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Filippo Gaia

Publications and source records attributed to Filippo Gaia.

10 recordsLinked to original sources
arXiv2026

Special Lagrangians with multiple isolated singularities

We extend the Caffarelli-Hardt-Simon perturbation argument for truncated regular minimal cones to the special Lagrangian setting and prove a bridge principle for regular special Lagrangian cones in the spirit of Nathan Smale. Our bridge principle yields a general existence theorem for conically singular special Lagrangian submanifolds with prescribed regular tangent cones: for any finite list of such cones in $\mathbb{C}^m$ having the same Lagrangian angle and suitably arranged, there exists a connected special Lagrangian submanifold with boundary and isolated conical singularities whose tangent cones at its singularities are precisely the prescribed cones. In particular, we obtain new special Lagrangian submanifolds in $\mathbb{C}^m$ with multiple prescribed isolated conical singularities.

math.DG↗
arXiv2026

Existence of constant mean curvature surfaces with controlled topology in 3-manifolds

We establish the existence of a non-trivial, branched immersion of a closed Riemann surface $\Sigma$ with constant mean curvature (CMC) $H$ into any closed, orientable 3-manifold $\mathcal{M}$, for almost every prescribed value of $H$. The genus of the surface $\Sigma$ is bounded from above by the Heegaard genus $h$ of $\mathcal{M}$. Starting from a family of sweep-outs of $\mathcal{M}$ by surfaces of genus $h$, we apply a min-max construction for a family $\{E_{H,\sigma}\}_\sigma$ of perturbations of the energy involving the second fundamental form of the immersions to produce almost-critical points $u_k$ of $E_{H,\sigma}$. We then show, following ideas introduced by Rivi\`ere and developed by Pigati and Rivi\`ere, that the maps $u_k$ converge to a "CMC-parametrized varifold". This limiting object is then shown to be a smooth, branched immersion with the prescribed mean curvature $H$.

math.DG↗
arXiv2025

IMProofBench: Benchmarking AI on Research-Level Mathematical Proof Generation

As the mathematical capabilities of large language models (LLMs) improve, it becomes increasingly important to evaluate their performance on research-level tasks at the frontier of mathematical knowledge. However, existing benchmarks are limited, as they focus solely on final-answer questions or high-school competition problems. To address this gap, we introduce IMProofBench, a private benchmark consisting of 77 peer-reviewed problems developed by expert mathematicians. Each problem requires a detailed proof and is paired with subproblems that have final answers, supporting both an evaluation by human experts and a large-scale quantitative analysis through automated grading. Furthermore, unlike prior benchmarks, the evaluation setup simulates a realistic research environment: models operate in an agentic framework with tools like web search for literature review and mathematical software such as SageMath. Our results show that current LLMs can already solve a significant percentage of research-level questions. IMProofBench will continue to evolve as a dynamic benchmark in collaboration with the mathematical community, ensuring its relevance for evaluating the next generation of LLMs.

cs.CL↗
arXiv2024

Free boundary Hamiltonian stationary Lagrangian discs in $\mathbb{C}^2$

Let $\Omega \subset \mathbb{C}^2$ be a smooth domain. We establish conditions under which a weakly conformal, branched $\Omega$-free boundary Hamiltonian stationary Lagrangian immersion $u$ of a disc in $\mathbb{C}^2$ is a $\Omega$-free boundary minimal immersion. We deduce that if $u$ is a weakly conformal, branched $B_1(0)$-free boundary Hamiltonian stationary Lagrangian immersion of a disc with Legendrian boundary data, then $u(D^2)$ must be a Lagrangian equatorial plane disc. We also present examples of $\Omega$-free boundary Hamiltonain stationary discs, demonstrating the optimality of our assumptions.

math.DG↗
arXiv2024

The fractional Hopf differential and a weak formulation of stationarity for the half Dirichlet energy

We obtain a weak formulation of the stationarity condition for the half Dirichlet energy, which can be expressed in terms of a fractional analogous to the Hopf differential. As an application we show that conformal harmonic maps from the disc are precisely the harmonic extensions of stationary points of the half Dirichlet energy on the circle. We also derive a Noether theorem and a Pohozaev identity for stationary points of the half Dirichlet energy.

math.AP↗
arXiv2023

A Variational Construction of Hamiltonian Stationary Surfaces with Isolated Schoen-Wolfson Conical Singularities

We construct using variational methods Hamiltonian Stationary Surfaces with Isolated Schoen-Wolfson Conical Singularities. We obtain these surfaces through a convergence process reminiscent to the Ginzburg-Landau asymptotic analysis in the strongly repulsive regime. We describe in particular how the prescription of Schoen Wolfson conical singularities is related to optimal Wente constants.

math.DG↗
arXiv2022

Weak and strong $L^p$-limits of vector fields with finitely many integer singularities in dimension $n$

For every given $p\in [1,+\infty)$ and $n\in\mathbb{N}$ with $n\ge 1$, the authors identify the strong $L^p$-closure $L_{\mathbb{Z}}^p(D)$ of the class of vector fields having finitely many integer topological singularities on a domain $D$ which is either bi-Lipschitz equivalent to the open unit $n$-dimensional cube or to the boundary of the unit $(n+1)$-dimensional cube. Moreover, for every $n\in\mathbb{N}$ with $n\ge 2$ the authors prove that $L_{\mathbb{Z}}^p(D)$ is weakly sequentially closed for every $p\in (1,+\infty)$ whenever $D$ is an open domain in $\mathbb{R}^n$ which is bi-Lipschitz equivalent to the open unit cube. As a byproduct of the previous analysis, a useful characterisation of such class of objects is obtained in terms of existence of a (minimal) connection for their singular set.

math.FA↗
arXiv2021

A variational approach to $S^1$-harmonic maps and applications

We present a renormalization procedure of the Dirichlet Lagrangian for maps from surfaces with or without boundary into $S^1$ and whose finite energy critical points are the $S^1-$harmonic maps with isolated singularities. We give some applications of this renormalization scheme in two different frameworks. The first application has to do with the renormalization of the Willmore energy for Lagrangian singular immersions into K\"ahler-Einstein Surfaces while the second application is dealing with frame energies for surfaces immersions into Euclidian spaces.

math.DG↗
arXiv2020

Noether Theorems for Lagrangians involving fractional Laplacians

In this work we derive Noether Theorems for energies of the form \begin{equation*} E(u)=\int_ΩL\left(x,u(x),(-Δ)^\frac{1}{4}u(x)\right)dx \end{equation*} for Lagrangians exhibiting invariance under a group of transformations acting either on the target or on the domain of the admissible functions $u$, in terms of fractional gradients and fractional divergences. Here $Ω$ stays either for an Euclidean space $\mathbb{R}^n$ or for the circle $\mathbb{S}^1$. We then discuss some applications of these results and related techniques to the study of nonlocal geometric equations and to the study of stationary points of the half Dirichlet energy on $\mathbb{S}^1$. In particular we introduce the $\frac{1}{2}$-fractional Hopf differential as a simple tool to characterize stationary point of the half Dirichlet energy in $H^\frac{1}{2}(\mathbb{S}^1,\mathbb{R}^m)$ and study their properties. Finally we show how the invariance properties of the half Dirichlet energy on $\mathbb{R}$ can be used to obtain Pohozaev identities.

math.AP↗