Energy capacity inequalities via an action selector

基于曲率的 $p$-容量界与容量--面积不等式

Curvature-Based Bounds for $p$-Capacity and Capacity--Area Inequalities

本文建立了凸域 $p$-容量基于曲率的上下界，确定了临界指数，并给出容量--面积不等式及刚性刻画。

质量、容函数与质容比

Mass, capacitary functions, and the mass-to-capacity ratio

该研究探讨非负标量曲率渐近平坦3维流形的ADM质量、调和函数与边界容量的关联，推导单调量、几何不等式与质容比恒等式，给出正质量定理新证明，并研究质容比约束下流形的性质。

Bounds for Minkowski Billiard Trajectories in Convex Bodies

In this paper we use the Ekeland-Hofer-Zehnder symplectic capacity to provide several bounds and inequalities for the length of the shortest periodic billiard trajectory in a smooth convex body in ${\mathbb R}^{n}$. Our results hold both for classical billiards, as well as for the more general case of Minkowski billiards.

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。

Optimal geometric estimates for fractional Sobolev capacities

This note develops certain sharp inequalities relating the fractional Sobolev capacity of a set to its standard volume and fractional perimeter.

On The Capacity of Surfaces in Manifolds with Nonnegative Scalar Curvature

Given a surface in an asymptotically flat 3-manifold with nonnegative scalar curvature, we derive an upper bound for the capacity of the surface in terms of the area of the surface and the Willmore functional of the surface. The capacity of a surface is defined to be the energy of the harmonic function which equals 0 on the surface and goes to 1 at infinity. Even in the special case of Euclidean space, this is a new estimate. More generally, equality holds precisely for a spherically symmetric sphere in a spatial Schwarzschild 3-manifold. As applications, we obtain inequalities relating the ca…

非负标量曲率的三维流形上的单调量

Monotone quantities on $3$-manifolds with nonnegative scalar curvature

本文通过调和函数在渐近平坦三维流形上推导出单调量，这些量在空间Schwarzschild流形上为常数，并应用该结果推广了几何不等式和质量-容量不等式。

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.

关于临界面积归一化电容器的ADM质量

On the ADM mass of critical area-normalized capacitors

本文证明临界面积归一化电容器的质量-容量不等式，并由此得到Schwarzschild度量的唯一性，改进相关时空与流形的唯一性定理。

第一边界 Dirac 特征值与边界容量势

First boundary Dirac eigenvalue and boundary capacity potential

本文研究非负标量曲率自旋渐近平坦流形中超曲面的第一 Dirac 特征值，利用边界容量势与容量建立新的几何下界，并通过奇异度量紧流形的特征值估计完成证明。

具有从下方有界的Ricci曲率的流形的p-容量估计

Estimates of $p$-capacity for manifolds with Ricci curvature bounded from below

研究具有Ricci曲率下界的完整非紧黎曼流形上p-容量的精确估计，通过建立比较不等式及刻画等号成立情况得到结果，还研究了电容器相对p-容量的归一化下界，引入尺度不变量确定其最优范围。