关于一个涉及法向数量曲率的不等式的注记

统计曲面的Wintgen不等式

Wintgen inequality for statistical surfaces

Wintgen不等式（1979）是四维欧几里得空间中曲面的一个尖锐的几何不等式，分别涉及Gauss曲率（内禀不变量）、法曲率和平方中曲率（外禀不变量）。本文得到了统计曲面的一个Wintgen不等式。

关于图的Li-Yau不等式的注记

Remarks on Li-Yau inequality on graphs

在Bauer等人在\cite{BHLLMY}中引入的曲率条件$CD(n,-K)$下，研究了图上热方程$\partial_tu=Δu$的解$u$的Li-Yau梯度估计.作为应用，我们得到了图的Harnack不等式和热核估计.同时给出了一类Hamilton梯度估计。

关于加权流形中闭超曲面的一个几何不等式的注记

Remark on a geometric inequality for closed hypersurfaces in weighted manifolds

本文在具有非负Bakry-Émery里奇曲率的非紧加权流形中建立闭超曲面的最优几何不等式，刻画了$N=\infty$情形的刚性条件，推广了经典Willmore型不等式。

含高斯曲率的紧致黎曼曲面上的Trudinger-Moser不等式

A Trudinger-Moser inequality on compact Riemannian surface involving Gaussian curvature

受X最近工作的启发。Chen和M. Zhu（Commun.数学统计，1（2013）369-385）中，我们建立了无边界紧致黎曼曲面上的Trudinger-Moser不等式。证明是基于爆破分析与Carleson-Chang的结果（公牛。110（1986）113-127）。这个不等式不同于经典的由L.丰塔纳（注释。数学。赫尔夫，68（1993）415-454），因为涉及高斯曲率。作为应用，我们改进了Chen-Zhu的结果：当共形黎曼度量的欧拉示性数不为零且共形曲面的体积有一致正的下界时，修正的Liouville能量有一致的下界.

The logarithmic Sobolev inequality for a submanifold in manifold with nonnegative sectional curvature

We prove a sharp logarithmic Sobolev inequality which holds for compact submanifolds without boundary in Riemannian manifold with nonnegative sectional curvature of arbitrary dimension and codimension, while the ambient manifold needs to have a specific Euclid-like property. Like the Michael-Simon Sobolev inequality, this inequality includes a term involving the mean curvature. This extends a recent result of S. Brendle with Euclidean setting.

单位球上凸自由边界超曲面的一个几何不等式

A geometric inequality for convex free boundary hypersurfaces in the unit ball

利用具有垂直于球面的自由边界的反中曲率流证明了边界在球面上的$n\geq 3$维凸超曲面的一个包含Willmore能量的几何不等式.

广义相对论中的彭罗斯不等式和涉及数量曲率的体积比较定理（论文）

The Penrose inequality in general relativity and volume comparison theorems involving scalar curvature (thesis)

在这篇论文中，我们描述了如何利用极小曲面技巧证明两类三维流形的广义相对论中的Penrose不等式。我们还描述了如何一个新的体积比较定理，涉及3-流形的数量曲率遵循这些相同的技术。

含加权$σ_k$积分的逆曲率流的单调量

Monotone quantities involving a weighted $σ_k$ integral along inverse curvature flows

给出了欧几里得空间中逆曲率流的一族沿光滑解的单调量。对于具有正k次中曲率的闭超曲面，我们也得到了一个相关的几何不等式。

一个尖锐不等式及其应用

A sharp inequality and its applications

建立了一个含指数权函数的带锐常数的类似Hardy不等式.这个不等式的特殊情况（n=2）导致了S^2上Onofri不等式的直接证明。

The logarithmic Sobolev inequality for a submanifold in Euclidean space

We prove a sharp logarithmic Sobolev inequality which holds for submanifolds in Euclidean space of arbitrary dimension and codimension. Like the Michael-Simon Sobolev inequality, this inequality includes a term involving the mean curvature.

分数阶GJMS算子的能量不等式

Some energy inequalities involving fractional GJMS operators

在Poincaré-Einstein流形的Laplacian的谱假设下，建立了一个关于$2γ\in(0,2)$或$2γ\in(2,4)$阶分数次GJMS算子的能量分别与加权共形Laplacian或加权Paneitz算子的能量之间的能量不等式.这个谱假设对于这样一个不等式成立是必要且充分的。通过引入与加权共形Laplacian算子和加权Paneitz算子相关联的共形共变边界算子，推广了Robin算子，证明了能量不等式.作为应用，我们建立了上半球上一个新的带权Sobolev迹不等式.

Anisotropic $p$-capacity and anisotropic Minkowski inequality

In this paper, we prove a sharp anisotropic $L^p$ Minkowski inequality involving the total $L^p$ anisotropic mean curvature and the anisotropic $p$-capacity, for any bounded domains with smooth boundary in $\mathbb{R}^n$. As consequences, we obtain an anisotropic Willmore inequality, a sharp anisotropic Minkowski inequality for outward $F$-minimising sets and a sharp volumetric anisotropic Minkowski inequality. For the proof, we utilize a nonlinear potential theoretic approach which has been recently developed in \cite{AFM1}.

A sharp isoperimetric inequality in metric measure spaces with non-negative Ricci curvature

We prove a sharp dimension-free isoperimetric inequality, involving the volume entropy, in non-compact metric measure spaces with non-negative synthetic Ricci curvature.