arXiv subjects

Marco Usula

Publications and source records attributed to Marco Usula.

6 recordsLinked to original sources

arXiv2026

Minimal surfaces, Knots, and Neural Networks

A recent conjecture by Joel Fine posits a relationship between the coefficients of the HOMFLY polynomial of a knot $K$ in the 3-sphere $S^3$, and the signed count of minimal surfaces in hyperbolic 4-space $\mathrm{H}^4$ meeting the sphere at infinity at $K$, with prescribed genus and self-intersection number. In this paper, we develop a novel machine learning framework based on Physics-Informed Neural Networks (PINNs) to solve the minimal surface equation in hyperbolic space. We utilise this framework to test Fine's Conjecture by constructing near-minimal surfaces bounding various families of knots in $S^3$. Furthermore, we develop an algorithmic method to find self-intersections and compute their sign. For every knot analysed, the computationally discovered minimal surfaces and their self-intersection numbers perfectly align with the predictions of Fine's Conjecture, providing empirical evidence for it.

math.DG↗

arXiv2026

Nahm Poles and 0-Instantons

We study self-dual 0-connections, or 0-instantons, on asymptotically hyperbolic 4-manifolds. These connections develop a uniform singularity along the conformal infinity, and are asymptotic, at each point of the boundary, to a "Nahm pole" model solution on $H^4$. Examples include the Levi-Civita spin connections on $S^+$ over spin Poincar\'e-Einstein 4-manifolds. Inspired by the Fefferman-Graham expansion for Poincar\'e-Einstein metrics, we study the asymptotic expansion of these 0-instantons. We prove that the expansion is log-smooth, and that the coefficient of the first log term - which we call the 0-instanton obstruction tensor - is a conformal invariant related to the Weyl curvature of the ambient conformal metric. We then show that this invariant vanishes if and only if the 0-instanton is smooth modulo gauge. Finally, we study the renormalized Yang-Mills energy: we prove that, if the metric is asymptotically Poincar\'e-Einstein to third order, then this energy is a well-defined conformal invariant, and equals the negative Chern-Simons invariant of the conformal infinity.

math.DG↗

arXiv2025

Isometric Embeddings of Conformally Compact Manifolds into Hyperbolic Spaces

The celebrated Nash Embedding Theorem asserts that every closed Riemannian manifold can be isometrically embedded into a sufficiently high-dimensional Euclidean space. In this paper, we prove an analogous result in the conformally compact context. Let $\left(M,g\right)$ be a conformally compact manifold whose sectional curvature at infinity is strictly bounded below by a negative constant $-\lambda^{2}$. We prove that $\left(M,g\right)$ can be realized as a submanifold, transverse to the sphere at infinity, of a sufficiently high-dimensional rescaled hyperbolic space of constant curvature $-\lambda^{2}$.

math.DG↗

arXiv2025

Biharmonic Maps Between Conformally Compact Manifolds

We study biharmonic maps between conformally compact manifolds, a large class of complete manifolds with bounded geometry, asymptotically negative curvature, and smooth compactification. These metrics provide a far-reaching generalization of hyperbolic space. We work on the class of simple $b$-maps, i.e. maps which send interior to interior, boundary to boundary, and are transversal to the boundary of the target manifold. The main result of this paper is a non-existence result: if a simple $b$-map $u:\left(M,g\right)\to\left(N,h\right)$ between conformally compact manifolds is biharmonic, its restriction to the boundary is non-constant, and moreover $\left(N,h\right)$ is non-positively curved, then $u$ is harmonic. We do not assume any integrability condition on $u$: in particular, $u$ is not required to have finite energy, nor is its tension field required to be in $L^{p}$ for any $p$. Our result implies the following version of the Generalized Chen's Conjecture: if $\left(N,h\right)$ is a non-positively curved conformally compact manifold, and $\Sigma\hookrightarrow N$ is a properly embedded submanifold with boundary meeting $\partial N$ transversely, then $\Sigma$ is biharmonic if and only if it is minimal.

math.DG↗

arXiv2024

Boundary value problems for 0-elliptic operators

Let $X$ be a manifold with boundary, and let $L$ be a 0-elliptic operator on X which is semi-Fredholm essentially surjective with infinite-dimensional kernel. Examples include Hodge Laplacians and Dirac operators on conformally compact manifolds. We construct left and right parametrices for L when supplemented with appropriate elliptic boundary conditions. The construction relies on a new calculus of pseudodifferential operators on functions over both $X$ and $\partial X$, which we call the "symbolic 0-calculus". This new calculus supplements the ordinary 0-calculus of Mazzeo--Melrose, enabling it to handle boundary value problems. In the original 0-calculus, operators are characterized as polyhomogeneous right densities on a blow-up of $X^2$. By contrast, operators in the symbolic 0-calculus are characterized (locally near each point of the boundary of the diagonal) as quantizations of polyhomogeneous symbols on appropriate blown-up model spaces.

math.AP↗

arXiv2020

Yang-Mills Connections on Conformally Compact Manifolds

We study the moduli space of Yang--Mills connections on bundles over a conformally compact manifold $\overline{M}$. We prove that, for every Yang--Mills connection $A$ that satisfies an appropriate nondegeneracy condition, and for every small deformation $\gamma$ of $A_{|\partial\overline{M}}$, there is a Yang--Mills connection in the interior that extends $A_{|\partial\overline{M}}+\gamma$. As a corollary, we confirm an expectation of Witten mentioned in his foundational paper about holography [arXiv:hep-th/9802150].

math.DG↗