arXiv subjects

John Loftin

Publications and source records attributed to John Loftin.

17 recordsLinked to original sources

arXiv2026

Equivariant finite energy proper minimal surfaces in $\mathbb{CH}^2$

Given a noncompact Riemann surface $\Sigma_0\,=\, \Sigma \setminus P$, where $P$ is a finite subset of a compact connected Riemann surface $\Sigma$, and a reductive representation $\rho\,:\,\pi_1(\Sigma_0)\,\longrightarrow\, \mathrm{PU}(2,1)$, we prove that any finite--energy $\rho$--equivariant conformal minimal immersion is proper around every cusp if and only if the peripheral holonomy of $\rho$ is parabolic. Assuming parabolic peripheral holonomy, we give an explicit parametrization of complete finite--energy immersions in the mixed case in terms of tame parabolic $\mathrm{PU}(2,1)$--Higgs bundles with nilpotent residues and satisfying concrete parabolic slope inequalities. We also discuss complete ends and construct explicit families of $\rho$ equivariant proper $\mathbb{CH}^2$ $n$--noids on $\mathbb{CP}^1\setminus P$ for $|P|\,\ge\, 5$.

math.DG↗

arXiv2022

Limits of Cubic Differentials and Buildings

In the Labourie-Loftin parametrization of the Hitchin component of surface group representations into SL(3,R), we prove an asymptotic formula for holonomy along rays in terms of local invariants of the holomorphic differential defining that ray. Globally, we show that the corresponding family of equivariant harmonic maps to a symmetric space converge to a harmonic map into the asymptotic cone of that space. The geometry of the image may also be described by that differential: it is weakly convex and a (one-third) translation surface. We define a compactification of the Hitchin component in this setting for triangle groups that respects the parametrization by Hitchin differentials.

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

Coordinates on the augmented moduli space of convex RP^2 structures

Let S be an orientable, finite type surface with negative Euler characteristic. The augmented moduli space of convex real projective structures on S was first defined and topologized by the first author. In this article, we give an explicit description of this topology using explicit coordinates. More precisely, given every point in this augmented moduli space, we find explicit continuous coordinates on the quotient of a suitable open neighborhood about this point by a suitable subgroup of the mapping class group of S. Using this, we give a simpler proof of the fact that the augmented moduli space of convex real projective structures on S is homeomorphic to the orbifold vector bundle of regular cubic differentials over the Deligne-Mumford compactification of the moduli space of Riemann surfaces homeomorphic to S.

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

Convex RP^2 Structures and Cubic Differentials under Neck Separation

Let S be a closed oriented surface of genus at least two. Labourie and the author have independently used the theory of hyperbolic affine spheres to find a natural correspondence between convex RP^2 structures on S and pairs (Σ,U) consisting of a conformal structure Σon S and a holomorphic cubic differential U over Σ. The pairs (Σ,U$, for Σvarying in moduli space, allow us to define natural holomorphic coordinates on the moduli space of convex RP^2 structures. We consider geometric limits of convex RP^2 structures on S in which the RP^2 structure degenerates only along a set of simple, non-intersecting, non-homotopic loops c. We classify the resulting RP^2 structures on S-c and call them regular convex RP^2 structures. We put a natural topology on the moduli space of all regular convex RP^2 structures on S and show that this space is naturally homeomorphic to the total space of the vector bundle over the Deligne-Mumford compactification of the moduli space of curves each of whose fibers over a noded Riemann surface is the space of regular cubic differentials. In other words, we can extend our holomorphic coordinates to bordify the moduli space of convex RP^2 structures along all neck pinches. The proof relies on previous techniques of the author, Benoist-Hulin, and Dumas-Wolf, as well as some details due to Wolpert of the geometry of hyperbolic metrics on conformal surfaces in the Deligne-Mumford compactification.

math.GT↗

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↗

arXiv2013

The vortex equation on affine manifolds

Let M be a compact connected special affine manifold equipped with an affine Gauduchon metric. We show that a pair (E, ϕ), consisting of a flat vector bundle E over M and a flat nonzero section ϕ of E, admits a solution to the vortex equation if and only if it is polystable. To prove this, we adapt the dimensional reduction techniques for holomorphic pairs on Kähler manifolds to the situation of flat pairs on affine manifolds.

math.DG↗

arXiv2012

Affine Yang-Mills-Higgs metrics

Let (E, φ) be a flat Higgs bundle on a compact special affine manifold M equipped with an affine Gauduchon metric. We prove that (E, φ) is polystable if and only if it admits an affine Yang-Mills-Higgs metric.

math.DG↗

arXiv2012

Holomorphic Cubic Differentials and Minimal Lagrangian Surfaces in CH2

Following earlier work of Loftin-McIntosh, we study minimal Lagrangian immersions of the universal cover of a closed surface (of genus at least 2) into CH2, with prescribed data of a conformal structure plus a holomorphic cubic differential. We show existence and non-uniqueness of such minimal Lagrangian immersions. We also establish the surface area with respect to the induced metric as a Weil-Petersson potential function for the space of holomorphic cubic differentials on the Riemann surface.

math.DG↗

arXiv2011

Hermitian-Einstein connections on principal bundles over flat affine manifolds

Let $M$ be a compact connected special flat affine manifold without boundary equipped with a Gauduchon metric $g$ and a covariant constant volume form. Let $G$ be either a connected reductive complex linear algebraic group or the real locus of a split real form of a complex reductive group. We prove that a flat principal $G$-bundle $E_G$ over $M$ admits a Hermitian-Einstein structure if and only if $E_G$ is polystable. A polystable flat principal $G$--bundle over $M$ admits a unique Hermitian-Einstein connection. We also prove the existence and uniqueness of a Harder-Narasimhan filtration for flat vector bundles over $M$.

math.DG↗

arXiv2008

Affine Manifolds, SYZ Geometry, and the "Y" Vertex

We study the real Monge-Ampère equation in two and three dimensions, both from the point of view of the SYZ conjecture, where solutions give rise to semi-flat Calabi-Yau's and in affine differential geometry, where solutions yield parabolic affine sphere hypersurfaces. We find explicit examples, connect the holomorphic function representation to Hitchin's description of special Lagrangian moduli space, and construct the developing map explicitly for a singularity corresponding to the type $I_n$ elliptic fiber (after hyper-Kähler rotation). Following Baues and Cortés, we show that various types of metric cones over two-dimensional elliptic affine spheres generate solutions of the Monge-Ampère equation in three dimensions. We then prove a local and global existence theorem for an elliptic affine two-sphere metric with prescribed singularities. The metric cone over the two-sphere minus three points yields a parabolic affine sphere with singularities along a "Y"-shaped locus. This gives a semi-flat Calabi-Yau metric in a neighborhood of the "Y" vertex. In the erratum, we correct a gap in the solution of the elliptic affine sphere equation (with the small caveat that a parameter in the equation, the cubic differential, must be smaller than we assumed before). The new proof still allows us to construct semi-flat Calabi-Yau metrics in a neighborhood of the "Y" vertex by appealing to the result of Baues-Cortés. We also give an alternate construction of such semi-flat Calabi-Yau metrics by using a result on hyperbolic affine spheres due to the first author.

math.DG↗

arXiv2008

Survey on Affine Spheres

We give a survey of the theory of affine spheres, emphasizing the convex cases and relationsships to Monge-Ampere equations and geometric structures on manifolds.

math.DG↗

arXiv2008

Limits of Solutions to a Parabolic Monge-Ampere Equation

We present the results from our earlier paper (arXiv:math/0602484) on the affine normal flow on noncompact convex hypersurfaces in affine space from a more PDE point of view, emphasizing the estimates involved. Our results concern the limits of solutions to a parabolic Monge-Ampere equation on $S^n$, where a sequence of smooth strictly convex initial value functions increase monotonically to a limiting initial value function which is infinite on at least a hemisphere of $S^n$. We prove long-time existence and instantaneous smoothing for quite general initial data, and we characterize ancient solutions as ellipsoids or paraboloids. We make essential use of estimates of Andrews and Gutierrez-Huang, and barriers due to Calabi.

math.AP↗

arXiv2007

Affine Hermitian-Einstein Metrics

We develop a theory of stable bundles and affine Hermitian-Einstein metrics for flat vector bundles over a special affine manifold (a manifold admitting an atlas whose gluing maps are all locally constant volume-preserving affine maps). Our paper presents a parallel to Donaldson-Uhlenbeck-Yau's proof of the existence of Hermitian-Einstein metrics on Kähler manifolds, and the extension of this theorem by Li-Yau to the non-Kähler complex case of Gauduchon metrics. Our definition of stability involves only flat vector subbundles (and not singular subsheaves), and so is simpler than the complex case in some places.

math.DG↗

arXiv2006

Ancient Solutions of the Affine Normal Flow

We construct noncompact solutions to the affine normal flow of hypersurfaces, and show that all ancient solutions must be either ellipsoids (shrinking solitons) or paraboloids (translating solitons). We also provide a new proof of the existence of a hyperbolic affine sphere asymptotic to the boundary of a convex cone containing no lines, which is originally due to Cheng-Yau. The main techniques are local second-derivative estimates for a parabolic Monge-Ampere equation modeled on those of Ben Andrews and Gutierrez-Huang, a decay estimate for the cubic form under the affine normal flow due to Ben Andrews, and a hypersurface barrier due to Calabi.

math.DG↗