arXiv subjects

Thibault Langlais

Publications and source records attributed to Thibault Langlais.

5 recordsLinked to original sources

arXiv2026

Deformations of harmonic maps with conical singularities

We study the deformation theory of harmonic maps with isolated singularities between compact Riemannian manifolds, in the case where all the tangent maps are smooth (away from the origin) and the decay to the tangents is polynomial. We will refer to these as conically singular harmonic maps. Under certain conditions on the Morse index and the nullity of the tangent maps and a non-degeneracy assumption on a weighted kernel of the Jacobi operator, we prove that such harmonic maps persist under a small $C^2$-perturbation of the background metric. In the second part of the paper, we construct examples of conically singular harmonic maps satisfying the assumptions of our deformation theorem. We first prove that the desired conditions on the tangent maps are satisfied by the radial projections $\mathbb{R}^m \setminus \{0\} \to \mathbb{S}^{m-1}$ ($m \geq 4$) and the complex Hopf fibrations $\mathbb{C}^{n+1} \setminus \{0\} \to \mathbb{CP}^{n}$ ($n \geq 1$). We then construct explicit examples of conically singular maps $\mathbb{S}^{m} \setminus \{N,S\} \to \mathbb{S}^{m-1}$, $\mathbb{S}^{4} \setminus \{N,S\} \to \mathbb{S}^2$ and $\mathbb{CP}^2 \setminus \{[0:0:1]\} \to \mathbb{CP}^1$ which are harmonic and satisfy the assumptions of the deformation theorem with respect to an appropriate metric on the domain and the round metric on the target.

math.DG↗

arXiv2026

Degenerations of exotic Calabi-Yau metrics through Atiyah's flop

We construct new families of complete Calabi-Yau metrics with maximal volume growth on the small resolutions of the conifold $\mathcal{Z} = \{z_1^2 + z_2^2 + z_3^2 + z_4^2 = 0\} \subset \mathbb{C}^4$. These metrics have tangent cone $\mathbb{C} \times (\mathbb{C}^2 / \mathbb{Z}_2)$ at infinity and are parametrised by their Kähler class. As the Kähler class degenerates, the metrics converge in the pointed Gromov-Hausdorff sense to a Calabi-Yau metric on $\mathcal{Z}$ with an isolated conical singularity modelled on the Stenzel metric at the ordinary double point and tangent cone at infinity $\mathbb{C} \times (\mathbb{C}^2/\mathbb{Z}_2)$, thereby providing a new metric realisation of the Atiyah flop.

math.DG↗

arXiv2025

Geometry and periods of $G_2$-moduli spaces

This paper is concerned with the geometry of the moduli space $\mathscr{M}$ of torsion-free $G_2$-structures on a compact $G_2$-manifold $M$, equipped with the volume-normalised $L^2$-metric $\mathscr{G}$. When $b^1(M) = 0$, this metric is known to be of Hessian type and to admit a global potential. Here we give a new description of the geometry of $\mathscr{M}$, based on the observation that there is a natural way to immerse the moduli space into a homogeneous space $\mathfrak{D}$ diffeomorphic to $GL(n+1)/ (\{\pm 1\} \times O(n))$, where $n = b^3(M) - 1$. We point out that the formal properties of this immersion $Φ: \mathscr{M} \rightarrow \mathfrak{D}$ are very similar to those of the period map defined on the moduli spaces of Calabi--Yau threefolds. With a view to understand the curvatures of $\mathscr{G}$, we also derive a new formula for the fourth derivative of the potential and relate it to the second fundamental form of $Φ(\mathscr{M}) \subset \mathfrak{D}$.

math.DG↗

arXiv2025

On the incompleteness of $G_2$-moduli spaces along degenerating families of $G_2$-manifolds

We derive a formula for the energy of a path in the moduli space of a compact $G_2$-manifold with vanishing first Betti number for the volume-normalised $L^2$-metric. This allows us to give simple sufficient conditions for a path of torsion-free $G_2$-structures to have finite energy and length. We deduce that the compact $G_2$-manifolds produced by the generalised Kummer construction have incomplete moduli spaces. Under some assumptions, we also state a necessary condition for the limit of a path of torsion-free $G_2$-structures to be at infinite distance in the moduli space.

math.DG↗

arXiv2024

Analysis and spectral theory of neck-stretching problems

We study the mapping properties of a large class of elliptic operators $P_T$ in gluing problems where two non-compact manifolds with asymptotically cylindrical geometry are glued along a neck of length $2T$. In the limit where $T \rightarrow \infty$, we reduce the question of constructing approximate solutions of $P_T u = f$ to a finite-dimensional linear system, and provide a geometric interpretation of the obstructions to solving this system. Under some assumptions on the real roots of the model operator $P_0$ on the cylinder, we construct a Fredholm inverse for $P_T$ with good control on the growth of its norm. As applications of our method, we study the decay rate and density of the low eigenvalues of the Laplacian acting on differential forms, and give improved estimates for compact $G_2$-manifolds constructed by twisted connected sum. We relate our results to the swampland distance conjectures in physics.

math.DG↗