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-Lee-Marsh 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

Scientific production

Publications

2. CR compactification for asymptotically locally complex hyperbolic almost Hermitian manifolds
   The Journal of Geometric Analysis 34(8), 238 (2024) [journal · arXiv:2307.04062]
   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.
3. Asymptotic strictly pseudoconvex CR structure for asymptotically locally complex hyperbolic manifolds
   Mathematische Zeitschrift 307(1), 8 (2024) [journal · arXiv:2201.12132]
   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.

Prepublications

0. Green's functions and a positive mass Theorem for asymptotically hyperbolic 3-manifolds
   with Klaus Kröncke and Francesca Oronzio
   In preparation
   Abstract

   We prove a monotonicity formula that holds along the level sets of the Green's function of an asymptotically hyperbolic 3-manifold. We then use it to prove a positive mass Theorem for the volume-renormalized mass, recently introduced by M. Dahl, K. Kröncke and S. McCormick.

PhD thesis

• Asymptotically complex hyperbolic geometry and curvature constraints
  Université de Montpellier [HAL]
  Abstract

  In this thesis, we investigate the asymptotic geometric properties of 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.

Physics

• Novel Consistency Conditions for Fixed Point Dynamics [arXiv:2504.05988]
  with Oleg Antipin, Francesco Sannino, and Shahram Vatani
  
  Abstract

  We gain insight on the fixed point dynamics of \(d\) dimensional quantum field theories by exploiting the critical behavior of the \(d-\epsilon\) sister theories. To this end we first derive a self-consistent relation between the \(d-\epsilon\) scaling exponents and the associated \(d\) dimensional \(\beta\)-functions. We then demonstrate that to account for an interacting fixed point in the original theory the related \(d-\epsilon\) scaling exponent must be multi-valued in \(\epsilon\). We elucidate our findings by discussing several examples such as the QCD Banks-Zaks infrared fixed point, QCD at large number of flavors, as well as the \(O(N)\) model in four dimensions. For the latter, we show that although the \(1/N\) corrections prevent the reconstruction of the renormalization group flow, this is possible when adding the \(1/N^2\) contributions.