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Martin Bridgeman

Publications and source records attributed to Martin Bridgeman.

At least 19 recordsLinked to original sources
arXiv2026

Bending, entropy and proper affine actions of surface groups

We show that for any closed surface $S$ there is an explict neighborhood $V$ of the fuchsian locus in quasifuchsian space $\mathsf{QF}(S)$ such that for every representation $\rho\in V$ which is not fuchsian, there is a proper affine action on $\mathfrak{sl}(2,\mathbb{C})$ with linear part $\mathsf{Ad}(\rho)$. We further show that there is a larger neighborhood $U$ of the Fuchsian locus so that every critical point of the entropy function in $U$ lies on the Fuchsian locus.

math.GT↗
arXiv2025

Schwarzian Bounds on Bending in Hyperbolic 3-Manifolds

The Schwarzian derivative provides a classical analytic measure of how far a holomorphic map of the disk is from being M\"obius, with Nehari's bounds giving sharp criteria for univalence. Independently, Thurston introduced a geometric parametrization of locally univalent maps via bending measured laminations on the hyperbolic plane, capturing deviation from roundness in hyperbolic three-space. While both approaches quantify the same phenomenon, their precise relationship has remained only implicit. In this paper we establish explicit quantitative bounds relating the Schwarzian norm $\|Sf\|_\infty$ and the bending norm $\|\beta_f\|_L$. In particular, for univalent maps with $\|Sf\|_\infty< 1/2$, we show that $\|\beta_f\|_L$ is controlled by an elementary function $B_L(\|Sf\|_L)$ that we compute explicitly. As an application, we obtain new effective bounds on the bending laminations of quasifuchsian manifolds in terms of the Teichm\"uller distance between their conformal boundary components. Our results sharpen the analytic-geometric correspondence between the Schwarzian derivative and hyperbolic geometry showing that just as small Schwarzian norm forces injectivity, it also forces controlled bending of convex hull boundaries.

math.GT↗
arXiv2024

Epstein Surfaces, $W$-Volume, and the Osgood-Stowe Differential

In a seminal paper, Epstein introduced the theory of what are now called Epstein surfaces, which construct surfaces in $\mathbb{H}^3$ associated to a conformal metric on a domain in $\hat{\mathbb{C}}$. More recently, these surfaces have been used by Krasnov-Schlenker to define the W-volume and renormalized volume associated with a convex co-compact hyperbolic 3-manifold. In this paper we consider Epstein surfaces, W-volume and renormalized volume in two main parts. In the first, we develop an alternate construction of Epstein surfaces using the Osgood-Stowe differential, a generalization of the Schwarzian derivative. Krasnov-Schlenker showed that the metric and shape operator of a surface in hyperbolic space is naturally dual to a conformal metric and shape operator on a projective structure via the hyperbolic Gauss map. We show that this projective shape operator can be derived from the Osgood-Stowe differential. This approach allows us to give a comprehensive and self-contained development of Epstein surfaces, W-volume, and renormalized volume. In the second part, we use the theory developed in the first to prove a number of new results including a generalization of Epstein's univalence criterion, a variational formula for W-volume in terms of the Osgood-Stowe differential, and a use of W-volume to relate the length of the bending lamination of the convex core to the norm of the Schwarzian derivative of the associated univalent map on the conformal boundary.

math.DG↗
arXiv2023

Universal Liouville action as a renormalized volume and its gradient flow

The universal Liouville action (also known as the Loewner energy for Jordan curves) is a K\"ahler potential on the Weil-Petersson universal Teichm\"uller space, which is identified with the family of Weil-Petersson quasicircles via conformal welding. Our main result shows that, under regularity assumptions, the universal Liouville action equals the renormalized volume of the hyperbolic $3$-manifold bounded by the two Epstein-Poincar\'e surfaces associated with the quasicircle. We also study the gradient descent flow of the universal Liouville action for the Weil-Petersson metric and show that the flow always converges to the origin (the circle). This provides a bound of the Weil-Petersson distance to the origin by the universal Liouville action.

math.DG↗
arXiv2023

$L^2$-bounds for drilling short geodesics in convex co-compact hyperbolic 3-manifolds

We give $L^2$-bounds on the change in the complex projective structure on the boundary of conformally compact hyperbolic 3-manifold with incompressible boundary after drilling short geodesics. We show that the change is bounded by a universal constant times the square root of the length of the drilled geodesics. While $L^\infty$-bounds of this type where obtained by the second author (2004), our bounds here do not depend on the injectivity radius of the boundary.

math.GT↗
arXiv2023

Ghost polygons, Poisson bracket and convexity

The moduli space of Anosov representations of a surface group in a semisimple group, which is an open set in the character variety, admits many more natural functions than the regular functions. We will study in particular length functions and, correlation functions. Our main result is a formula that computes the Poisson bracket of those functions using some combinatorial devices called {\em ghost polygons} and {\em ghost bracket} encoded in a formal algebra called {\em ghost algebra} related in some cases to the swapping algebra introduced by the second author. As a consequence of our main theorem, we show that the set of those functions -- length and correlation -- is stable under the Poisson bracket. We give two applications: firstly in the presence of positivity we prove the convexity of length functions, generalising a result of Kerckhoff in Teichm\"uller space, secondly we exhibit subalgebras of commuting functions. An important tool is the study of {\em uniformly hyperbolic bundles} which is a generalisation of Anosov representations beyond periodicity.

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arXiv2022

A bound on the $L^2$-norm of a projective structure by the length of the bending lamination

One can associate to a complex projective structure on a surface holomorphic quadratic differential $Φ$ via the Schwarzian derivative and a bending lamination $λ$ via the Thurston parameterization. In this note we obtain upper bounds on the $L^2$-norm of $Φ$ in terms of the length of $λ$. The proof uses the theory of $W$-volume introduced by Krasnov-Schlenker.

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arXiv2021

Lower bounds for volumes and orthospectra of hyperbolic manifolds with geodesic boundary

In this paper we derive explicit estimates for the functions which appear in the previous work of Bridgeman and Kahn. As a consequence, we obtain an explicit lower bound for the length of the shortest orthogeodesic in terms of the volume of a hyperbolic manifold with totally geodesic boundary. We also give an alternative derivation of a lower bound for the volumes of these manifolds as a function of the dimension.

math.GT↗
arXiv2021

The Weil-Petersson gradient flow of renormalized volume and 3-dimensional convex cores

In this paper, we use the Weil-Petersson gradient flow for renormalized volume to study the space $CC(N;S,X)$ of convex cocompact hyperbolic structures on the relatively acylindrical 3-manifold $(N;S)$. Among the cases of interest are the deformation space of an acylindrical manifold and the Bers slice of quasi-Fuchsian space associated to a fixed surface. To treat the possibility of degeneration along flow-lines to peripherally cusped structures, we introduce a surgery procedure to yield a surgered gradient flow that limits to the unique structure $M_{\rm geod} \in CC(N;S,X)$ with totally geodesic convex core boundary facing $S$. Analyzing the geometry of structures along a flow line, we show that if $V_R(M)$ is the renormalized volume of $M$, then $V_R(M)-V_R(M_{\rm geod})$ is bounded below by a linear function of the Weil-Petersson distance $d_{\rm WP}(\partial_c M, \partial_c M_{\rm geod})$, with constants depending only on the topology of $S$. The surgered flow gives a unified approach to a number of problems in the study of hyperbolic 3-manifolds, providing new proofs and generalizations of well-known theorems such as Storm's result that $M_{\rm geod}$ has minimal volume for $N$ acylindrical and the second author's result comparing convex core volume and Weil-Petersson distance for quasifuchsian manifolds.

math.GT↗
arXiv2020

Strata Separation for the Weil-Petersson Completion and Gradient Estimates for Length Functions

In general, it is difficult to measure distances in the Weil-Petersson metric on Teichmüller space. Here we consider the distance between strata in the Weil-Petersson completion of Teichmüller space of a surface of finite type. Wolpert showed that for strata whose closures do not intersect, there is a definite separation independent of the topology of the surface. We prove that the optimal value for this minimal separation is a constant $δ_{1,1}$ and show that it is realized exactly by strata whose nodes intersect once. We also give a nearly sharp estimate for $δ_{1,1}$ and give a lower bound on the size of the gap between $δ_{1,1}$ and the other distances. A major component of the paper is an effective version of Wolpert's upper bound on $ \langle \nabla \ell_α,\nabla \ell_β\rangle$, the inner product of the Weil-Petersson gradient of length functions. We further bound the distance to the boundary of Teichmüller space of a hyperbolic surface in terms of the length of the systole of the surface. We also obtain new lower bounds on the systole for the Weil-Petersson metric on the moduli space of a punctured torus.

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arXiv2020

Hessian of Hausdorff dimension on purely imaginary directions

We extend classical results of Bridgeman-Taylor and McMullen on the Hessian of the Hausdorff dimension on quasi-Fuchsian space to the class of (1,1,2)-hyperconvex representations, a class introduced in arXiv:1902.01303 which includes small complex deformations of Hitchin representations and of $Θ$-positive representations. We also prove that the Hessian of the Hausdorff dimension of the limit set at the inclusion $Γ\to PO(n,1) \to PU(n,1)$ is positive definite when $Γ$ is co-compact in $PO(n,1)$ (unless $n=2$ and the deformation is tangent to $\mathfrak{X}(Γ,PO(2,1))).$

math.DG↗
arXiv2019

Uniform bounds on harmonic Beltrami differentials and Weil-Petersson curvatures

In this article we show that for every finite area hyperbolic surface $X$ of type $(g,n)$ and any harmonic Beltrami differential $μ$ on $X$, then the magnitude of $μ$ at any point of small injectivity radius is uniform bounded from above by the ratio of the Weil-Petersson norm of $μ$ over the square root of the systole of $X$ up to a uniform positive constant multiplication. We apply the uniform bound above to show that the Weil-Petersson Ricci curvature, restricted at any hyperbolic surface of short systole in the moduli space, is uniformly bounded from below by the negative reciprocal of the systole up to a uniform positive constant multiplication. As an application, we show that the average total Weil-Petersson scalar curvature over the moduli space is uniformly comparable to $-g$ as the genus $g$ goes to infinity.

math.DG↗
arXiv2019

Dilogarithm identities for solutions to Pell's equation in terms of continued fraction convergents

In this paper we give describe a new connection between the dilogarithm function and solutions to Pell's equation $x^2-ny^2 = \pm 1$. For each solution $x,y$ to Pell's equation we obtain a dilogarithm identity whose terms are given by the continued fraction expansion of the associated unit $x+y\sqrt{n} \in \Z[\sqrt{n}]$. We further show that Ramanujan's dilogarithm value-identities correspond to an identity for the regular ideal hyperbolic hexagon.

math.GT↗
arXiv2018

Schwarzian derivatives, projective structures, and the Weil-Petersson gradient flow for renormalized volume

To a complex projective structure $Σ$ on a surface, Thurston associates a locally convex pleated surface. We derive bounds on the geometry of both in terms of the norms $\|ϕ_Σ\|_\infty$ and $\|ϕ_Σ\|_2$ of the quadratic differential $ϕ_Σ$ of $Σ$ given by the Schwarzian derivative of the associated locally univalent map. We show that these give a unifying approach that generalizes a number of important, well known results for convex cocompact hyperbolic structures on 3-manifolds, including bounds on the Lipschitz constant for the nearest-point retraction and the length of the bending lamination. We then use these bounds to begin a study of the Weil-Petersson gradient flow of renormalized volume on the space $CC(N)$ of convex cocompact hyperbolic structures on a compact manifold $N$ with incompressible boundary, leading to a proof of the conjecture that the renormalized volume has infimum given by one-half the simplicial volume of $DN$, the double of $N$.

math.DG↗
arXiv2018

Simple Length Rigidity for Hitchin Representations

We show that a Hitchin representation is determined by the spectral radii of the images of simple, non-separating closed curves. As a consequence, we classify isometries of the intersection function on Hitchin components of dimension 3 and on the self-dual Hitchin components in all dimensions. As an important tool in the proof, we establish a transversality result for positive quadruples of flags.

math.GT↗
arXiv2017

Simple root flows for Hitchin representations

We study simple root flows and Liouville currents for Hitchin representations. We show that the Liouville current is associated to the measure of maximal entropy for a simple root flow, derive a Liouville volume rigidity result, and construct a Liouville pressure metric on the Hitchin component.

math.DG↗