Sub-Riemannian calculus and monotonicity of the perimeter for graphical strips

RCD空间中周长极小化集合奇异集的单调性公式与分层

Monotonicity formula and stratification of the singular set of perimeter minimizers in RCD spaces

本文在RCD(0,N)度量测度锥中建立周长极小化集合的单调性公式及刚性陈述，并由此得到非塌缩RCD空间中奇异分层的尖锐Hausdorff维数估计，以及非负Ricci曲率流形中全局周长极小化子blow-down锥的存在性。

凸体的加权周长的连续性与单调性

Continuity and monotonicity of weighted perimeters of convex bodies

该研究针对$\boldsymbol{\text{R}}^n$中凸体的加权周长，证明其关于Hausdorff度量的连续性、特定径向权下包含关系的单调性，将结果用于证明形状优化问题极小元的存在性。

平面带状等周问题的比较定理

A comparison theorem for the planar strip isoperimetric problem

本文通过构造三弧竞争区域，证明了在条带内外密度不同的平面等周问题中，四弧区域非最优，解决了相关猜想。

Weighted monotonicity theorems and applications to minimal surfaces in $\mathbb{H}^n$ and $S^n$

We prove that in a Riemannian manifold $M$, each function whose Hessian is proportional the metric tensor yields a weighted monotonicity theorem. Such function appears in the Euclidean space, the round sphere $S^n$ and the hyperbolic space $\mathbb{H}^n$ as the distance function, the Euclidean coordinates of $\mathbb{R}^{n+1}$ and the Minkowskian coordinates of $\mathbb{R}^{n,1}$. Then we show that weighted monotonicity theorems can be compared and that in the hyperbolic case, this comparison implies three $SO(n,1)$-distinct unweighted monotonicity theorems. From these, we obtain upper bounds …

Isoperimetric, Sobolev and Poincaré inequalities on hypersurfaces in sub-Riemannian Carnot groups

In this paper we shall study smooth submanifolds immersed in a k-step Carnot group G of homogeneous dimension Q. Among other results, we shall prove an isoperimetric inequality for the case of a $C^2$-smooth compact hypersurface S with - or without - boundary $\partial S$; S and $\partial S$ are endowed with their homogeneous measures, actually equivalent to the intrinsic (Q-1)-dimensional and (Q-2)-dimensional Hausdorff measures with respect to some homogeneous metric $\varrho$ on G; see Section 5. This generalizes a classical inequality, involving the mean curvature of the hypersurface, prov…