arXiv subjects

Ian McIntosh

Publications and source records attributed to Ian McIntosh.

13 recordsLinked to original sources

arXiv2025

The geometric Toda equations for noncompact symmetric spaces

This paper has two purposes. The first is to classify all those versions of the Toda equations which govern the existence of $τ$-primitive harmonic maps from a surface into a homogeneous space $G/T$ for which $G$ is a noncomplex noncompact simple real Lie group, $τ$ is the Coxeter automorphism which Drinfel'd \& Sokolov assigned to each affine Dynkin diagram, and $T$ is the compact torus fixed pointwise by $τ$. Here $τ$ may be either an inner or an outer automorphism. We interpret the Toda equations over a compact Riemann surface $Σ$ as equations for a metric on a holomorphic principal $T^\mathbb{C}$-bundle $Q^\mathbb{C}$ over $Σ$ whose Chern connection, when combined with holomorphic field $φ$, produces a $G$-connection which is flat precisely when the Toda equations hold. The second purpose is to establish when stability criteria for the pair $(Q^\mathbb{C},φ)$ can be used to prove the existence of solutions. We classify those real forms of the Toda equations for which this pair is a principal pair and we call these \emph{totally noncompact} Toda pairs: stability theory then gives algebraic conditions for the existence of solutions. Every solution to the geometric Toda equations has a corresponding $G$-Higgs bundle. We explain how to construct this $G$-Higgs bundle directly from the Toda pair and show that Baraglia's cyclic Higgs bundles arise from a very special case of totally noncompact cyclic Toda pairs.

math.DG↗

arXiv2021

On the connected components of the moduli space of equivariant minimal surfaces in $\mathbb{CH}^2$

An equivariant minimal surface in $\mathbb{CH}^n$ is a minimal map of the Poincaré disc into $\mathbb{CH}^n$ which intertwines two actions of the fundamental group of a closed surface $Σ$: a Fuchsian representation on the disc and an irreducible action by isometries on $\mathbb{CH}^n$. The moduli space of these can been studied by relating it to the nilpotent cone in each moduli space of $PU(n,1)$-Higgs bundles over the conformal surface corresponding to the map. By providing a necessary condition for points on this nilpotent cone to be smooth this article shows that away from the points corresponding to branched minimal immersions or $\pm$-holomorphic immersions the moduli space is smooth. The argument is easily adapted to show that for $\mathbb{RH}^n$ the full space of (unbranched) immersions is smooth. For $\mathbb{CH}^2$ we show the connected components of the moduli space of minimal immersions are indexed by the Toledo invariant and the Euler number of the normal bundle of the immersion. This is achieved by studying the limit points of the $\mathbb{C}^\times$-action on the nilpotent cone. It is shown that the limit points as $t\to 0$ lead only to branched minimal immersions or $\pm$-holomorphic immersions. In particular, the Euler number of the normal bundle can only jump by passing through branched minimal maps.

math.DG↗

arXiv2020

Equivariant minimal surfaces in $\mathbb{CH}^2$ and their Higgs bundles

This paper gives a construction for all minimal immersions $f$ of the Poincaré disc into the complex hyperbolic plane $\mathbb{CH}^2$ which are equivariant with respect to an irreducible representation $ρ$ of a hyperbolic surface group into $PU(2,1)$. We exploit the fact that each such immersion is a twisted conformal harmonic map and therefore has a corresponding Higgs bundle. We identify the structure of these Higgs bundles and show how each is determined by properties of the map, including the induced metric and a holomorphic cubic differential on the surface. We show that the moduli space of pairs $(ρ,f)$ is a disjoint union of finitely many complex manifolds, whose structure we fully describe. The holomorphic (or anti-holomorphic) maps provide multiple components of this union, as do the non-holomorphic maps. Each of the latter components has the same dimension as the representation variety for $PU(2,1)$, and is indexed by the number of complex and anti-complex points of the immersion. These numbers determine the Toledo invariant and the Euler number of the normal bundle of the immersion. We show that there is an open set of quasi-Fuchsian representations of Toledo invariant zero for which the minimal surface is unique and Lagrangian.

math.DG↗

arXiv2018

The moduli spaces of equivariant minimal surfaces in $\mathbb{RH}^3$ and $\mathbb{RH}^4$ via Higgs bundles

In this article we introduce a definition for the moduli space of equivariant minimal immersions of the Poincaré disc into a non-compact symmetric space, where the equivariance is with respect to representations of the fundamental group of a compact Riemann surface of genus at least two. We then study this moduli space for the non-compact symmetric space $\mathbb{RH}^n$ and show how $SO_0(n,1)$-Higgs bundles can be used to parametrise this space, making clear how the classical invariants (induced metric and second fundamental form) figure in this picture. We use this parametrisation to provide details of the moduli spaces for $\mathbb{RH}^3$ and $\mathbb{RH}^4$, and relate their structure to the structure of the corresponding Higgs bundle moduli spaces.

math.DG↗

arXiv2017

Cubic Differentials in the Differential Geometry of Surfaces

We discuss the local differential geometry of convex affine spheres in $\re^3$ and of minimal Lagrangian surfaces in Hermitian symmetric spaces. In each case, there is a natural metric and cubic differential holomorphic with respect to the induced conformal structure: these data come from the Blaschke metric and Pick form for the affine spheres and from the induced metric and second fundamental form for the minimal Lagrangian surfaces. The local geometry, at least for main cases of interest, induces a natural frame whose structure equations arise from the affine Toda system for $\mathfrak a^{(2)}_2$. We also discuss the global theory and applications to representations of surface groups and to mirror symmetry.

math.DG↗

arXiv2015

Minimal Lagrangian submanifolds via the geodesic Gauss map

For an oriented isometric immersion $f:M\to S^n$ the spherical Gauss map is the Legendrian immersion of its unit normal bundle $UM^\perp$ into the unit sphere subbundle of $TS^n$, and the geodesic Gauss map $γ$ projects this into the manifold of oriented geodesics in $S^n$ (the Grassmannian of oriented 2-planes in $\mathbb{R}^{n+1}$), giving a Lagrangian immersion of $UM^\perp$ into a Kaehler-Einstein manifold. We give expressions for the mean curvature vectors for both the spherical and geodesic Gauss maps in terms of the second fundamental form of $f$, and show that when $f$ has conformal shape form this depends only on the mean curvature of $f$. In particular we deduce that the geodesic Gauss map of every minimal surface in $S^n$ is minimal Lagrangian. We also give simple proofs that: deformations of $f$ always correspond to Hamiltonian deformations of $γ$; the mean curvature vector of $γ$ is always a Hamiltonian vector field. This extends work of Palmer on the case when $M$ is a hypersurface.

math.DG↗

arXiv2015

Minimal Lagrangian Surfaces in CH2 and Representations of Surface Groups into SU(2,1)

We use an elliptic differential equation of Tzitzeica type to construct a minimal Lagrangian surface in CH2 from the data of a compact hyperbolic Riemann surface and a small holomorphic cubic differential. The minimal Lagrangian surface is invariant under an SU(2,1) action of the fundamental group. We further parameterise a neighborhood of the R-Fuchsian representations in the representation space by pairs consisting of a point in Teichmuller space and a small cubic differential. By constructing a fundamental domain, we show these representations are complex-hyperbolic quasi-Fuchsian, thus recovering a result of Guichard and Parker-Platis. Our proof involves using the Toda lattice framework to construct an SU(2,1) frame corresponding to a minimal Lagrangian surface. Then the equation of Tzitzeica type is an integrability condition. A very similar equation to ours governs minimal surfaces in hyperbolic 3-space, and our paper can be interpreted as an analog of the theory of minimal surfaces in quasi-Fuchsian manifolds, as first studied by Uhlenbeck.

math.DG↗

arXiv2011

The quaternionic KP hierarchy and conformally immersed 2-tori in the 4-sphere

The quaternionic KP hierarchy is the integrable hierarchy of p.d.e obtained by replacing the complex numbers with the quaternions, mutatis mutandis, in the standard construction of the KP hierarchy equations and solutions; it is equivalent to what is often called the Davey-Stewartson II hierarchy. This article studies its relationship with the theory of quaternionic holomorphic 2-tori in HP^1 (which are equivalent to conformally immersed 2-tori in S^4). After describing how the Sato-Segal-Wilson construction of KP solutions (particularly solutions of finite type) carries over to this quaternionic setting, we compare three different notions of "spectral curve": the QKP spectral curve, which arises from an algebra of commuting differential operators; the (unnormalised) Floquet multiplier spectral curve for the related Dirac operator; and the curve parameterising Darboux transforms of a conformal 2-torus in S^4 (in the sense of Bohle, Leschke, Pedit and Pinkall). The latter two are shown to be images of the QKP spectral curve, which need not be smooth. Moreover, it is a singularisation of this QKP spectral curve, rather than the normalised Floquet multiplier curve, which determines the classification of conformally immersed 2-tori of finite spectral genus.

math.DG↗

arXiv2011

The classification of Hamiltonian stationary Lagrangian tori in the complex projective plane by their spectral data

It is known that all weakly conformal Hamiltonian stationary Lagrangian immersions of tori in the complex projective plane may be constructed by methods from integrable systems theory. This article describes the precise details of a construction which leads to a form of classification. The immersion is encoded as spectral data in a similar manner to the case of minimal Lagrangian tori in the complex projective plane, but the details require a careful treatment of both the "dressing construction" and the spectral data to deal with a loop of flat connexions which is quadratic in the loop parameter.

math.DG↗

arXiv2009

The spectral data for Hamiltonian stationary Lagrangian tori in R^4

This article determines the spectral data, in the integrable systems sense, for all weakly conformally immersed Hamiltonian stationary Lagrangian in $\R^4$. This enables us to describe their moduli space and the locus of branch points of such an immersion. This is also an informative example in integrable systems geometry, since the group of ambient isometries acts non-trivially on the spectral data and the relevant energy functional (the area) need not be constant under deformations by higher flows.

math.DG↗

arXiv2004

Minimal Lagrangian 2-tori in CP^2 come in real families of every dimension

We show that for every non-negative integer n there is a real n-dimensional family of minimal Lagrangian tori in CP^2, and hence of special Lagrangian cones in C^3 whose link is a torus. The proof utilises the fact that such tori arise from integrable systems, and can be described using algebro-geometric (spectral curve) data.

math.DG↗

arXiv2002

Special Lagrangian cones in C^3 and primitive harmonic maps

In this article I show that every special Lagrangian cone in C^3 determines, and is determined by, a primitive harmonic surface in the 6-symmetric space SU_3/SO_2. For cones over tori, this allows us to use the classification theory of harmonic tori to describe the construction of all the corresponding special Lagrangian cones. A parameter count is given for the space of these, and some examples found recently by Joyce are put into this context.

math.DG↗

arXiv2000

Harmonic tori and generalised Jacobi varieties

This article shows that every non-isotropic harmonic 2-torus in complex projective space factors through a generalised Jacobi variety related to the spectral curve. Each map is composed of a homomorphism into the variety and a rational map off it. The same ideas allow one to construct (pluri)-harmonic maps of finite type from Euclidean space into Grassmannians and the projective unitary groups. Further, some of these maps will be purely algebraic. For maps into complex projective space the algebraic maps of the plane are always doubly periodic i.e. they yield 2-tori. The classification of all these algebraic maps remains open.

math.DG↗