有限维嵌入的Bi-Lipschitz逼近

Approximations of Lipschitz maps via Ehresmann fibrations and Reeb's sphere theorem for Lipschitz functions

We show, as our main theorem, that if a Lipschitz map from a compact Riemannian manifold $M$ to a connected compact Riemannian manifold $N$, where $\dim M \geq \dim N$, has no singular points on $M$ in the sense of F.H. Clarke, then the map admits a smooth approximation via Ehresmann fibrations. We also show the Reeb sphere theorem for Lipschitz functions, i.e., if a closed Riemannian manifold admits a Lipschitz function with exactly two singular points in the sense of Clarke, then the manifold is homeomorphic to the sphere.

Regularity theory for $2$-dimensional almost minimal currents I: Lipschitz approximation

We construct Lipschitz $Q$-valued functions which approximate carefully integral currents when their cylindrical excess is small and they are almost minimizing in a suitable sense. This result is used in two subsequent works to prove the discreteness of the singular set for the following three classes of $2$-dimensional integral currents: area minimizing in Riemannian manifolds, semicalibrated and spherical cross sections of $3$-dimensional area minimizing cones.

黎曼流形上Lipschitz函数的光滑逼近

Smooth Approximation of Lipschitz functions on Riemannian manifolds

我们证明了对于定义在可分黎曼流形$M$（可能是无穷维的）上的Lipschitz函数$f$，对于每个连续的$ε:M\to (0,+\infty)$，对于每个正数$r>0$，存在$C^\infty$光滑Lipschitz函数$g:M\to\mathbb{R}$，使得对于每个$p\in M$和$\textrm{Lip}(g)\leq\textrm{Lip}(f)+r$，$|f(p)-g(p)|\leqε(p)$.因此，每个可分黎曼流形都是一致可凸的。我们也给出了这个结果的一些应用，例如Deville-Goeley-Zizler光滑变分原理对于可分黎曼流形的一般形式。

Lipschitz区域的光滑逼近、弱曲率与等容估计

Smooth approximation of Lipschitz domains, weak curvatures and isocapacitary estimates

本文提出用光滑有界区域列逼近Lipschitz区域的新方法，支持内、外逼近，适用于边界具有弱曲率（局部为$W^{2,q}$函数图像）的区域，并建立关于初始区域的一致等容估计。

丰富群中几乎 $\mathbb G$-周长最小化边界的Lipschitz逼近

Lipschitz approximation of almost $\mathbb G$-perimeter minimizing boundaries in plentiful groups

本文证明丰富群中内蕴周长几乎最小化边界可由内蕴Lipschitz图逼近，首次将正则性理论推广至Heisenberg群之外。

关于Banach-Finsler流形上Lipschitz函数的光滑逼近与光滑扩张的若干问题

On some problems on smooth approximation and smooth extension of Lipschitz functions on Banach-Finsler Manifolds

让我们考虑一个黎曼流形$M$（可分或不可分）。证明了对每个$ε>0$，每个Lipschitz函数$f:M\rightarrow\mathbb R$都能被Lipschitz，$C^1$ -光滑函数$g$和$\Lip(g)\le \Lip(f)+ε$一致逼近.因此，每个黎曼流形都是一致可凸的。结果的上下文中的$C^\ell$ Finsler流形建模的Banach空间。在Finsler流形$M$（以及$M$被建模的Banach空间$X$）上给出了充分条件，使得每个Lipschitz函数$f:M\rightarrow \mathbb R$都可以被Lipschitz，$C^k$ -光滑函数$g$和$\Lip(g)\le C \Lip(f)$一致逼近（对于某些$C$只依赖于$X$）.给出了这些结果的一些应用，以及在可分情形下，满足上述逼近性质的$C^\ell$ Finsler流形类的一个特征.最后，给出了$C^1$ Finsler流形$M$和$X$上的充分条件，以保证定义在$M$子流形$N$上的实函数$f$在$N$和Lipschitz上的Lipschitz和$C^1$ -光滑扩张的存在性，只要$f$在$M$诱导的度量下是$C^1$ -光滑的.