Vincent Borrelli | Searcharxiv

arXiv2026

Submanifolds of class $C^{1,\alpha}$ and sets with positive $\mu$-reach

It is well-known since the seminal work of Herbert Federer [Trans. of the AMS, 1959] that submanifolds of class $C^{1,1}$ have positive reach. In this paper, we extend this property to less regular submanifolds by using the notion of $\mu$-reach that was introduced in the 2000's. We first show that every compact $C^1$ submanifold of the Euclidean space $\E^n$ has positive $\mu$-reach for all $\mu<1$. We then show that intermediate regularities $C^{1,\alpha}$ induce more quantitative results on the norm $\|\nabla \d_M\|$ of the generalized gradient of the distance function~$\d_M$ to the submanifold. More precisely, if $M\subset \E^n$ is a submanifold of class $C^{1,\alpha}$, with $\alpha<1$, then there exists a constant $C>0$ such that $$\forall p\in\E^n\setminus M,\quad 1 - \| \nabla \d_M(p) \|^2 \leq C ~ \d_M(p)^{\frac{2 \alpha}{1- \alpha}}.$$ We finally show that the exponent $2\alpha/(1-\alpha)$ in this estimate is sharp.

math.DG↗

arXiv2023

The Hyperbolic Plane in $\mathbb{E}^3$

We build an explicit $C^1$ isometric embedding $f_{\infty}:\mathbb{H}^2\to\mathbb{E}^3$ of the hyperbolic plane whose image is relatively compact. Its limit set is a closed curve of Hausdorff dimension 1. Given an initial embedding $f_0$, our construction generates iteratively a sequence of maps by adding at each step $k$ a layer of $N_{k}$ corrugations. To understand the behavior of $df_\infty$ we introduce a $formal$ $corrugation$ $process$ leading to a $formal$ $analogue$ $\Phi_{\infty}:\mathbb{H}^2\to \mathcal{L}(\mathbb{R}^2,\mathbb{R}^3)$. We show a self-similarity structure for $\Phi_{\infty}$. We next prove that $df_\infty$ is close to $\Phi_{\infty}$ up to a precision that depends on the sequence $N_*:= (N_{k})_k$. We then introduce the $pattern$ $maps$ $\boldsymbol{\nu}_{\infty}^\Phi$ and $\boldsymbol{\nu}_{\infty}$, of respectively $\Phi_{\infty}$ and $df_\infty$, that together with $df_0$ entirely describe the geometry of the Gauss maps associated to $\Phi_{\infty}$ and $df_\infty$. For well chosen sequences of corrugation numbers, we finally show an asymptotic convergence of $\boldsymbol{\nu}_{\infty}$ towards $\boldsymbol{\nu}_{\infty}^\Phi$ over circles of rational radii.

math.DG↗