulas

本文研究非线性势的细特性，统一了单调性公式的视角，证明了$p$-容量势与逆平均曲率流的关联，并推导了高斯-博内型定理。

On the Curvature of Monotone Metrics and a Conjecture Concerning the Kubo-Mori Metric

It is the aim of this article to determine curvature quantities of an arbitrary Riemannian monotone metric on the space of positive matrices resp. nonsingular density matrices. Special interest is focused on the scalar curvature due to its expected quantum statistical meaning. The scalar curvature is explained in more detail for three examples, the Bures metric, the largest monotone metric and the Kubo-Mori metric. In particular, we show an important conjecture of Petz concerning the Kubo-Mori metric up to a formal proof of the concavity of a certain function on R_+^3. This concavity seems to …

Capacity, quasi-local mass, and singular fill-ins

We derive new inequalities between the boundary capacity of an asymptotically flat 3-manifold with nonnegative scalar curvature and boundary quantities that relate to quasi-local mass; one relates to Brown--York mass and the other is new. We argue by recasting the setup to the study of mean-convex fill-ins with nonnegative scalar curvature and, in the process, we consider fill-ins with singular metrics, which may have independent interest. Among other things, our work yields new variational characterizations of Riemannian Schwarzschild manifolds and new comparison results for surfaces in them.

Coarse and Precise $L^p$-Green Potential Estimates on Noncompact Riemannian Manifolds

We are concerned about the coarse and precise aspects of a priori estimates for Green's function of a regular domain for the Laplacian-Betrami operator on any $3\le n$-dimensional complete non-compact boundary-free Riemannian manifold through the square Sobolev/Nash/logarithmic-Sobolev inequalities plus the rough and sharp Euclidean isoperimetric inequalities. Consequently, we are led to evaluate the critical limit of an induced monotone Green's functional using the asymptotic behavior of the Lorentz norm deficit of Green's function at the infinity, as well as the harmonic radius of a regular …