CR compactification for asymptotically locally complex hyperbolic almost Hermitian
manifolds
The Journal of Geometric Analysis34(8), 238 (2024)
[journal🔓, arXiv]
Abstract
This article extends the result of [1] to the case of almost Hermitian manifolds
\((M,g,J)\), for which the almost complex structure \(J\) is not parallel (and in fact \(J\)
is not integrable, and the almost symplectic form \(\omega = g(J\cdot,\cdot)\) is not
closed).
We also lower the assumption on the decay rate \(a\).
We show that if
\[
\|R-R^0\|_g,\quad \|\nabla R\|_g,\quad \|\nabla J\|_g,\quad \|\nabla^2 J\|_g =
\mathcal{O}(e^{-ar}), \quad a>1,
\]
then \((M,J)\) arises as the interior of an almost complex manifold with strictly
pseudoconvex, integrable, CR boundary of class \(\mathcal{C}^1\).
In addition, the metric \(g\) is asymptotically complex hyperbolic, and one recovers the CR
structure of the boundary by analysing the asymptotic development of the metric near
infinity.
This yields a geometric characterisation of asymptotically complex hyperbolic almost
Hermitian manifolds.
Asymptotic strictly pseudoconvex CR structure for asymptotically locally complex hyperbolic
manifolds
Mathematische Zeitschrift307(1), 8 (2024)
[journal🔓, arXiv]
Abstract
Let \((M,g,J)\) be a complete noncompact Kähler manifold of dimension greater or equal to
\(4\), \(R\) its curvature tensor, and \(R^0\) that of the complex hyperbolic space.
We show that if there exists \(a>3/2\) with
\[
\|R-R^0\|_g, \quad \|\nabla R \|_g = \mathcal{O}(e^{-ar}),
\]
then \((M,g,J)\) has a boundary at infinity, which is constructed geometrically, and which
is a strictly pseudoconvex CR manifold of class \(\mathcal{C}^1\).
Moreover, the Riemannian metric \(g\) is asymptotically complex hyperbolic : the CR
structure at infinity is recovered by analysing the asymptotic development of \(g\) near
infinity.
This result yields a geometric characterisation of asymptotically complex hyperbolic Kähler
manifolds.
PhD thesis
Asymptotically complex hyperbolic geometry and curvature constraints
Université de Montpellier
[HAL]
Abstract
In this thesis, we investigate the asymptotic geometric properties a class of complete and
non compact Kähler manifolds we call asymptotically locally complex hyperbolic manifolds.
The local geometry at infinity of such a manifold is modeled on that of the complex
hyperbolic space, in the sense that its curvature is asymptotic to that of the model space.
The natural geometric assumptions, we show that this constraint on the curvature ensures the
existence of a rich geometry at infinity: we can endow it with a strictly pseudoconvex CR
boundary at infinity.
Scientific interests
Overview
During my PhD, I constructed geometric boundaries at infinity for some asymptotically complex hyperbolic
Kähler manifolds.
This result is the analogue of that obtained by Bahuaud-Gicquaud-Marsh-Lee in the real hyperbolic
setting.
I then generalised this result to the almost Hermitian case.
In Stockholm, with Klaus Kröncke and Francesca Oronzio, we studied a positive mass theorem for
asymptotically hyperbolic 3-manifolds via the theory of potentials (this is still an ongoing
project).
I am currently interested in the construction of 1-parameter families of asymptotically hyperbolic
Einstein metrics degenerating to asymptotically complex hyperbolic Einstein metrics.
At the level of the boundaries at infinity, the CR boundary of the limit is obtained as an adiabatic
limit of the conformal infinities of the 1-parameter family.
Keywords
Differential and Riemannian geometry
Negatively curved manifolds
Einstein metrics
Geometric analysis
Elliptic theory
Interactions between groups and geometry
Talks
Conferences
08/12/2023: Einstein Spaces and Special Geometry, Mittag-Leffler Institute, Stockholm (slides)
Seminars
02/12/2025: Geometry seminar, Marseille
12/16/2024: Geometry seminar, Nancy
11/29/2024: Geometry seminar, Brest
09/14/2024: Geometry seminar, Université Libre de Bruxelles
06/11/2024: Geometry seminar, Max Planck Institute, Leipzig
01/08/2024: Geometry seminar, Université Libre de Bruxelles
04/06/2023: Geometry seminar, Marseille
03/24/2023: Geometry seminar, Nantes
03/06/2023: Geometry seminar, Francfort
03/02/2023: Differential geometry seminar, KTH Stockholm
11/20/2022: Differential geometry seminar, KTH Stockholm
05/24/2022: Geometry seminar, Tours
05/13/2022: Differential geometry seminar, KTH Stockholm
02/16/2022: Geometry seminar, groupes et dynamique, ENS de Lyon
01/27/2022: Spectral theory and geometry seminar, Grenoble
12/17/2021: Darboux seminar, Montpellier
Workshops
11/20/2024: Workshop Homologie de Floer, Les Plantiers
05/10/2022: Block seminar on Einstein 4-manifold, Sulzbürg
02/08/2019: Workshop Scalar curvature and rigidity, Montpellier
02/06/2019: Workshop Scalar curvature and rigidity, Montpellier
