arXiv subjects

Willem Adriaan Salm

Publications and source records attributed to Willem Adriaan Salm.

7 recordsLinked to original sources

arXiv2026

Prescribed Singular Sets for Z/2-Harmonic 1-Forms on R^n

Z/2 harmonic 1-forms arise naturally as singular limits in gauge theory and calibrated geometry, but the topology that can occur in their singular sets is poorly understood. In this paper we study the flexibility of these singular sets when the ambient Riemannian metric is allowed to vary. We prove that every compact smoothly embedded codimension-two submanifold with trivial normal bundle in a Euclidean space of dimension at least three can be realized as the singular set of a nondegenerate Z/2 harmonic 1-form for a complete metric that is Euclidean outside a compact set. As an application, we use Calabi surgery to construct Z/2-harmonic 1-forms with prescribed local singular sets on closed manifolds of positive first Betti number.

math.DG↗

arXiv2026

Construction of gravitational instantons with non-maximal volume growth via codimension-1 collapse

In this paper, we construct families of gravitational instantons of type ALG, ALG*, ALH and ALH* using a gluing construction. Away from a finite set of exceptional points, the metric collapses with bounded curvature to a quotient of $\mathbb{R}^3$ by $\mathbb{Z}_2$ and a lattice of rank one or two. Depending on whether the gravitational instantons are of type ALG/ALG* or ALH/ALH*, there are either two or four exceptional points respectively that are modelled on the Atiyah-Hitchin manifold. The other exceptional points are modelled on the Taub-NUT metric. There are at most four, respectively eight, of these points in each case.

math.DG↗

arXiv2026

A stabilization result for Z/2-harmonic 1-forms by constructing solutions on closed 3-manifolds with long cylindrical necks

In this paper, we give an explicit construction of families of $\mathbb{Z}_2$-harmonic 1-forms that degenerate to manifolds with cylindrical ends. We do this by considering certain linear combinations of $L^2$-bounded $\mathbb{Z}_2$-harmonic 1-forms and by modifying the metric near the link. This construction works if number of $L^2$-bounded $\mathbb{Z}_2$-harmonic 1-forms is strictly more than twice the number of connected components of the link. This can always be done if we consider a connected sum with a 3-manifold with sufficiently large $b_1$.

math.DG↗

arXiv2026

Deformation rigidity for Z/2 eigensections

We prove a rigidity result for certain critical Z/2 eigensections of the Laplacian on S^2 associated to a flat real line bundle determined by a branch-point configuration. More precisely, we show that every minimal non-degenerate critical eigensection is deformation rigid: any sufficiently small deformation of the configuration that still admits a critical eigensection must come from an SO(3)-rotation. This generalizes the rigidity phenomenon previously discovered in symmetric examples of Taubes-Wu.

math.DG↗

arXiv2026

Metric perturbations and deformations of k-nondegenerate Z/2-harmonic 1-forms

We study metric perturbations and deformation theory for degenerate Z/2-harmonic 1-forms. For a natural class of degenerate examples, we prove that after a suitable perturbation of the ambient Riemannian metric, the form can be deformed to a nearby non-degenerate Z/2-harmonic 1-form. Our argument combines analysis of the leading coefficients in the local expansion under metric perturbations with a quantitative Nash-Moser implicit function theorem.

math.DG↗

arXiv2025

Search for Z/2 eigenfunctions on the sphere using machine learning

We use machine learning to search for examples of Z/2 eigenfunctions on the 2-sphere. For this we created a multivalued version of a feedforward deep neural network, and we implemented it using the JAX library. We found Z/2 eigenfunctions for three cases: In the first two cases we fixed the branch points at the vertices of a tetrahedron and at a cube respectively. In a third case, we allowed the AI to move the branch points around and, in the end, it positioned the branch points at the vertices of a squashed tetrahedron.

math.DG↗

arXiv2024

Elliptic analysis on collapsing gravitational instantons modelled using the Gibbons-Hawking ansatz

In this paper, we will establish elliptic regularity estimates on families of gravitational instantons whose model metric near infinity collapse to a flat 3 dimensional space. We show that the constants in these estimates can be chosen uniformly for the whole family of metrics. We apply these results to the Laplacian and show it is Fredholm. We also determine between which weighted spaces it is an isomorphism.

math.DG↗