arXiv subjects

Joel Fine

Publications and source records attributed to Joel Fine.

At least 19 recordsLinked to original sources

arXiv2025

Hypersymplectic Structures Invariant Under an Effective Circle Action

A hypersymplectic structure on a 4-manifold is a triple of symplectic forms for which any non-zero linear combination is again symplectic. In 2006, Donaldson conjectured that on a compact 4-manifold any hypersymplectic structure can be deformed through cohomologous hypersymplectic structures to a hyperk\"ahler triple. We prove this under the assumption that the initial structure is invariant under an effective $S^1$-action. In particular we show that the underlying 4-manifold is diffeomorphic to $\mathbb{T}^4$.

math.SG↗

arXiv2024

Seiberg-Witten equations in all dimensions

Starting with an $n$-dimensional oriented Riemannian manifold with a Spin-c structure, we describe an elliptic system of equations which recover the Seiberg-Witten equations when $n=3,4$. The equations are for a U(1)-connection $A$ and spinor $\phi$, as usual, and also an odd degree form $\beta$ (generally of inhomogeneous degree). From $A$ and $\beta$ we define a Dirac operator $D_{A,\beta}$ using the action of $\beta$ and $*\beta$ on spinors (with carefully chosen coefficients) to modify $D_A$. The first equation in our system is $D_{A,\beta}(\phi)=0$. The left-hand side of the second equation is the principal part of the Weitzenb\"ock remainder for $D^*_{A,\beta}D_{A,\beta}$. The equation sets this equal to $q(\phi)$, the trace-free part of projection against $\phi$, as is familiar from the cases $n=3,4$. In dimensions $n=4m$ and $n=2m+1$, this gives an elliptic system modulo gauge. To obtain a system which is elliptic modulo gauge in dimensions $n=4m+2$, we use two spinors and two connections, and so have two Dirac and two curvature equations, that are then coupled via the form $\beta$. We also prove a collection of a priori estimates for solutions to these equations. Unfortunately they are not sufficient to prove compactness modulo gauge, instead leaving the possibility that bubbling may occur.

math.DG↗

arXiv2024

Convergence of the hypersymplectic flow on $T^4$ with $T^3$-symmetry

A hypersymplectic structure on a 4-manifold is a triple $\omega_1, \omega_2, \omega_3$ of 2-forms for which every non-trivial linear combination $a^1\omega_1 + a^2 \omega_2 + a^3 \omega_3$ is a symplectic form. Donaldson has conjectured that when the underlying manifold is compact, any such structure is isotopic in its cohomolgy class to a hyperk\"ahler triple. We prove this conjecture for a hypersymplectic structure on $T^4$ which is invariant under the standard $T^3$ action. The proof uses the hypersymplectic flow, a geometric flow which attempts to deform a given hypersymplectic structure to a hyperk\"ahler triple. We prove that on $T^4$, when starting from a $T^3$-invariant hypersymplectic structure, the flow exists for all time and converges modulo diffeomorphisms to the unique cohomologous hyperk\"ahler structure.

math.DG↗

arXiv2022

Non-degeneracy of Poincaré-Einstein four-manifolds satisfying a chiral curvature inequality

A Poincaré-Einstein metric $g$ is called non-degenerate if there are no non-zero infinitesimal Einstein deformations of $g$, in Bianchi gauge, that lie in $L^2$. We prove that a 4-dimensional Poincaré-Einstein metric is non-degenerate if it satisfies a certain chiral curvature inequality. Write $R_+$ for the part of the curvature operator of g which acts on self-dual 2-forms. We prove that if $R_+$ is negative definite then $g$ is non-degenerate. This is a chiral generalisation of a result due to Biquard and Lee, that a Poincaré-Einstein metric of negative sectional curvature is non-degenerate

math.DG↗

arXiv2022

Knots, minimal surfaces and J-holomorphic curves

Let $K$ be a knot in the 3-sphere, viewed as the ideal boundary of hyperbolic 4-space $\mathbb{H}^4$. We prove that the number of minimal discs in $\mathbb{H}^4$ with ideal boundary $K$ is a knot invariant. I.e.\ the number is finite and doesn't change under isotopies of $K$. In fact this gives a family of knot invariants, indexed by an integer describing the extrinsic topology of how the disc sits in $\mathbb{H}^4$. These invariants can be seen as Gromov--Witten invariants counting $J$-holomorphic discs in the twistor space $Z$ of $\mathbb{H}^4$. Whilst Gromov--Witten theory suggests the general scheme for defining the invariants, there are substantial differences in how this must be carried out in our situation. These are due to the fact that the geometry of both $\mathbb{H}^4$ and $Z$ becomes singular at infinity, and so the $J$-holomorphic curve equation is degenerate, rather than elliptic, at the boundary. This means that both the Fredholm and compactness arguments involve completely new features.

math.DG↗

arXiv2020

An ambient approach to conformal geodesics

Conformal geodesics are distinguished curves on a conformal manifold, loosely analogous to geodesics of Riemannian geometry. One definition of them is as solutions to a third order differential equation determined by the conformal structure. There is an alternative description via the tractor calculus. In this article we give a third description using ideas from holography. A conformal $n$-manifold $X$ can be seen (formally at least) as the asymptotic boundary of a Poincaré--Einstein $(n+1)$-manifold $M$. We show that any curve $γ$ in $X$ has a uniquely determined extension to a surface $Σ_γ$ in $M$, which which we call the \emph{ambient surface of $γ$}. This surface meets the boundary $X$ in right angles along $γ$ and is singled out by the requirement that it it be a critical point of renormalised area. The conformal geometry of $γ$ is encoded in the Riemannian geometry of $Σ_γ$. In particular, $γ$ is a conformal geodesic precisely when $Σ_γ$ is asymptotically totally geodesic, i.e. its second fundamental form vanishes to one order higher than expected. We also relate this construction to tractors and the ambient metric construction of Fefferman and Graham. In the $(n+2)$-dimensional ambient manifold, the ambient surface is a graph over the bundle of scales. The tractor calculus then identifies with the usual tensor calculus along this surface. This gives an alternative compact proof of our holographic characterisation of conformal geodesics.

math.DG↗

arXiv2020

Local rigidity of Einstein 4-manifolds satisfying a chiral curvature condition

Let (M,g) be a compact oriented Einstein 4-manifold. Write R-plus for the part of the curvature operator of g which acts on self-dual 2-forms. We prove that if R-plus is negative definite then g is locally rigid: any other Einstein metric near to g is isometric to it. This is a chiral generalisation of Koiso's Theorem, which proves local rigidity of Einstein metrics with negative sectional curvatures. Our hypotheses are roughly one half of Koiso's. Our proof uses a new variational description of Einstein 4-manifolds, as critical points of the so-called poure connection action S. The key step in the proof is that when R-plus is negative definite, the Hessian of S is strictly positive modulo gauge.

math.DG↗

arXiv2020

Examples of compact Einstein four-manifolds with negative curvature

We give new examples of compact, negatively curved Einstein manifolds of dimension $4$. These are seemingly the first such examples which are not locally homogeneous. Our metrics are carried by a sequence of 4-manifolds $(X_k)$ previously considered by Gromov and Thurston. The construction begins with a certain sequence $(M_k)$ of hyperbolic 4-manifolds, each containing a totally geodesic surface $Σ_k$ which is nullhomologous and whose normal injectivity radius tends to infinity with $k$. For a fixed choice of natural number $l$, we consider the $l$-fold cover $X_k \to M_k$ branched along $Σ_k$. We prove that for any choice of $l$ and all large enough $k$ (depending on $l$), $X_k$ carries an Einstein metric of negative sectional curvature. The first step in the proof is to find an approximate Einstein metric on $X_k$, which is done by interpolating between a model Einstein metric near the branch locus and the pull-back of the hyperbolic metric from $M_k$. The second step in the proof is to perturb this to a genuine solution to Einstein's equations, by a parameter dependent version of the inverse function theorem. The analysis relies on a delicate bootstrap procedure based on $L^2$ coercivity estimates.

math.DG↗

arXiv2020

A report on the hypersymplectic flow

This article discusses a relatively new geometric flow, called the hypersymplectic flow. In the first half of the article we explain the original motivating ideas for the flow, coming from both 4-dimensional symplectic topology and 7-dimensional $G_2$-geometry. We also survey recent progress on the flow, most notably an extension theorem assuming a bound on scalar curvature. The second half contains new results. We prove that a complete torsion-free hypersymplectic structure must be hyperkähler. We show that a certain integral bound involving scalar curvature rules out a finite time singularity in the hypersymplectic flow. We show that if the initial hypersymplectic structure is sufficiently close to being point-wise orthogonal then the flow exists for all time. Finally, we prove convergence of the flow under some strong assumptions including, amongst other things, long time existence.

math.DG↗

arXiv2019

Symplectic domination

Let M be a compact oriented even-dimensional manifold. This note constructs a compact symplectic manifold S of the same dimension and a map f from S to M of strictly positive degree. The construction relies on two deep results: the first is a theorem of Ontaneda that gives a Riemannian manifold N of tightly pinched negative curvature which admits a map to M of degree equal to one; the second is a result of Donaldson on the existence of symplectic divisors. Given Ontaneda's negatively curved manifold N, the twistor space Z is symplectic. The manifold S is then a suitable multisection of the twistor space, found via Donaldson's theorem.

math.SG↗

arXiv2017

Hypersymplectic 4-manifolds, the $G_2$-Laplacian flow and extension assuming bounded scalar curvature

A hypersymplectic structure on a 4-manifold $X$ is a triple $\underlineω$ of symplectic forms which at every point span a maximal positive-definite subspace of $Λ^2$ for the wedge product. This article is motivated by a conjecture of Donaldson: when $X$ is compact $\underlineω$ can be deformed through cohomologous hypersymplectic structures to a hyperkähler triple. We approach this via a link with $G_2$-geometry. A hypersymplectic structure $\underlineω$ on a compact manifold $X$ defines a natural $G_2$-structure $ϕ$ on $X \times \mathbb{T}^3$ which has vanishing torsion precisely when $\underlineω$ is a hyperkähler triple. We study the $G_2$-Laplacian flow starting from $ϕ$, which we interpret as a flow of hypersymplectic structures. Our main result is that the flow extends as long as the scalar curvature of the corresponding $G_2$-structure remains bounded. An application of our result is a lower bound for the maximal existence time of the flow, in terms of weak bounds on the initial data (and with no assumption that scalar curvature is bounded along the flow).

math.DG↗

arXiv2017

The space of hyperkähler metrics on a 4-manifold with boundary

Let X be a compact 4-manifold with boundary. We study the space of hyperkähler triples on X, modulo diffeomorphisms which are the identity on the boundary. We prove that this moduli space is a smooth infinite-dimensional manifold and describe the tangent space in terms of triples of closed anti-self-dual 2-forms. We also explore the corresponding boundary value problem: a hyperkähler triple restricts to a closed framing of the bundle of 2-forms on the boundary; we identify the infinitesimal deformations of this closed framing that can be filled in to hyperkähler deformations of the original triple. Finally we study explicit examples coming from gravitational instantons with isometric actions of SU(2).

math.DG↗

arXiv2016

Limits of Riemannian 4-manifolds and the symplectic geometry of their twistor spaces

The twistor space of a Riemannian 4-manifold carries two almost complex structures, $J_+$ and $J_-$, and a natural closed 2-form $ω$. This article studies limits of manifolds for which $ω$ tames either $J_+$ or $J_-$. This amounts to a curvature inequality involving self-dual Weyl curvature and Ricci curvature, and which is satisfied, for example, by all anti-self-dual Einstein manifolds with non-zero scalar curvature. We prove that if a sequence of manifolds satisfying the curvature inequality converges to a hyperkähler limit X (in the $C^2$ pointed topology) then X cannot contain a holomorphic 2-sphere (for any of its hyperkähler complex structures). In particular, this rules out the formation of bubbles modelled on ALE gravitational instantons in such families of metrics.

math.DG↗

arXiv2016

A gauge theoretic approach to Einstein 4-manifolds

This article investigates a new gauge theoretic approach to Einstein's equations in dimension 4. Whilst aspects of the formalism are already explained in various places in the mathematics and physics literature, our first goal is to give a single coherent account of the theory in purely mathematical language. We then explain why the new approach may have important mathematical applications: the possibility of using the calculus of variations to find Einstein 4-manifolds, as well as links to symplectic topology. We also carry out some of the technical groundwork to attack these problems.

math.DG↗

arXiv2015

Asymptotically hyperbolic connections

General Relativity in 4 dimensions can be equivalently described as a dynamical theory of SO(3)-connections rather than metrics. We introduce the notion of asymptotically hyperbolic connections, and work out an analog of the Fefferman-Graham expansion in the language of connections. As in the metric setup, one can solve the arising "evolution" equations order by order in the expansion in powers of the radial coordinate. The solution in the connection setting is arguably simpler, and very straightforward algebraic manipulations allow one to see how the obstruction appears at third order in the expansion. Another interesting feature of the connection formulation is that the "counter terms" required in the computation of the renormalised volume all combine into the Chern-Simons functional of the restriction of the connection to the boundary. As the Chern-Simons invariant is only defined modulo large gauge transformations, the requirement that the path integral over asymptotically hyperbolic connections is well-defined requires the cosmological constant to be quantised. Finally, in the connection setting one can deform the 4D Einstein condition in an interesting way, and we show that asymptotically hyperbolic connection expansion is universal and valid for any of the deformed theories.

gr-qc↗

arXiv2014

Circle-invariant fat bundles and symplectic Fano 6-manifolds

We prove that a compact 4-manifold which supports a circle-invariant fat SO(3)-bundle is diffeomorphic to either S^4 or CP^2-bar. The proof involves studying the resulting Hamiltonian circle action on an associated symplectic 6-manifold. Applying our result to the twistor bundle of Riemannian 4-manifolds shows that S^4 and CP^2-bar are the only 4-manifolds admitting circle-invariant metrics solving a certain curvature inequality. This can be seen as an analogue of Hsiang-Kliener's theorem that only S^4 and CP^2 admit circle-invariant metrics of positive sectional curvature.

math.DG↗

arXiv2012

Quantisation and the Hessian of Mabuchi energy

Let L be an ample bundle over a compact complex manifold X. Fix a Hermitian metric in L whose curvature defines a Kähler metric on X. The Hessian of Mabuchi energy is a fourth-order elliptic operator D on functions which arises in the study of scalar curvature. We quantise D by the Hessian E(k) of balancing energy, a function appearing in the study of balanced embeddings. E(k) is defined on the space of Hermitian endomorphisms of H^0(X, L^k), endowed with the L^2-innerproduct. We first prove that the leading order term in the asymptotic expansion of E(k) is D. We next show that if Aut(X,L) is discrete modulo scalars, then the eigenvalues and eigenspaces of E(k) converge to those of D. We also prove convergence of the Hessians in the case of a sequence of balanced embeddings tending to a constant scalar curvature Kähler metric. As consequences of our results we prove that a certain estimate of Phong-Sturm is sharp and give a negative answer to a question of Donaldson. We also discuss some possible applications to the study of Calabi flow.

math.DG↗

arXiv2011

A gauge theoretic approach to the anti-self-dual Einstein equations

In [29], Plebanski reformulated the anti-self-dual Einstein equations with non-zero scalar curvature as a first order PDE for a connection in an SO(3)-bundle over the four-manifold. The aim of this article is to place this differential equation in a new framework, in which it is both elliptic and a stationary point of a parabolic flow. To do this, we exploit a link with definite connections (introduced in [12]) to draw an analogy with instantons and the Yang-Mills flow. This picture leads to a natural conjecture, analogous to one made by Donaldson concerning hyperkähler 4-manifolds [9]. It also provides a moment-map description of the anti-self-dual Einstein equations with non-zero scalar curvature.

math.DG↗