arXiv subjects
Publications and source records attributed to Dorian Martino.
We prove that the renormalised volume between the two Epstein surfaces generated by a Weil--Petersson quasicircle is continuous with respect to the Weil--Petersson topology. As a consequence, we prove that the universal Liouville action is proportional to this renormalised volume for any Weil--Petersson quasicircle.
For every $n\geq 3$ we construct a bounded discontinuous map $u\in W^{1,n}(\mathbb{B}^n,\mathbb{R}^{n+2})$ solving an $n$-Laplace system with an antisymmetric potential $Ω\in L^n$. This shows that a recent regularity result on $n$-Laplace systems with antisymmetric potentials in Lorentz spaces by the authors is sharp in the sense that we cannot move from Lorentz spaces to classical Lebesgue spaces. This gives in particular a negative answer to a question by Rivière.
We show that $id:\mathbb{S}^1 \to \mathbb{S}^1$ is \emph{not} a minimizing $W^{s,\frac{1}{s}}$-harmonic map for $s \in (0,\frac{1}{8}$). On the other hand, for $s \in (\frac{1}{3},1)$ it is a local minimizing map, and for $s\in [\frac{1}{2}-\varepsilon,\frac{1}{2}+\varepsilon]$ it is a global minimizer. The usual extension or Fourier techniques being unavailable, our argument relies instead of stability analysis in $s$.
For $s \in (1/4,1)$ and any degree the only $W^{s,\frac{1}{s}}$-minimizers for $\mathbb{S}^1 \to \mathbb{S}^1$ maps are Blaschke products. This gives a resolution of Open Problems 23 and 24 in Brezis-Mironescu's mappings to the circle book, as well as Brezis' Favorite Open Problem 5.4 in this $s$-range. Previous results of this type were partial and restricted only to a small neighborhood of $s=\frac{1}{2}$. In particular, Brezis' Favorite Open Problem 5.1 is completely settled. Moreover, as a consequence of the argument, one also obtains linearized stability results.
Let $n\geq 3$ and let $Ω\subset \mathbb{R}^n$ be a $\mathcal{C}^1$ bounded domain which is diffeomorphic to a ball. We investigate here the problem of finding critical points of the $n$-energy in the space $\mathcal{I}=\{v\in W^{1,n}(Ω,\mathbb{R}^n) ; \ |\mathrm{tr}_{|\partial Ω}v|=1\}$. Maps in $\mathcal{I}$ have a well-defined topological degree on $\partial Ω$ but this degree is not continuous for the weak convergence in $W^{1,n}$. Hence finding critical points with prescribed degrees results in a problem of lack of compactness. We first prove that minimizers of the $n$-energy exist only when $Ω$ is a round ball and when the prescribed degree is $-1,0$ or $1$. We then develop a mountain pass approach for the $(n+α)$-energies and study the convergence, when $α$ goes to zero, of the resulting critical points via a bubbling analysis. We exclude the existence of bubbles in the case where $Ω$ is close to a ball by proving an energy gap result for free boundary $n$-harmonic maps from $\mathbb{B}^n$ to $\mathbb{B}^n$. We thus obtain the existence of critical points of the $n$-energy with prescribed degree $1$ when $Ω$ is close to a ball.
We review recent progress concerning the analysis of Lagrangians on immersions into $\mathbb{R}^d$ depending on the first and second fundamental forms and their covariant derivatives.
We consider a class of scale-invariant curvature energies defined on immersed $4$-dimensional manifolds and prove that weak immersions that are critical points of such energies are analytic in any given local harmonic chart. Because of the criticality of this variational problem, the regularity result is obtained through the identification of conservation laws by applying Noether theorem. The resulting identities generate a lower order elliptic system of PDEs to which methods from integrability by compensation and interpolation theory are applied.
For any $n \geq 3$ and any closed manifold $\mathcal{N}$ with $π_{n+k}(\mathcal{N}) \neq \{0\}$ for some $k \geq 0$, we establish the existence of nontrivial $n$-harmonic maps from $\mathbb{S}^n$ into $\mathcal{N}$. When $k\geq 1$, these maps naturally appear as bubbling limits of $p$-harmonic maps with $p > n$, obtained by min-max constructions in the limit $p \to n^+$.
We study the second Huber theorem in dimensions 2 and 4. In dimension 2, we prove a new version assuming that the Gauss curvature lies in a negative Sobolev space using Coulomb frames. In dimension $4$, given a metric having a pointwise singularity with $L^p$-bounds on the Bach tensor, we construct a conformal metric which is regular across the singularity. To do so, we introduce another Coulomb-type condition, similar to the case of Yang--Mills connections. This enables us to obtain a conformal metric satisfying an $\varepsilon$-regularity property. We obtain a generalization of the two-dimensional case that can be applied to study the singularities of Bach-flat metrics and immersions with second fundamental forms in $W^{2,\frac{4}{3}+\varepsilon}$.
We develop the analysis of Lipschitz immersions of $n$-dimensional manifolds into $\mathbb{R}^d$ having their second fundamental forms bounded in the critical Sobolev space $W^{\frac{n}{2}-1,2}$ in dimension $n\geq 4$ even and any codimension. We prove that, while such a weak immersion is not necessary $C^1$, it generates a $C^1$ differential structure on the domain. More precisely, for any such an immersion, there is an atlas in which the first fundamental form is continuous and the transition maps are $C^1$. We prove that this $C^1$ structure is diffeomorphic to the original one. This result is the starting point of the analysis of the behavior of sequences of weak immersions with second fundamental forms uniformly bounded in the critical Sobolev space $W^{\frac{n}{2}-1,2}$. In the second part of the paper we establish a weakly sequential closure theorem for such sequences. This analysis is motivated by the study of conformally invariant Lagrangian of immersions in dimension larger than two such as generalized Willmore energies, for instance the Graham--Reichert functional obtained in the computation of renormalized volumes of five-dimensional minimal submanifolds of the hyperbolic space $\mathbb{H}^{d+1}$.
We prove that any weak immersion in the critical Sobolev space $W^{\frac{n}{2}+1,2}(\mathbb{R}^n;\mathbb{R}^d)$ in even dimension $n\geq 4$, has global harmonic coordinates if its second fundamental form is small in the Sobolev space $W^{\frac{n}{2}-1,2}(\mathbb{R}^n;\mathbb{R}^d)$. This is a generalization to arbitrary even dimension $n\ge 4$ of a famous result of Müller--Sverak \cite{muller1995} for $n=2$. The existence of such coordinates is a key tool used by the authors in \cite{MarRiv20252} for the analysis of scale-invariant Lagrangians of immersions, such as the Graham--Reichert functional. From a purely intrinsic perspective, the proof of the main result leads to a general local existence theorem of harmonic coordinates for general metrics with Riemann tensor in $L^p$ for any $p>n/2$ in any dimension $n\geq 3$.
We study the umbilic points of Willmore surfaces in codimension 1 from the viewpoint of the conformal Gauss map. We first study the local behaviour of the conformal Gauss map near umbilic curves and prove that they are geodesics up to a conformal transformation if and only if the Willmore immersion is, up to a conformal transformation, the gluing of minimal surfaces in the 3-dimensional hyperbolic space. Then, we prove a Gauss--Bonnet formula for the conformal Gauss map of Willmore surfaces which turns out to be an asymptotic expansion involving the length of the umbilic curves in the spirit of renormalized volume expansions. We interpret this formula as a unified version for the different expressions of the value of the Willmore energy for conformally minimal surfaces in each space-form.
In this article, we show that sequences of $(n+α)$-harmonic maps with a free boundary in $\mathbb S^{d-1}$, where $α$ is a parameter tending to zero, converge to a bubble tree. For such sequences, we prove in detail that the limiting energy is equal to the energy of the macroscopic limit plus the sum of the energies of certain ``bubbles'', each multiplied by a corresponding coefficient.
Given $p\geq 2$ and a map $g : B^n(0,1)\to S_n^{++}$, where $S_n^{++}$ is the group of positively definite matrices, we study critical points of the following functional: $$ v\in W^{1,p}\left(B^n(0,1);\mathbb{R}^N \right) \mapsto \int_{B^n(0,1)} |\nabla v|^p_g\, d\mathrm{vol}_g = \int_{B^n(0,1)} \left( g^{αβ}(x) \left\langle \partial_αv(x), \partial_βv(x) \right\rangle \right)^{\frac{p}{2}}\, \sqrt{\det g(x)}\, dx. $$ We show that if $g$ is uniformly close to a constant matrix, then $v$ is locally Hölder-continuous. If $g$ is Hölder-continuous, we show that $\nabla v$ is locally Hölder-continuous. As an application, we prove that any Hölder-continuous solution to $|Δ_{g,p}u|\lesssim |\nabla u|^p_g$ satisfies additional regularity properties depending on the regularity of $g$. In the case $p=n$, only the continuity is assumed \textit{a priori}.
We consider rotated $n$-Laplace systems on the unit ball $B_1 \subset \mathbb{R}^n$ of the form \begin{align*} -\mathrm{div}\left( Q|\nabla u|^{n-2} \nabla u\right) = \mathrm{div}(G), \end{align*} where $u\in W^{1,n}(B_1;\mathbb{R}^N)$, $Q\in W^{1,n}(B_1;SO(N))$ and $G\in L^{\left( \frac{n}{n-1},q \right)}(B_1;\mathbb{R}^n\otimes \mathbb{R}^N)$ for some $0<q<\frac{n}{n-1}$. We prove that $\nabla u\in L^{(n,q(n-1))}_{loc}$ with estimates. As a corollary, we obtain that solutions to $Δ_n u \in \mathcal{H}^1$, where $\mathcal{H}^1$ is the Hardy space, have a higher integrability, namely $\nabla u \in L^{(n,n-1)}_{loc}$.
Given a branched Willmore immersion from a closed Riemann surface, we show that Bryant's quartic is holomorphic. Consequently, this quartic vanishes when the underlying surface is a sphere and we obtain the full classification of branched Willmore spheres. To do so, we show that the asymptotic expansion in the $C^2$-topology of the conformal Gauss map at a branched point is a null straight line.
We prove an analog of Bryant's duality theorem for a four dimensional Willmore energy $\mathcal{E}_{GR}$ obtained by Graham-Reichert and Zhang. We show that for an immersion $Φ$ from a four dimensional compact manifold without boundary $Σ$ into $\mathbb{R}^5$, the energy $\mathcal{E}_{GR}(Φ)$ is equal to two energies on its conformal Gauss map $Y$. One defined only in terms of the image of $Y$, which is the analog of the area functional for Willmore surfaces, and an other one defined on maps from $Σ$ into the De Sitter space $\mathbb{S}^{5,1}$, which is the analog of the Dirichlet energy for Willmore surfaces. We prove that even when restricted to immersions of a given topological manifold $Σ^4$, $\mathcal{E}_{GR}$ is never bounded from below on the set of immersions from $Σ$ into $\mathbb{R}^5$. We exhibit a second conformally invariant energy $\mathcal{E}_P$ which is bounded from below and whose construction is closer to the two dimensional Willmore energy.
We prove an energy quantization result for Willmore surfaces with bounded index, whether the underlying Riemann surfaces degenerates in the moduli space or not. To do so, we translate the question on the conformal Gauss map's point of view. In particular, we prove that in a neck or a collar region, the conformal Gauss map converges to a light-like geodesic in the De Sitter space.