Gilles Carron
Current Organizations / Institutes
The organizations/institutes from the scholar's latest article
Previous Organizations / Institutes
The organizations / institutes from the scholar's other articles in the past ten years
Research Subjects
Gilles Carron has a total of 42 co-authors. Here are the top co-authors by number of shared publications:
- Ilaria Mondello
Laboratoire d’Analyse et Mathématiques appliquées, Université Paris Est Créteil, Créteil, France
5 shared publications - Rafe Mazzeo
Department of Mathematics, Stanford University, Stanford, USA
4 shared publications Werner Ballmann
4 shared publications- David Tewodrose
Vrije Universiteit Brussel, Belgium
3 shared publications - Jochen Brüning
Humboldt Universität zu Berlin
3 shared publications - Thierry Coulhon
Université de Cergy-Pontoise
3 shared publications
Latest Publications
- Open Access
- 16 July 2025
- Published by Finnish Mathematical Society in Annales Fennici Mathematici
- Vol. 50 (2) , 623–663
Limits of manifolds with a Kato bound on the Ricci curvature. II
We prove that metric measure spaces obtained as limits of closed Riemannian manifolds with Ricci curvature satisfying a uniform Kato bound are rectifiable. In the case of a non-collapsing assumption and a strong Kato bound, we additionally show that for any \(\alpha \in (0,1)\) the regular part of the space lies in an open set with the structure of a \(C^\alpha\)-manifold.- 16 July 2025
- Published by Finnish Mathematical Society in Annales Fennici Mathematici
- Vol. 50 (2) , 623–663
Open AccessLimits of manifolds with a Kato bound on the Ricci curvature. II
We prove that metric measure spaces obtained as limits of closed Riemannian manifolds with Ricci curvature satisfying a uniform Kato bound are rectifiable. In the case of a non-collapsing assumption and a strong Kato bound, we additionally show that for any \(\alpha \in (0,1)\) the regular part of the space lies in an open set with the structure of a \(C^\alpha\)-manifold. - 28 June 2025
- Published by Springer Nature in Calculus of Variations and Partial Differential Equations
- Vol. 64 (6) , 1-29
Kato meets Bakry–Émery
- 28 June 2025
- Published by Springer Nature in Calculus of Variations and Partial Differential Equations
- Vol. 64 (6) , 1-29
Kato meets Bakry–Émery
- 18 February 2025
- Published by European Mathematical Society - EMS - Publishing House GmbH in Journal of Spectral Theory
- Vol. 15 (1) , 279-303
Type problem, the first eigenvalue and Hardy inequalities
In this paper, we study the relationship between the type problem and the asymptotic behaviour of the first (Dirichlet) eigenvalues \lambda_{1}(B_{r}) of “balls” B_{r}:=\{\rho0 for all r>r_{0} , we obtain a sharp estimate of the volume growth: |B_{r}|\ge cr^{\mu(\gamma)}. Moreover, when \gamma>j_{0}^{2}\approx 5.784 , where j_{0} denotes the first positive zero of the Bessel function J_{0} , then M is non-parabolic and we have a Hardy-type inequality. In the case where r_{0}=0 , a sharp Hardy-type inequality holds. These spectral conditions are satisfied if one assumes that \Delta\rho^{2}\geq2\mu(\gamma)>0 . In particular, when \inf_{M}\Delta\rho^{2}>4 , M is non-parabolic and we get a sharp Hardy-type inequality. Related results for finite volume case are also studied.
- 18 February 2025
- Published by European Mathematical Society - EMS - Publishing House GmbH in Journal of Spectral Theory
- Vol. 15 (1) , 279-303
Type problem, the first eigenvalue and Hardy inequalities
In this paper, we study the relationship between the type problem and the asymptotic behaviour of the first (Dirichlet) eigenvalues \lambda_{1}(B_{r}) of “balls” B_{r}:=\{\rho0 for all r>r_{0} , we obtain a sharp estimate of the volume growth: |B_{r}|\ge cr^{\mu(\gamma)}. Moreover, when \gamma>j_{0}^{2}\approx 5.784 , where j_{0} denotes the first positive zero of the Bessel function J_{0} , then M is non-parabolic and we have a Hardy-type inequality. In the case where r_{0}=0 , a sharp Hardy-type inequality holds. These spectral conditions are satisfied if one assumes that \Delta\rho^{2}\geq2\mu(\gamma)>0 . In particular, when \inf_{M}\Delta\rho^{2}>4 , M is non-parabolic and we get a sharp Hardy-type inequality. Related results for finite volume case are also studied.
- Open Access
- 21 October 2024
- Published by Mathematical Sciences Publishers in Geometry & Topology
- Vol. 28 (6) , 2635-2745
Limits of manifolds with a Kato bound on the Ricci curvature
We study the structure of Gromov-Hausdorff limits of sequences of Riemannian manifolds f.M n ˛; g ˛/g ˛2A whose Ricci curvature satisfies a uniform Kato bound. We first obtain Mosco convergence of the Dirichlet energies to the Cheeger energy, and show that tangent cones of such limits satisfy the RCD.0; n/ condition. Under a noncollapsing assumption, we introduce a new family of monotone quantities, which allows us to prove that tangent cones are also metric cones. We then show the existence of a well-defined stratification in terms of splittings of tangent cones. We finally prove volume convergence to the Hausdorff n-measure.
- 21 October 2024
- Published by Mathematical Sciences Publishers in Geometry & Topology
- Vol. 28 (6) , 2635-2745
Open AccessLimits of manifolds with a Kato bound on the Ricci curvature
We study the structure of Gromov-Hausdorff limits of sequences of Riemannian manifolds f.M n ˛; g ˛/g ˛2A whose Ricci curvature satisfies a uniform Kato bound. We first obtain Mosco convergence of the Dirichlet energies to the Cheeger energy, and show that tangent cones of such limits satisfy the RCD.0; n/ condition. Under a noncollapsing assumption, we introduce a new family of monotone quantities, which allows us to prove that tangent cones are also metric cones. We then show the existence of a well-defined stratification in terms of splittings of tangent cones. We finally prove volume convergence to the Hausdorff n-measure.
- Open Access
Type problem, the first eigenvalue and Hardy inequalities
In this paper, we study the relationship between the type problem and the asymptotic behaviour of the first (Dirichlet) eigenvalues $λ_1(B_r)$ of ``balls'' $B_r:=\{ρr_0$ \[ r^2 λ_1(B_r)\ge γ>0, \] we obtain a sharp estimate of the volume growth: $|B_r|\ge cr^{μ(γ)}.$ Moreover when $γ>j_0^2\approx 5.784$, where $j_0$ denotes the first positive zero of the Bessel function $J_0$, then $M$ is hyperbolic and we have a Hardy type inequality. In the case where $r_0=0$, a sharp Hardy type inequality holds. These spectral conditions are satisfied if one assumes that $Δρ^2\geq2μ(γ)>0$. In particular, when $\inf_MΔρ^2>4$, $M$ is hyperbolic and we get a sharp Hardy type inequality. Related results for finite volume case are also studied.Open AccessType problem, the first eigenvalue and Hardy inequalities
In this paper, we study the relationship between the type problem and the asymptotic behaviour of the first (Dirichlet) eigenvalues $λ_1(B_r)$ of ``balls'' $B_r:=\{ρr_0$ \[ r^2 λ_1(B_r)\ge γ>0, \] we obtain a sharp estimate of the volume growth: $|B_r|\ge cr^{μ(γ)}.$ Moreover when $γ>j_0^2\approx 5.784$, where $j_0$ denotes the first positive zero of the Bessel function $J_0$, then $M$ is hyperbolic and we have a Hardy type inequality. In the case where $r_0=0$, a sharp Hardy type inequality holds. These spectral conditions are satisfied if one assumes that $Δρ^2\geq2μ(γ)>0$. In particular, when $\inf_MΔρ^2>4$, $M$ is hyperbolic and we get a sharp Hardy type inequality. Related results for finite volume case are also studied. - 22 December 2022
- Published by Elsevier in Journal of Functional Analysis
- Vol. 284 (6)
The Yamabe flow on asymptotically Euclidean manifolds with nonpositive Yamabe constant
- 22 December 2022
- Published by Elsevier in Journal of Functional Analysis
- Vol. 284 (6)
The Yamabe flow on asymptotically Euclidean manifolds with nonpositive Yamabe constant
- Open Access
Boundedness of Schr{ö}dinger operator in energy space
On a complete weighted Riemannian manifold $(M^n,g,\mu)$ satisfying the doubling condition and the Poincar{\'e} inequalities, we characterize the class of function $V$ such that the Schr{\"o}dinger operator $\Delta-V$ maps the homogeneous Sobolev space $W_o^{1,2} (M)$ to its dual space. On Euclidean space, this result is due to Maz'ya and Verbitsky. In the proof of our result, we investigate the weighted $L^2$-boundedness of the Hodge projector.Open AccessBoundedness of Schr{ö}dinger operator in energy space
On a complete weighted Riemannian manifold $(M^n,g,\mu)$ satisfying the doubling condition and the Poincar{\'e} inequalities, we characterize the class of function $V$ such that the Schr{\"o}dinger operator $\Delta-V$ maps the homogeneous Sobolev space $W_o^{1,2} (M)$ to its dual space. On Euclidean space, this result is due to Maz'ya and Verbitsky. In the proof of our result, we investigate the weighted $L^2$-boundedness of the Hodge projector. - 25 November 2022
- Published by European Mathematical Society - EMS - Publishing House GmbH in Oberwolfach Reports
- Vol. 18 (4) , 3135-3186
Mini-Workshop: Variable Curvature Bounds, Analysis and Topology on Dirichlet Spaces
A Dirichlet form $\mathcal{E}$ is a densely defined bilinear form on a Hilbert space of the form $L^2(X, \mu)$, subject to some additional properties, which make sure that $\mathcal{E}$ can be considered as a natural abstraction of the usual Dirichlet energy $\mathcal{E}(f1, f2) = \int\_D (\nabla f1,\nabla f2)$ on a domain $D$ in $\mathbb{R}^m$. The main strength of this theory, however, is that it allows also to treat nonlocal situations such as energy forms on graphs simultaneously. In typical applications, $X$ is a metrizable space, and the theory of Dirichlet forms makes it possible to define notions such as curvature bounds on $X$ (although $X$ need not be a Riemannian manifold), and also to obtain topological information on $X$ in terms of such geometric information.- 25 November 2022
- Published by European Mathematical Society - EMS - Publishing House GmbH in Oberwolfach Reports
- Vol. 18 (4) , 3135-3186
Mini-Workshop: Variable Curvature Bounds, Analysis and Topology on Dirichlet Spaces
A Dirichlet form $\mathcal{E}$ is a densely defined bilinear form on a Hilbert space of the form $L^2(X, \mu)$, subject to some additional properties, which make sure that $\mathcal{E}$ can be considered as a natural abstraction of the usual Dirichlet energy $\mathcal{E}(f1, f2) = \int\_D (\nabla f1,\nabla f2)$ on a domain $D$ in $\mathbb{R}^m$. The main strength of this theory, however, is that it allows also to treat nonlocal situations such as energy forms on graphs simultaneously. In typical applications, $X$ is a metrizable space, and the theory of Dirichlet forms makes it possible to define notions such as curvature bounds on $X$ (although $X$ need not be a Riemannian manifold), and also to obtain topological information on $X$ in terms of such geometric information.- Open Access
Torus stability under Kato bounds on the Ricci curvature
We show two stability results for a closed Riemannian manifold whose Ricci curvature is small in the Kato sense and whose first Betti number is equal to the dimension. The first one is a geometric stability result stating that such a manifold is Gromov-Hausdorff close to a flat torus. The second one states that, under a stronger assumption, such a manifold is diffeomorphic to a torus: this extends a result by Colding and Cheeger-Colding obtained in the context of a lower bound on the Ricci curvature.
Open AccessTorus stability under Kato bounds on the Ricci curvature
We show two stability results for a closed Riemannian manifold whose Ricci curvature is small in the Kato sense and whose first Betti number is equal to the dimension. The first one is a geometric stability result stating that such a manifold is Gromov-Hausdorff close to a flat torus. The second one states that, under a stronger assumption, such a manifold is diffeomorphic to a torus: this extends a result by Colding and Cheeger-Colding obtained in the context of a lower bound on the Ricci curvature.
- Open Access
- 17 November 2021
- Published by MathDoc/Centre Mersenne in Journal de l’École polytechnique — Mathématiques
- Vol. 9, 101-154
A rigidity result for metric measure spaces with Euclidean heat kernel
We prove that a metric measure space equipped with a Dirichlet form admitting an Euclidean heat kernel is necessarily isometric to the Euclidean space. This helps us providing an alternative proof of Colding’s celebrated almost rigidity volume theorem via a quantitative version of our main result. We also discuss the case of a metric measure space equipped with a Dirichlet form admitting a spherical heat kernel.
- 17 November 2021
- Published by MathDoc/Centre Mersenne in Journal de l’École polytechnique — Mathématiques
- Vol. 9, 101-154
Open AccessA rigidity result for metric measure spaces with Euclidean heat kernel
We prove that a metric measure space equipped with a Dirichlet form admitting an Euclidean heat kernel is necessarily isometric to the Euclidean space. This helps us providing an alternative proof of Colding’s celebrated almost rigidity volume theorem via a quantitative version of our main result. We also discuss the case of a metric measure space equipped with a Dirichlet form admitting a spherical heat kernel.