arXiv subjects
Publications and source records attributed to Mélanie Theillière.
We introduce a method of mixed convex integration and demonstrate its suitability for solving a particular class of semilinear second-order partial differential relations. As an application, we provide a new proof of a result on scalar curvature originally established by Lohkamp.
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 $formal$ $analogue$ $Φ_{\infty}:\mathbb{H}^2\to \mathcal{L}(\mathbb{R}^2,\mathbb{R}^3)$. We show a self-similarity structure for $Φ_{\infty}$. We next prove that $df_\infty$ is close to $Φ_{\infty}$ up to a precision that depends on the sequence $N_*:= (N_{k})_k$. We then introduce the $pattern$ $maps$ $\boldsymbolν_{\infty}^Φ$ and $\boldsymbolν_{\infty}$, of respectively $Φ_{\infty}$ and $df_\infty$, that together with $df_0$ entirely describe the geometry of the Gauss maps associated to $Φ_{\infty}$ and $df_\infty$. For well chosen sequences of corrugation numbers, we finally show an asymptotic convergence of $\boldsymbolν_{\infty}$ towards $\boldsymbolν_{\infty}^Φ$ over circles of rational radii.
Convex integration and the holonomic approximation theorem are two well-known pillars of flexibility in differential topology and geometry. They may each seem to have their own flavor and scope. The goal of this paper is to bring some new perspective on this topic. We explain how to prove the holonomic approximation theorem for first order jets using convex integration. More precisely we first prove that this theorem can easily be reduced to proving flexibility of some specific relation. Then we prove this relation is open and ample, hence its flexibility follows from off-the-shelf convex integration.
Convex Integration is a theory developed in the '70s by M. Gromov. This theory allows to solve families of differential problems satisfying some convex assumptions. From a subsolution, the theory iteratively builds a solution by applying a series of convex integrations. In a previous paper arXiv:1909.04908, we proposed to replace the usual convex integration formula by a new one called Corrugation Process. This new formula is of particular interest when the differential problem under consideration has the property of being of Kuiper. In this paper, we consider the differential problem of $ε$-isometric maps and we prove that it is Kuiper in codimension 1. As an application, we construct $ε$-isometric maps from a short map having a conical singularity.
We replace the usual Convex Integration formula by a Corrugation Process and introduce the notion of Kuiper differential relations. This notion provides a natural framework for the construction of solutions with self-similarity properties. We consider the case of the totally real relation, we prove that it is Kuiper and we state a totally real isometric embedding theorem. We then show that the totally real isometric embeddings obtained by the Corrugation Process exhibits a self-similarity property. Kuiper relations also enable a uniform expression of the Corrugation Process that no longer involves integrals. This expression generalizes the ansatz used in arXiv:0905.0370 to generate isometric maps. We apply it to build a new explicit immersion of the real projective plane inside R^3.