Gilles Carron
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View article: Limits of manifolds with a Kato bound on the Ricci curvature. II
Limits of manifolds with a Kato bound on the Ricci curvature. II Open
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…
View article: Type problem, the first eigenvalue and Hardy inequalities
Type problem, the first eigenvalue and Hardy inequalities Open
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}:=\{\rho
View article: Strong Kato limit can be branching
Strong Kato limit can be branching Open
We provide an example of a non-collapsed strong Kato limit that is branching, essentially branching, and satisfies neither the $\mathrm{CD}(K,\infty)$ nor the $\mathrm{MCP}(K,N)$ conditions for any $K \in \mathbb{R}$ and $N \in [1,+\infty)…
View article: Type problem, the first eigenvalue and Hardy inequalities
Type problem, the first eigenvalue and Hardy inequalities Open
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 g…
View article: Convergence of the Yamabe flow on singular spaces with positive Yamabe constant
Convergence of the Yamabe flow on singular spaces with positive Yamabe constant Open
54 pages, 1 figure
View article: Kato meets Bakry-Émery
Kato meets Bakry-Émery Open
We prove that any complete Riemannian manifold with negative part of the Ricci curvature in a suitable Dynkin class is bi-Lipschitz equivalent to a finite-dimensional $\mathrm{RCD}$ space, by building upon the transformation rule of the Ba…
View article: Boundedness of Schr{\\"o}dinger operator in energy space
Boundedness of Schr{\\"o}dinger operator in energy space Open
On a complete weighted Riemannian manifold $(M^n,g,\\mu)$ satisfying the\ndoubling condition and the Poincar{\\'e} inequalities, we characterize the class\nof function $V$ such that the Schr{\\"o}dinger operator $\\Delta-V$ maps the\nhomog…
View article: Boundedness of Schr{ö}dinger operator in energy space
Boundedness of Schr{ö}dinger operator in energy space Open
On a complete weighted Riemannian manifold $(M^n,g,μ)$ satisfying the doubling condition and the Poincar{é} inequalities, we characterize the class of function $V$ such that the Schr{ö}dinger operator $Δ-V$ maps the homogeneous Sobolev spa…
View article: Torus stability under Kato bounds on the Ricci curvature
Torus stability under Kato bounds on the Ricci curvature Open
24 pages. Comments are welcome!
View article: The Yamabe flow on asymptotically Euclidean manifolds with nonpositive Yamabe constant
The Yamabe flow on asymptotically Euclidean manifolds with nonpositive Yamabe constant Open
We study the Yamabe flow on asymptotically flat manifolds with non-positive Yamabe constant $Y\leq 0$. Previous work by the second and third named authors \cite{ChenWang} showed that while the Yamabe flow always converges in a global weigh…
View article: Torus stability under Kato bounds on the Ricci curvature
Torus stability under Kato bounds on the Ricci curvature Open
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 manif…
View article: Limits of manifolds with a Kato bound on the Ricci curvature. II
Limits of manifolds with a Kato bound on the Ricci curvature. II Open
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…
View article: A rigidity result for metric measure spaces with Euclidean heat kernel
A rigidity result for metric measure spaces with Euclidean heat kernel Open
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 vo…
View article: Convergence of the Yamabe flow on singular spaces with positive Yamabe constant
Convergence of the Yamabe flow on singular spaces with positive Yamabe constant Open
In this work, we study the convergence of the normalized Yamabe flow with positive Yamabe constant on a class of pseudo-manifolds that includes stratified spaces with iterated cone-edge metrics. We establish convergence under a low energy …
View article: Limits of manifolds with a Kato bound on the Ricci curvature
Limits of manifolds with a Kato bound on the Ricci curvature Open
We study the structure of Gromov-Hausdorff limits of sequences of Riemannian manifolds $\{(M_α^n,g_α)\}_{α\in A}$ whose Ricci curvature satisfies a uniform Kato bound. We first obtain Mosco convergence of the Dirichlet energies to the Chee…
View article: $A_\infty$ weights and compactness of conformal metrics under $L^{n/2}$ curvature bounds
$A_\infty$ weights and compactness of conformal metrics under $L^{n/2}$ curvature bounds Open
International audience
View article: Euclidean Volume Growth for Complete Riemannian Manifolds
Euclidean Volume Growth for Complete Riemannian Manifolds Open
View article: Geometric inequalities for manifolds with Ricci curvature in the Kato class
Geometric inequalities for manifolds with Ricci curvature in the Kato class Open
We obtain Euclidean volume growth results for complete Riemannian manifolds satisfying a Euclidean Sobolev inequality and a spectral type condition on the Ricci curvature. We also obtain eigenvalue estimates, heat kernel estimates, and Bet…
View article: Euclidean volume growth for complete Riemannian manifolds
Euclidean volume growth for complete Riemannian manifolds Open
We provide an overview of technics that lead to an Euclidean upper bound on the volume of geodesic balls.
View article: Euclidean volume growth for complete Riemannian manifolds
Euclidean volume growth for complete Riemannian manifolds Open
We provide an overview of technics that lead to an Euclidean upper bound on\nthe volume of geodesic balls.\n
View article: A rigidity result for metric measure spaces with Euclidean heat kernel
A rigidity result for metric measure spaces with Euclidean heat kernel Open
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 vo…
View article: $A\_\infty$ weights and compactness of conformal metrics under $L^{n/2}$ curvature bounds
$A\_\infty$ weights and compactness of conformal metrics under $L^{n/2}$ curvature bounds Open
We study sequences of conformal deformations of a smooth closed Riemannian manifold of dimension $n$, assuming uniform volume bounds and $L^{n/2}$ bounds on their scalar curvatures. Singularities may appear in the limit. Nevertheless, we s…
View article: Geometric and spectral estimates based on spectral Ricci curvature assumptions
Geometric and spectral estimates based on spectral Ricci curvature assumptions Open
We obtain a Bonnet-Myers theorem under a spectral condition: a closed Riemannian manifold $(M^n,g)$ for which the lowest eigenvalue of the Ricci tensor $ρ$ is such that the Schrödinger operator $(n-2)Δ+ ρ$ is positive has finite fundamenta…
View article: Geometric and spectral estimates based on spectral Ricci curvature assumptions
Geometric and spectral estimates based on spectral Ricci curvature assumptions Open
We obtain a Bonnet-Myers theorem under a spectral condition: a closed Riemannian manifold $(M^n,g)$ for which the lowest eigenvalue of the Ricci tensor $\rho$ is such that the Schr\"odinger operator $(n-2)\Delta + \rho$ is positive has fin…
View article: Riesz transform on manifolds with quadratic curvature decay
Riesz transform on manifolds with quadratic curvature decay Open
We investigate the L^p -boundedness of the Riesz transform on Riemannian manifolds whose Ricci curvature has quadratic decay. Two criterions for the L^p -unboundedness of the Riesz transform are given. We recover known results about manifo…
View article: Geometric inequalities for manifolds with Ricci curvature in the Kato class
Geometric inequalities for manifolds with Ricci curvature in the Kato class Open
We obtain an Euclidean volume growth results for complete Riemannian manifolds satisfying a Euclidean Sobolev inequality and a spectral type condition on the Ricci curvature. We also obtain eigenvalue estimates, heat kernel estimates, Bett…
View article: Harmonic functions on Manifolds whose large spheres are small.
Harmonic functions on Manifolds whose large spheres are small. Open
We study the growth of harmonic functions on complete Riemannian manifolds where the extrinsic diameter of geodesic spheres is sublinear. It is an generalization of a result of A. Kasue. Our estimates also yields a result on the boundednes…
View article: Harmonic Functions On Manifolds Whose Large Sphere Are Small
Harmonic Functions On Manifolds Whose Large Sphere Are Small Open
We study the growth of harmonic functions on complete Riemann-ian manifolds where the extrinsic diameter of geodesic spheres is sublinear. It is an generalization of a result of A. Kazue. We also get a Cheng and Yau estimates for the gradi…
View article: Hölder regularity of solutions for Schrödinger operators on stratified spaces
Hölder regularity of solutions for Schrödinger operators on stratified spaces Open