# On the problem of comparing graded metamaterials

Bryn Davies Department of Mathematics, Imperial College London, London SW7
2AZ, UK (email<EMAIL_ADDRESS>Lili Fehertoi-Nagy Department of
Physics, Imperial College London, London SW7 2AZ, UK Henry J.
Putley11footnotemark: 1

###### Abstract

We use a simple effective model, obtained through the application of high-
frequency homogenisation, to tackle the fundamental question of how the choice
of gradient function affects the performance of a graded metamaterial. This
approach provides a unified framework for comparing gradient functions
efficiently and in a way that allows us to draw conclusions that apply to a
range of different wave regimes. We consider the specific problem of single-
frequency localisation, for which the appropriate effective model is a one-
dimensional Schrödinger equation. Our analytic results both corroborate those
of existing studies (which use either expensive full-field wave simulations or
black-box numerical optimisation algorithms) and extend them to other
metamaterial regimes. Based on our analysis, we are able to propose a design
strategy for optimising monotonically graded metamaterials and offer an
explanation for the lack of a universal optimal gradient function.

Keywords: rainbow trapping, wave localisation, high-frequency homogenisation,
energy harvesting, reflection and transmission, machine hearing, frequency
separation

##  1 Introduction

Structures with periodically modulated material parameters are ubiquitous in
wave physics, thanks to their diverse and exotic wave propagation properties
[1, 2, 3]. Such materials appear in far-reaching applications, ranging from
waveguides [4, 5] to perfect lenses [6], invisibility cloaks [7] and other
metamaterials [2, 8]. Fundamental to most of these fascinating applications is
the existence of band gaps: ranges of frequencies that are unable to propagate
through the material and for which the wave energy is reflected. Being able to
predict and manipulate these band gaps has facilitated the development of many
of the wave control devices now available.

Graded metamaterials function by slowly varying the parameters of a band gap
material in order to spatially modulate the local band gap. For example,
varying geometric parameters such that the local band gap slowly shifts causes
different frequencies to be reflected at different positions in the material.
This phenomenon is often known as _rainbow trapping_ and was first
demonstrated in an optical setting [9]. It has since been implemented in many
other systems, including acoustics [10], elasticity [11, 12], seismic waves
[13] and water waves [14]. A simple example of such a system is shown in the
upper part of Figure 1, where we have modelled an array of cylinders with
sound-hard (Neumann) boundary conditions. Varying the distances between the
cylinders causes the local band gap to sweep upwards. This means that low
frequencies will be reflected early on in the structure, while higher
frequencies will be able to propagate further along the array before they
encounter a local band gap. This simple phenomenon is referred to in this work
as the _rainbow effect_ (to acknowledge the subtle differences between graded
devices that “trap” or “reflect” energy, as discussed in [15]).

The rainbow effect has found a range of significant applications, including
energy harvesting and machine hearing. In the former case, the ability of a
graded metamaterial to strongly focus waves of a specific frequency to a known
location means a relatively small harvesting device (_e.g._ a piezoelectric
patch) can extract a relatively large amount of energy [16, 17]. Conversely,
applications to machine hearing exploit the fact that a graded metamaterial
exhibits a continuous relationship between the incoming frequency and the
position of maximum amplitude [18, 19, 20]. This replicates the action of the
cochlea in mammalian hearing, and means that the frequency of a single-
frequency wave can be identified by measuring the location at which the
maximum amplitude occurs.

If we allow the gradient of a graded metamaterial to not be monotonic, then we
can produce other interesting phenomena. For example, if the gradient function
is symmetric this causes the local band gap to shift in one direction and then
back again. This is known to create localised eigenmodes in the centre of the
array, as certain frequencies cannot propagate past a certain point in either
direction [21]. The corresponding example for sound-hard cylinders is shown in
Figure 1. We see that the widened separation distances in the centre of the
array mean that the band gap shifts downwards and then back up again. This
means there is a range of frequencies (_i.e._ around $\Omega=4$) that can
propagate only in a central region of the array, so will be localised there.

Low frequenciesHigh frequencies.Monotonic gradient

Localisation.Symmetric gradient

Figure 1: Slowly varying the separation distances between scatterers causes
the local band gap to shift up or down. The top example is a monotonically
graded metamaterial, where the separation distances gradually decrease causing
the band gap to shift upwards. This configuration exhibits a rainbow effect.
The lower example is a symmetrically graded metamaterial, where the separation
distances increase before decreasing again. This configuration will lead to
wave localisation at the centre of the array. These results are for arrays of
cylinders with Neumann boundary conditions, modelled using multipole
expansions. The cylinders have radius 0.25 and the horizontal axes in the
plots show the separation distance between the cylinders’ centres.

A natural question to ask in the setting of graded metamaterials is what is
the best gradient to choose? We need the spatial modulation to be on a slow
scale, so that the physics of local band gaps applies, but otherwise are only
limited by manufacturing constraints. As we will see in this work, this
comparison is a somewhat subtle question. For this reason, many studies simply
opt to use the obvious example of a linear gradient [17, 10, 13, 22]. A
notable exception to this is the field of cochlea-inspired graded
metamaterials, where exponential gradients are the obvious choice to replicate
the spatial frequency separation that occurs in the cochlea [23, 20, 19].

There have been some previous studies comparing the effect of different
gradients on metamaterial function. For example, [24, 25] compare the
localisation of sound waves in graded acoustic metamaterials and make the
(qualitative) observation that an exponentially graded crystal yields “a
stronger wave enhancement” than a linear gradient. Conversely, [16] observes
that in a flexural system of graded beam-like resonators a cube-root gradient
yields a “stronger bandgap” and “higher energy density” than both linear and
cubic gradients. Similarly, [26] shows that the “largest power is harvested”
by a square-root gradient, compared to a linear or quadratic gradient. These
examples demonstrate the lack of consensus in the field, particularly since
the gradients they settle on are so different (the exponential gradient is
convex whereas the square and cube roots are concave, for example). This
motivates the present study, where we develop a concise approach for
systematically characterising the localisation due to different gradient
functions.

A natural approach when seeking to optimise the performance of a graded
metamaterial is to use machine learning and numerical optimisation. For
example, [27] used constrained optimisation to maximise the broadband
absorption of a graded array of Helmholtz resonators and [28] extended this
approach to study water waves trapped by a graded array of vertical barriers.
Similarly, [29] used reinforcement learning to optimise a graded array of
resonators on an elastic plate. It is worth noting that these sophisticated
numerical studies often lead to simple answers: for example, the final
configuration obtained by [27] for a broadband rainbow absorber is an array of
Helmholtz resonators whose resonant frequencies vary (approximately) linearly
as a function of their position. In this work, we will favour an approach that
reveals the underlying mechanisms to explain the relative performance of
different gradient functions. Further, the effective model we consider means
we are able to draw conclusions that transcend the specific physical setting.

There have been some other recent breakthroughs in the field of graded
metamaterials that should be mentioned, although they are not central to this
study. While subwavelength graded metamaterials inherently have some
robustness to imperfections [30], the principle of “topological protection”
has been used to create graded structures with enhanced robustness properties
[31]. Further, it has been shown that using periodic materials with zero group
velocity modes _within_ the first Brillouin zone leads to graded structures
that genuinely “trap” the energy, rather than just reflecting it, increasing
the harvested energy [15]. Additionally, it has been shown that
quasicrystalline arrangements of scatterers can be graded to produce devices
that display a rainbow effect with a fractal collection of many band gaps,
giving a very broadband response [32].

A central question underpinning this study is to decide on a quantitative
measure of wave localisation in graded media. Indeed, a possible explanation
for the lack of consensus over the optimal gradient function is that there are
different ideas on the quantity that should be optimised. Suggestions include
maximising the energy density at a specific location [16, 28] or maximising
the total absorption over a range of frequencies [28, 27]. In the present
study, we will consider the problem of maximising the focusing of a single-
frequency wave. In doing so, we are optimising the extent to which energy is
localised, ready to be harvested. Similarly, stronger localisation increases
the resolution of any frequency sensor based on a graded metamaterial. Hence,
this motivates the problem of studying the profile of the localised
eigenmodes, measured by normalising the amplitude and measuring the width
using the full width at half maximum.

We will take advantage of a multi-scale approximation method known as _high-
frequency homogenisation_ [33] to study the profile of localised eigenmodes.
This is a two-scale approach, similar to the work of _e.g._ [34, 35], that can
be used to model graded materials as perturbations of the original periodic
materials [21, 36]. It does so by characterising the localised eigenmodes as
being, at leading order in the asymptotic parameter, the product of a slowly
varying amplitude function (sometimes referred to as the _envelope_) and the
Bloch mode that exists in the unperturbed periodic structure. It is already
well established that the homogenised model gives an accurate prediction of a
localised eigenmode’s amplitude; see _e.g._ [33, 21, 36] and references
therein, as well as the convergence analyses of [37, 38]. This approach means
we can make meaningful comparisons between different gradient functions
without the need for any direct numerical studies of the wave scattering
problem. Additionally, it means that our conclusions apply to a wide range of
physical systems, as several different wave scattering problems have been
shown to reduce to the same effective model.

This article will begin by summarising the key steps in the derivation of the
effective model through high-frequency homogenisation in Section 2, before
moving on to studying solutions of the resulting ordinary differential
equation. After stating the precise formulation of the localisation problem we
will consider in Section 3, including the appropriate reformulation of the
full width at half maximum, we will move on to studying symmetric gradients in
Section 4 and monotonic gradients in Section 5. In both cases, we will present
both numerical solutions (computed using spectral differentiation) and
analytic solutions (subject to appropriate approximations, in some cases).

##  2 Effective model

The analysis in this paper will all be concerned with understanding the
localised solutions to an effective model. This model is a one-dimensional
Schrödinger equation that characterises wave propagation in perturbed periodic
media. Given a gradient function $g:\mathbb{R}\to\mathbb{R}$ and parameters
$T,\alpha>0$, which depend on the unperturbed periodic material, we consider
solutions $f$ to the eigenvalue problem

$-Tf^{\prime\prime}(X)+\alpha g(X)f(X)=\Omega_{2}^{2}f(X).$ (2.1)

The eigenvalue $\Omega_{2}^{2}$ corresponds to the difference between the
eigenfrequency of the localised eigenmode and that of the Bloch mode at the
cutoff frequency of the unperturbed periodic structure.

The Schrödinger model (2.1) arises naturally in several settings, such as
through the application of high-frequency homogenisation to modelling
localised eigenmodes in perturbed periodic media [33]. We will present two
such examples of this below. The first is for a one-dimensional periodic
material that is graded by scaling the wave speed according to the function
$g$. Since this analysis is strictly novel (but similar to previous works) we
will briefly present the details of the derivation. In two dimensions, similar
analyses were conducted in [21], where the gradient is applied as a re-scaling
of the material geometry, in the direction perpendicular to the axis of
periodicity. This was shown to apply for a range of physical systems,
including a diffraction grating composed of cylinders and a metasurface with
protruding comb-like teeth, as sketched in Figure 3. We will summarise the key
assumptions of their analysis here, for completeness. In both cases, the
interpretation of (2.1) is that $f$ is the slowly varying amplitude of the
eigenmode, $T$ is the effective group velocity and $\alpha$ is a parameter
that describes (in conjunction with the eigenvalue $\Omega_{2}^{2}$) how the
eigenfrequency of the localised eigenmode is perturbed away from the edge of
the band gap.

Both the two simple examples we outline below in Sections 2.1 and 2.2 have
material parameters that are non-dispersive (the local wave speed $c$ does not
depend on the frequency $\Omega$). However, high-frequency homogenisation can
also be applied to locally dispersive materials [39]. In this case, the
resulting effective equations for $f$ are the same, with the dispersive
material parameters appearing in the expressions for the constants $T$ and
$\alpha$. This means that the effective model (2.1) can also be used to
describe systems with local resonances, as is the case for several of the
graded metamaterials explored in the literature, _e.g._ [18, 13, 17, 19]. The
information about the local resonances is contained within the expressions for
$T$ and $\alpha$.

###  2.1 One-dimensional analysis

We study a one-dimensional system, which describes the propagation of waves
through an alternating sequence of materials

$l^{2}\frac{\mathrm{d}^{2}u}{\mathrm{d}x^{2}}+\frac{\Omega^{2}}{c^{2}}u=0,$
(2.2)

where $\Omega$ is the non-dimensionalised frequency and $l$ is the length of
the periodicity of the problem. If $l$ is the “short” scale, then we let $L$
be the “long” scale, which characterises the overall length of the structure.
We suppose that $\epsilon=l/L\ll 1$ and assume that the wave speed $c=c(x/l)$
varies as a function of the short-scale variable.

We suppose that the material is graded according to some function $g$, in the
sense that the material parameters are scaled by a varying factor
$1-\epsilon^{2}g(X)$. If we define the coordinates

$\xi=\frac{x}{l},\quad X=\frac{x}{L},$ (2.3)

then we suppose that the distribution of the material parameters $c$ is graded
according to

$c(\xi)\mapsto\left(1-\epsilon^{2}g(X)\right)^{-1}{c}(\xi).$ (2.4)

Under this transformation, the equation (2.2) becomes

$\frac{\partial^{2}u}{\partial\xi^{2}}+2\epsilon\frac{\partial^{2}u}{\partial
X\partial\xi}+\epsilon^{2}\frac{\partial^{2}u}{\partial
X^{2}}+\left(1-2\epsilon^{2}g(X)+\epsilon^{4}g(X)^{2}\right)\frac{\Omega^{2}}{{c}^{2}(\xi)}u=0.$
(2.5)

We subsequently seek a multi-scale solution of the form

$\displaystyle u(\hat{\xi},X)$ $\displaystyle=u_{0}(\hat{\xi},X)+\epsilon
u_{1}(\xi,X)+\epsilon^{2}u_{2}(\xi,X)+\dots,$ (2.6)

where the key step in the high-frequency homogenisation methodology [33] is to
make assumptions on the leading-order terms. In particular, we are interested
in the localisation of waves that occurs at the edges of a band gap as that
gap shifts due to the $O(\epsilon^{2})$ perturbations. Thus, we assume that
$\Omega$ is a perturbation of $\Omega_{0}$, which is the cutoff frequency at
the edge of the band gap:

$\displaystyle\Omega^{2}$
$\displaystyle=\Omega_{0}^{2}+\epsilon\Omega_{1}^{2}+\epsilon^{2}\Omega_{2}^{2}+\dots.$
(2.7)

We assume that $\Omega_{0}^{2}$ is a simple eigenvalue, which is not a
restrictive assumption since we want it to be at the edge of a band gap.
Seeking a solution of the form (2.6), we find that the leading-order equation
in $\epsilon$ is

$\frac{\partial^{2}u_{0}}{\partial\xi^{2}}+\frac{\Omega_{0}^{2}}{c^{2}(\xi)}u_{0}=0,$
(2.8)

from which we see that

$u_{0}(\xi,X)=U_{0}(\xi)f(X),$ (2.9)

where $U_{0}$ is the standing wave that exists at the cutoff frequency and
$f(X)$ is a slowly varying function that describes the envelope of the mode.

Moving to the $O(\epsilon)$ terms in the hierarchy of equations, we find that

$\frac{\partial^{2}u_{1}}{\partial\xi^{2}}+\frac{\Omega_{0}^{2}}{c^{2}(\xi)}u_{1}=-2\frac{\partial^{2}u_{0}}{\partial\xi\partial
X}-\frac{\Omega_{1}^{2}}{c^{2}(\xi)}u_{0}.$ (2.10)

If (2.10) is multiplied by $u_{0}$ and integrated for $\xi$ in the periodic
unit cell $\xi\in[-1,1]$, then the periodic or anti-periodic boundary
conditions at either end imply that $\Omega_{1}=0$. Thus, we conclude that
$u_{1}$ can be written as
$u_{1}(\xi,X)=f^{\prime}(X)U_{1}(\xi)+h(X)U_{0}(\xi)$ where $f^{\prime}$ is
the derivative of the envelope function $f(X)$ defined in (2.9), $h$ is some
unknown function and $U_{1}$ satisfies the equation

$\frac{\mathrm{d}^{2}U_{1}}{\mathrm{d}\xi^{2}}+\frac{\Omega_{0}^{2}}{c^{2}(\xi)}U_{1}=-2\frac{\mathrm{d}U_{0}}{\mathrm{d}\xi}.$
(2.11)

This can be solved to find $U_{1}$ explicitly111In particular, if $W(\xi)$ is
any non-periodic solution of the leading-order equation (2.8), then we have
that $U_{1}(\xi)=AW(\xi)-\xi U_{0}(\xi)$, where $A$ is a constant given by
$A=2U_{0}(1)/(W(1)+W(-1))$. See [33] for details..

Finally, we turn to the $O(\epsilon^{2})$ terms in (2.5). The corresponding
equation is

$\frac{\partial^{2}u_{2}}{\partial\xi^{2}}+\frac{\Omega_{0}^{2}}{c^{2}(\xi)}u_{2}=2g(X)\frac{\Omega_{0}^{2}}{c^{2}(\xi)}u_{0}-2\frac{\partial^{2}u_{1}}{\partial\xi\partial
X}-\frac{\partial^{2}u_{0}}{\partial
X^{2}}-\frac{\Omega_{2}^{2}}{c^{2}(\xi)}u_{0}.$ (2.12)

Assuming the periodicity in $c(\xi)$ is with respect to the unit cell
$\xi\in[-1,1]$, we can multiply (2.12) by $U_{0}(\xi)$ and integrate over
$\xi\in[-1,1]$ to find that

$\int_{-1}^{1}2\frac{\Omega_{0}^{2}}{c^{2}(\xi)}g(X)f(X)U_{0}(\xi)^{2}-2f^{\prime\prime}(X)U_{1}^{\prime}(\xi)U_{0}(\xi)-f^{\prime\prime}(X)U_{0}(\xi)^{2}-\frac{\Omega_{2}^{2}}{c^{2}(\xi)}f(X)U_{0}(\xi)^{2}\>\mathrm{d}\xi=0.$
(2.13)

Rearranging the terms, we see that $f$ satisfies the eigenvalue problem (2.1)
with $T$ given by

$T=\frac{\int_{-1}^{1}2U_{1}^{\prime}(\xi)U_{0}(\xi)+U_{0}^{2}(\xi)\>\mathrm{d}\xi}{\int_{-1}^{1}U_{0}^{2}(\xi)/c^{2}(\xi)\>\mathrm{d}\xi}$
(2.14)

and $\alpha=2\Omega_{0}^{2}$.

Figure 2: The localised defect mode for a one-dimensional two-phase periodic
material that is perturbed by the symmetric gradient
$g(X)=1-\mathrm{sech}^{2}X$. In this case, $\epsilon=0.05$ and the homogenised
envelope function, obtained by solving (2.1), accurately predicts the decay of
the amplitude.

An example of a localised mode and the accompanying homogenised envelope
$f(X)$ are shown in Figure 2, for the case where $g(X)=1-\mathrm{sech}^{2}X$.
We can see that the envelope, which was obtained by solving (2.1) with the
appropriate values for $T$ and $\alpha$, predicts the rate of decay of the
localised defect mode well. In this case, the asymptotic parameter has the
value $\epsilon=0.05$. In other works, this method has been shown to be
(surprisingly) robust with respect to the choice of $\epsilon$ and gives a
good approximation even when it is comparatively large, see _e.g._ [21, 36].

###  2.2 Two-dimensional analysis

Figure 3: High-frequency homogenisation can be used to derive the effective
equation that is the basis of this work from a range of different graded wave
scattering problems. For example, the two-dimensional analysis presented in
Section 2.2 has been shown to apply both to a metasurface with comb-like
teeth, as depicted in (a) and (c), and to a diffraction grating composed of
elliptical cylinders, as depicted in (b) and (d). We show examples of both
monotonic and symmetric, with the gradient function indicated with a dashed
line in each case.

In two-dimensions, we consider time-harmonic waves propagating through a
homogeneous medium that occupies the exterior of some scattering material.
This material could be, for example, a metasurface with protruding comb-like
teeth or a diffraction grating composed of elliptical cylinders, as sketched
in Figure 3. This model gives a good description of the scattering of
transverse electric (TE) polarised electromagnetic waves by a perfectly
conducting material if we apply Neumann boundary conditions on the surface of
the scattering material.

We suppose that the unperturbed material is periodic in the direction parallel
to the $x_{1}$ axis. The governing equation is

$l^{2}\Delta u+\Omega^{2}u=0,$ (2.15)

where $\Omega$ is the non-dimensionalised frequency. Once again, we let $l$ be
the “short” scale and $L$ be the “long” scale, supposing that $\epsilon=l/L\ll
1$. In this case, the gradient will be applied by rescaling the geometry in
the direction perpendicular to the direction of periodicity. Again, this will
be done by multiplying by a factor $1-\epsilon^{2}g$ (this time, in the
$x_{2}$-direction). We can define the new coordinates

$\xi_{1}=\frac{x_{1}}{l},\quad\xi_{2}=\frac{x_{2}}{l}\left(1-\epsilon^{2}g(X)\right),\quad
X=\frac{x_{1}}{L},$ (2.16)

and subsequently seek a multi-scale solution to (2.1) of the form

$\displaystyle u(\hat{\xi},X)=u_{0}(\hat{\xi},X)+\epsilon
u_{1}(\hat{\xi},X)+\epsilon^{2}u_{2}(\hat{\xi},X)+\dots,$ (2.17)

where $\hat{\xi}=(\xi_{1},\xi_{2})$. As in the one-dimensional case, we
suppose that
$\Omega^{2}=\Omega_{0}^{2}+\epsilon\Omega_{1}^{2}+\epsilon^{2}\Omega_{2}^{2}+\dots$,
where $\Omega_{0}$ is the cutoff frequency, corresponding to a simple
eigenvalue whose eigenmode takes the form of a Bloch mode that is either
periodic or anti-periodic. In this case, solving the leading-order equation in
$\epsilon$ once again tells us that

$u_{0}(\hat{\xi},X)=U_{0}(\hat{\xi})f(X),$ (2.18)

where $U_{0}$ is the standing wave that exists at the cutoff frequency and
$f(X)$ is a slowly varying function that describes the envelope of the mode.
Proceeding to solve the hierarchy of equations in $\epsilon$, we find that
$\Omega_{1}=0$ and that $f$ and $\Omega_{2}$ must satisfy the Schrödinger
equation (2.1), where $T$ and $\alpha$ are parameters which capture the
microstructure of the unperturbed periodic problem, given by

$T=\dfrac{-\int\int_{S}(U_{0}^{2}+U_{0}\frac{\partial}{\partial\xi_{1}}U_{1}-U_{1}\frac{\partial}{\partial\xi_{1}}U_{0}){\rm
d}S}{\int\int_{S}U_{0}^{2}{\rm d}S}$ (2.19)

and

$\alpha=\dfrac{\int\int_{S}(\frac{\partial}{\partial\xi_{2}}U_{0}^{2}-U_{0}\frac{\partial^{2}}{\partial\xi_{2}^{2}}U_{0}){\rm
d}S}{\int\int_{S}U_{0}^{2}{\rm d}S},$ (2.20)

where $U_{0}=U_{0}(\xi_{1},\xi_{2})$ is the standing wave that exists at the
cutoff frequency $\Omega_{0}$, $S$ is the exterior of the scattering material
within unit cell and $U_{1}(\xi_{1},\xi_{2})$ is the solution to the
inhomogeneous Helmholtz problem

$\Delta_{\hat{\xi}}U_{1}+\Omega_{0}^{2}U_{1}=-2\partial_{\xi_{1}}U_{0},$
(2.21)

with the appropriate periodic or anti-periodic boundary conditions. See [21]
for details of this derivation.

###  2.3 Admissible parameter values

In the rest of this paper, we will focus solely on solving the Schrödinger
equation (2.1) that we have motivated through high-frequency homogenisation.
It has been shown repeatedly in previous works that this model gives good
approximations to the one- and two-dimensional physical problems described
above, even when $\epsilon$ is relatively large. An example is shown in Figure
2, where a one-dimensional periodic material is perturbed with the gradient
function $g(X)=1-\mathrm{sech}^{2}(X)$ to create a localised mode. Many
further examples can also be found in _e.g._ [33, 37, 38, 36, 40, 21] and
references therein.

Before we start solving the homogensised Schrödinger equation (2.1) for
different gradient functions $g(X)$, it is important to understand the range
of values of the parameters $T$ and $\alpha$ that are relevant. As a
demonstrative example, consider the simple one-dimensional problem (2.2) with
wave speed given by

$c(\xi)=\begin{cases}1&\text{for }-1\leq\xi<0,\\\ 1/r&\text{for
}0\leq\xi<1,\end{cases}$ (2.22)

where $r\neq 1$ is some real-valued parameter. The dispersion relation
relating the frequency $\Omega$ and the Bloch parameter $\kappa$ is well known
[41]:

$\cos\Omega\cos\Omega r-\frac{1+r^{2}}{2r}\sin\Omega\sin\Omega r=\cos
2\kappa.$ (2.23)

This can be solved when $\kappa=0,\pi$ to find the appropriate cutoff
frequencies $\Omega_{0}$. This example was studied in detail in [33, Section
3a] where it was calculated that the formula (2.14) yields the expression

$T=\frac{\pm
4\Omega_{0}\sin\Omega_{0}\sin\Omega_{0}r}{(r\sin\Omega_{0}\mp\sin\Omega_{0}r)(\cos\Omega_{0}\mp\cos\Omega_{0}r)},$
(2.24)

where the upper/lower signs should be chosen depending on whether the Bloch
mode at the cutoff frequency $\Omega_{0}$ is periodic/anti-periodic.

In general, when $\Omega_{0}$ is at the lower edge of a band gap, it will be
the case that $T>0$. Some examples of $T$ (plotted as a function of the
material contrast parameter $r$) are shown in Figure 4, corresponding to the
cutoff frequencies at the lower edge of the second and third band gaps. An
example of the Bloch spectrum is given in the left-hand plot of Figure 4, for
$r=2$, where the corresponding cutoff frequencies are indicated by red
circles. We can see that $T$ is singular for some values of $r$ (including
$r=1$, for example). These are the values at which the corresponding band gap
closes. This violates our assumption that $\Omega_{0}$ is a simple root and
means the effective model must be modified. This can be done (we typically no
longer have $\Omega_{1}=0$, see _e.g._ [33]), however the model for simple
eigenvalues is sufficient for our purposes, since we are interested in
eigenvalues $\Omega_{0}$ that lie at the edges of band gaps.

The main conclusion to draw from Figure 4 for this study is that $T$ ranges
over several orders of positive values, including taking small values when $r$
is large. If we instead took examples of $\Omega_{0}$ which are at the upper
edge of a band gap, then $T$ would take negative values. However, in this case
we would also have that $\Omega_{2}^{2}<0$ and would need to make the change
$g\mapsto-g$ to achieve the same rainbow effect. Hence, it is sufficient to
study (2.1) with $T,\alpha>0$ and we need to consider $T/\alpha$ ranging over
several orders of magnitude.

Figure 4: For a simple example of a one-dimensional two-phase material, the
dispersion relation and the homogenised coefficients can be computed
explicitly. Here, we show the Bloch spectral bands for the material (2.22) in
the specific case when the material contrast parameter $r=2$. Then, we show
the high-frequency homogenisation coefficient $T$, given by (2.24), as a
function of $r$, for $\Omega_{0}$ being at the lower edge of the second and
third band gaps, respectively.

##  3 Localisation problem statement

The specific problem we will consider in this study is how the choice of
gradient function affects the ability of a graded metamaterial to localise a
wave of a single frequency. We will use the canonical effective model (2.1) as
a unified framework for comparing metamaterials with different gradient
functions $g$, independent of the specific physical setting. That is, given a
gradient function $g(X)$, we will explore the extent to which the envelope
function $f(X)$, obtained by solving (2.1), is localised.

To quantify the localisation of the slowly varying envelope function $f(X)$,
we will use a version of the _full width at half maximum_ (FWHM). This
measures the width of a function by taking the distance between the furthest
two points at which the function is equal to at least half of its maximum
value. Our version will be modified such that we only measure the width of the
largest peak. Then, assuming the function $f$ is continuous and has a unique
$x_{\mathrm{max}}:=\mathrm{argmax}_{x\in\mathbb{R}}|f(x)|$ we define the FWHM
as

$\mathrm{FWHM}(f)=\inf\big{\\{}x>x_{\mathrm{max}}:|f(x)|<\tfrac{1}{2}|f(x_{\mathrm{max}})|\big{\\}}-\sup\big{\\{}x<x_{\mathrm{max}}:|f(x)|<\tfrac{1}{2}|f(x_{\mathrm{max}})|\big{\\}}.$
(3.1)

Then, given a metamaterial with high-frequency homogenisation coefficients $T$
and $\alpha$, our quantitative measure of wave localisation in a graded
metamaterial will be

$\mathrm{loc}(g)_{T,\alpha}=\frac{1}{\mathrm{FWHM}(f)},$ (3.2)

where the corresponding $f$ is the first eigenmode of (2.1), with appropriate
boundary conditions.

Since we wish to neglect the effect of boundary conditions in our analysis, we
will solve (2.1) on an unbounded domain with appropriate decay conditions as
$|X|\to\infty$. We will study two different classes of gradient function in
this work. First, we will consider symmetric gradient functions $g$, where we
expect waves to be localised at the centre of the structure. In this case, the
amplitude $f$ will be assumed to decay as $X\to\pm\infty$. Subsequently, we
will study the more classical example of monotonic gradient functions $g$,
where we expect the wave to propagate until it reaches a position of
localisation. In this case, we assume that $f$ and its derivative decay as
$X\to\infty$.

##  4 Comparison of localisation for symmetric gradients

We begin by considering examples where the gradient function $g$ is symmetric.
In which case, waves will be localised in the centre of the structure with
decay in either direction. This can be thought of either as attaching two
classical monotonic graded metamaterials back to back, or simply as an
interesting wave localisation problem in its own right. Structures of this
kind could be used as the starting point for building multi-dimensional
waveguides that guide waves along specified paths.

The class of symmetric gradients is a natural starting point as the subtlety
of identifying the position of localisation is negated, since we expect waves
to be localised at the centre. We will assume that the wave is at the cutoff
frequency at the middle of the array, which equates to assuming that $g(0)=0$
and means we expect the wave amplitude to decay immediately in both
directions. An additional advantage is that there are known explicit solutions
to the Schrödinger equation (2.1) for several symmetric gradient functions
$g$, allowing us to validate our numerical approach.

Altogether, we will consider seven different gradient functions. These include
the examples considered previously in the literature: linear [17], exponential
[25, 18], cubic [16] and cube root [16]. We supplement this with three further
examples, for which the Schrödinger equation (2.1) has explicit analytic
solutions. The seven gradient functions are shown in Figure 5, in the first
column of subplots. For the sake of calibrating the comparisons, the gradient
functions are each normalised so that $g(0)=0$ and $g(\pm 1)=1$, meaning that
each gradient function has the same average gradient over the interval
$[0,1]$.

###  4.1 Numerical results

Numerical solutions to the eigenvalue problem (2.1) can be computed using a
spectral method with periodic spectral differentiation [42]. In Figure 5,
examples of the localised mode profiles are shown. In each row of subplots,
the gradient function is plotted followed by the first eigenmode for two
different values of $T/\alpha$. The full width at half maximum (FWHM) is
indicated on each eigenmode. Subsequently, the localisation (3.2) can be
studied as a function of $T/\alpha$, which is shown in the lower part of
Figure 5. The same data points are repeated on a log–log scale inset, where it
is apparent that each line is approximately a straight line. Below the main
plot, we show the gradient that gave the strongest localisation. It is notable
that multiple symbols are shown for some values of $T/\alpha$, which
corresponds to the modes being equally localised (at the resolution of the
numerical scheme).

A striking conclusion to draw from Figure 5 is that the relative strengths of
localisation of the different gradient functions changes as $T/\alpha$ varies.
The straight lines in the inset log–log plot have different slopes for
different gradient functions, meaning the range of $T/\alpha$ is divided into
three regimes, which is clear from looking at the bottom display in Figure 5.
For very small values of $T/\alpha$, the cube root gradient gives the
strongest localisation. As $T/\alpha$ increases, we then enter an intermediate
regime where the concave exponential gradient performs best. Finally, for
large values of $T/\alpha$, we see that the trigonometric and linear gradients
give the strongest localisation. This suggests that, for the case of symmetric
gradient functions, the gradient which gives the strongest localisation will
depend on the specific physical setting and choice of metamaterial (as this
determines the values of $T$ and $\alpha$).

Figure 5: Symmetric gradient functions give symmetric eigenmodes whose
localisation can be quantified using the full width at half maximum (FWHM). In
the upper displays, we plot the first eigenmode of (2.1) for seven different
symmetric gradient functions (and two different values of $T/\alpha$). On each
mode, the length corresponding to the FWHM is shown. Below this, the
localisation (given by the reciprocal of the FWHM) is plotted as a function of
$T/\alpha$ for each symmetric gradient function (the inset plot shows the same
data on a log–log plot). Finally, the lower display shows the gradient with
the most localised first eigenmode at each $T/\alpha$ (with multiple symbols
shown in the event of a tie).

###  4.2 Analytic results

The Schrödinger equation (2.1) has known analytic solutions for some symmetric
potentials. This means the numerical results of Figure 5 can be partially
verified analytically. The first example which admits an explicit solution is
$g(X)=1-\mathrm{sech}^{2}X$. The localised eigenmode, and associated
homogenised envelope function $f$, was shown in Figure 2. In this case, the
(normalised) solution $f$ of the ODE (2.1) is given by [43]

$f(X)=\cosh^{1/2-\gamma}X,$ (4.1)

where $\gamma=\sqrt{1/4+\alpha/T}$. Additionally, the case $g(X)=X^{2}$ admits
the well-known solution

$f(X)=\exp\bigg{(}-\frac{1}{2}\sqrt{\frac{\alpha}{T}}X^{2}\bigg{)}.$ (4.2)

Finally, the Schrödinger equation (2.1) can be reduced to the Bessel equation
under appropriate transformations. For example, if $g(X)=1-\exp(-|X|)$ then
this approach can be used to see that [44]

$f(X)=J_{\nu}\bigg{(}2\sqrt{\frac{\alpha}{T}}\exp\Big{(}-\tfrac{|X|}{2}\Big{)}\bigg{)},$
(4.3)

where $J_{\nu}$ is the standard Bessel function of the first kind and the
order $\nu$ is given by $\nu=2\sqrt{\alpha/T}(\Omega_{2}^{2}+1)$. The order
$\nu$ can also be found directly, without knowing the value of the eigenvalue
$\Omega_{2}^{2}$, by solving the implicit equation
$J_{\nu-1}(2\sqrt{\alpha/T})-\nu/2\sqrt{T/\alpha}J_{\nu}(2\sqrt{\alpha/T})=0$
[44].

Given these explicit solutions (4.1)–(4.3), it is possible to compute the
localisation (3.2) explicitly. We have that

$\mathrm{loc}(1-\mathrm{sech}^{2}X)_{T,\alpha}=\left(2\cosh^{-1}\big{(}2^{\frac{2}{2\gamma-1}}\big{)}\right)^{-1},$
(4.4)

where $\cosh^{-1}:[1,\infty)\to[0,\infty)$ is the inverse hyperbolic cosine
function. Additionally, the convenient form of the solution (4.2) for the
quadratic potential leads to the expression

$\mathrm{loc}(X^{2})_{T,\alpha}=\left(2\sqrt{2\sqrt{\tfrac{T}{\alpha}}\log
2}\right)^{-1}.$ (4.5)

This is particularly informative, as it shows that
$\mathrm{loc}(X^{2})_{T,\alpha}\propto(T/\alpha)^{-1/4}$, explaining the shape
of the curve observed in Figure 5. The rate triangle on the inset log–log plot
in Figure 5 shows the expected
$\mathrm{loc}_{T,\alpha}\propto(T/\alpha)^{-1/4}$ rate, for comparison. A
similar formula can be obtained from (4.3), however it relies on inverse
Bessel functions so is difficult to state concisely, hence we omit it for
brevity.

##  5 Comparison of localisation for monotonic gradients

We now consider the classical case of gradient functions $g$ that are
monotonically increasing. This case is widely used in energy harvesting and
machine hearing settings, as it allows waves to couple with and propagate
through the metamaterial up until the point at which they are localised (which
depends on their frequency). Our definition of the (modified) full width at
half maximum (3.1) is designed so that it only measures the width of the
largest peak, which will be the final peak before the mode’s amplitude decays
quickly. An important subtlety for monotonic gradient functions is that, in
contrast to the symmetric gradients studied in the previous setting, it is not
immediately clear _where_ the wave will be localised. Indeed, characterising
the position at which it is localised will be a key step in our analysis,
below.

We consider seven different gradient functions, which are the monotonic
versions of the symmetric gradient functions studied in the previous section.
In order to make for a fair comparison between the different gradients, we
normalise the average gradient to be 1 across the interval $[-1,1]$, achieved
by setting $g(-1)=-1$ and $g(1)=1$. For physical reality, so we don’t have
arbitrarily large physical structures, the gradient is truncated such that
$g(X)=-1$ for $X<-1$ and $g(X)=1$ for $X>1$.

###  5.1 Numerical results

The numerical results for monotonic gradients are shown in Figure 6. In this
case, it should be noted immediately that the differences between the
different gradient functions are much smaller than for the symmetric
gradients. Whereas it was possible to judge by eye which modes in Figure 5
were most or least strongly localised, the differences are much smaller in
Figure 6. Another difference from the symmetric case is that there are no
apparent transitions between the relative localisation of different gradients.
The lines in the log–log plot in Figure 6 are (approximately) parallel and
there appears to be a consistent hierarchy of the localisation due to each
gradient. This hierarchy is that the cubic, cube root and concave exponential
gradients produce the strongest localisation (with minimal differences between
them), followed by the quadratic gradient and subsequently by the convex
exponential. Finally, the trigonometric and linear gradients give the weakest
localisation (again, with differences at the resolution of our numerical
scheme).

These results, firstly, validate the conclusions of [25, 24] that exponential
gradients lead to stronger localisation than linear gradients (it turns out,
this is irrespective of whether the exponential is convex or concave).
Additionally, they also confirm the conclusions of [16] that cube root
gradients similarly outperform linear gradients. That these conclusions hold
consistently across a range of values of $T/\alpha$ suggests that these
conclusions will hold for a broad range of physical systems (so long as they
fall into the multi-scale framework of the high-frequency homogenisation
methodology).

Figure 6: Monotonic gradient functions give eigenmodes that transition from
bring a propagating mode to a decaying mode. Their localisation can be
quantified using the full width at half maximum (FWHM) of the last peak. In
the upper displays, we plot the first eigenmode of (2.1) for seven different
monotonic gradient functions (and two different values of $T/\alpha$). On each
mode, the length corresponding to the FWHM is shown. Below this, the
localisation (given by the reciprocal of the FWHM) of the first mode is
plotted for different values of $T/\alpha$ for each gradient function.
Finally, the lower display shows the gradient with the most localised first
eigenmode at each $T/\alpha$ (with multiple symbols shown in the event of a
tie).

###  5.2 Analytic results

In the case of monotonic gradient functions, analytic solutions to the
Schrödinger equation (2.1) are harder to come by. The notable exception to
this is the case of a linear gradient, for which (2.1) is a transformed
version of the famous Airy equation [45]. We can see this by making the change
of variables

$Y=\left(\frac{T}{\alpha}\right)^{2/3}\frac{\alpha X-\Omega_{2}^{2}}{T},$
(5.1)

which transforms (2.1) into the Airy equation

$f^{\prime\prime}(Y)=Yf(Y).$ (5.2)

Since we are interested in solutions for which $f(X)\to 0$ as $X\to\infty$,
this determines the correct solution of (5.2) and we find that the solution of
(2.1) with $g(X)=X$ is given by

$f(x)=\mathrm{Ai}\bigg{[}\left(\frac{T}{\alpha}\right)^{2/3}\frac{\alpha
X-\Omega_{2}^{2}}{T}\bigg{]},$ (5.3)

where $\mathrm{Ai}[\,\cdot\,]$ is the Airy function of the first kind [45].

For non-linear monotonic gradient functions we will use approximation
strategies in order to make useful analytic comparisons222In reality, the
solution to the Schrödinger equation can be expressed in terms of special
functions for many different non-linear monotonic gradient functions. For
example, parabolic cylinder functions can be used for quadratic gradients and
modified Bessel functions of the first kind can be used for cubic gradients.
However, such solutions are of limited use for drawing comparisons and it will
be much more insightful to approximate each solution locally by an Airy
function.. Since our aim is to compare the extent of localisation, our
approach will be to linearise $g$ around the point at which it is localised,
and then formulate an approximate solution in terms of Airy functions. Before
doing this, we can use a _WKB approximation_ to characterise the overall
properties of the solution. This approach, named after Wentzel, Kramers and
Brillouin, is a popular method for approximating the solution to the
Schrödinger equation in the case that the gradient function is slowly varying
[45]. The idea is that, if the gradient varies slowly, the solution $f(X)$
will be similar to the oscillating function $\exp(\mathrm{i}kX)$ that would
solve the equation if the gradient $g$ was constant. In particular, the WKB
ansatz is

$f(X)=A(X)e^{\mathrm{i}\Phi(X)},$ (5.4)

for some real-valued functions $A$ and $\Phi$ that are to be found.
Substituting (5.4) into (2.1) and equating real and imaginary parts gives the
equations $(A^{2}\Phi^{\prime})^{\prime}=0$ and
$A^{\prime\prime}=A((\Phi^{\prime})^{2}-k^{2})$, where $k=k(X)$ is a spatially
varying local wavenumber, given by

$k(X)=\sqrt{\frac{\Omega_{2}^{2}-\alpha g(X)}{T}}.$ (5.5)

The first of these two equations implies that $A=C/\sqrt{\Phi^{\prime}}$ for
some constant $C$. The second of these can be simplified by assuming that
$A^{\prime\prime}/A$ is much smaller than both $(\Phi^{\prime})^{2}$ and
$k^{2}$. In which case, we have that $\Phi^{\prime}=\pm k$ and we reach the
solution

$f(X)\approx\frac{1}{\sqrt{k(X)}}\exp\bigg{(}\pm\mathrm{i}\int^{X}k(s)\>\mathrm{d}s\bigg{)}.$
(5.6)

This approximate solution can be used to understand the overall behaviour of
$f$. When $\alpha g(X)<\Omega_{2}^{2}$ the wavenumber $k(X)$ is real-valued
meaning that the approximate solution (5.6) for $f$ corresponds to an
oscillatory propagating wave. Conversely, when $\alpha g(X)>\Omega_{2}^{2}$
the wavenumber $k(X)$ is purely imaginary meaning that the solution has an
exponentially decaying profile. This matches the behaviour we see in Figure 6.

It is clear that the approximate WKB solution (5.6) is not valid near to the
point where $\alpha g(X)=\Omega_{2}^{2}$, as the wavenumber $k(X)$ vanishes
there. In this region, we will linearise the gradient about this point, which
reduces the Schrödinger equation (2.1) to an Airy equation (5.2). In
particular, we have that

$f^{\prime\prime}(X)=\frac{\alpha
g^{\prime}(X_{0})}{T}(X-X_{0})f(X)+O\big{(}(X-X_{0})^{2}\big{)},$ (5.7)

as $X-X_{0}\to 0$, where $X_{0}$ is the point such that $\alpha
g(X_{0})=\Omega_{2}^{2}$. This leads to the approximate solution

$f(X)=\mathrm{Ai}\bigg{[}\left(\frac{\alpha
g^{\prime}(X_{0})}{T}\right)^{1/3}(X-X_{0})\bigg{]}+O\big{(}(X-X_{0})^{2}\big{)}.$
(5.8)

Based on this analysis, it is possible to form an approximate solution for $f$
on the whole real line by stitching together the WKB solution (5.6) and the
Airy solution (5.8) near to the singularity. This approach is similar to
conventional boundary layer analyses [45, 46] or the matched HFH-WKB method
developed by [47]. For our purposes, however, the expressions (5.6) and (5.8)
are already sufficient.

As was the case for the symmetric gradients, we can use our analysis to
estimate the localisation factor $\mathrm{loc}(g)_{T,\alpha}$. For the linear
gradient, (5.3) shows us immediately that
$\mathrm{loc}(X)_{T,\alpha}\propto(T/\alpha)^{-1/3}$. The approximate solution
(5.8) suggests that this relationship will hold for other gradients also,
which is confirmed by the inset log–log plot in Figure 6 and explains the
observation that these lines all have the same slope.

The approximation (5.8) reveals some insight into the significant role that
the quantity $g^{\prime}(X_{0})$ plays in determining the strength of
localisation. It suggests that the localisation can be maximised by maximising
$g^{\prime}(X_{0})$. We can explore this hypothesis for the seven gradient
functions considered in this work by comparing $\mathrm{loc}(g)_{T,\alpha}$
and $g^{\prime}(X_{0})$, which is shown in Figure 7 for $T=0.2$ and $\alpha=1$
(the plot is qualitatively the same for other values of $T$ and $\alpha$). It
is noticeable that the five convex gradient functions do indeed follow an
approximate linear trend (on this log–log plot); the dashed line shown has
been fitted to these five data points (in a least-squares sense) and has an
$R$-squared value of $R^{2}=0.86$. This suggests that a viable strategy for
maximising the localisation due to convex monotonic gradient functions is to
maximise the slope of the gradient function $g$ at the point at which $\alpha
g(X_{0})=\Omega_{2}^{2}$. The cube root and concave exponential gradient
functions do not appear to fit the trend that is otherwise present in Figure
7. Our interpretation of this is that it is a consequence of their concavity
as it means the gradient function increases very quickly at the start of the
graded region. This effectively leads to an impedance mismatch that localises
the wave through reflection, which is a different mechanism to that which is
employed by the gradient functions that sweep upwards more gradually.

Figure 7: A comparison of the localisation $\mathrm{loc}(g)_{0.2,1}$ and the
slope of the gradient function $g^{\prime}(X_{0})$, evaluated. Both axes use
logarithmic scales. In this figure, we use $T=0.2$, $\alpha=1$ but note that
the behaviour is the same for other parameter values. The dashed line is a
straight line that has been fitted (in a least-squares sense) to the data
points for the five convex gradient functions.

##  6 Concluding remarks

High-frequency homogenisation provides a simple ordinary differential model
that accurately describes wave localisation by a graded metamaterial. We have
used this framework to compare the effect of the choice of gradient function
in an efficient way, without the need for expensive wave scattering
simulations. Additionally, the effective model used in this work means that
our conclusions span several different physical settings and wave regimes (as
we demonstrated in Section 2). Through this approach, we observed that the
localisation due to symmetrically graded metamaterials varies depending on the
effective parameters. This means that the symmetric gradient function that
yields the strongest localisation will be different for different physical
settings. For monotonically graded metamaterials the picture is slightly
clearer, as there is a hierarchy of different gradient functions that is
stable as the effective parameters change. Further, our analysis revealed that
the localisation could be maximised (at least within the space of convex
gradient functions) by maximising the slope of the gradient function at a
specified transition point. However, it is important to note that, even in
spite of this insight, the differences between some of the monotonic gradient
functions are negligibly small.

The fact that no clear universal optimal choice of gradient function exists
even in this simplified setting, where we have considered single frequency
localisation within a homogenised model, provides important context for future
studies. In particular, our results show the need to think carefully about the
choice of objective function before applying optimisation or machine learning
algorithms and that any such approach is likely to encounter a great number of
local minima. Conversely, a more positive interpretation of these results is
that graded metamaterials perform well for a wide range of gradient functions
(and it doesn’t seem to matter too much which gradient function is chosen).
The fundamental mechanism behind the function of graded metamaterials (as
depicted in Figure 1) is sufficiently simple and powerful that it is (within
reason) robust with respect to the choice of gradient.

## Acknowledgements

The authors would like to thank Richard Craster and Marie Touboul for helpful
advice and suggestions.

## Funding

The work of B.D. was funded by EPSRC through a fellowship with grant number
EP/X027422/1. H.J.P. is grateful for EPSRC funding provided through the Centre
for Doctoral Training ‘Fluid Dynamics Across Scales’ Programme (grant no.
EP/L016230/1. R.V).

## Data Accessibility

The software used to perform the numerical simulations presented in this work
is available for download at https://doi.org/10.5281/zenodo.7864009.

## Conflicts of interest

The authors declare that they have no conflicts of interest.

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We now show that if the map $H^{1}(C,T_{C}(-D))\to H^{1}(C,E_{\star}^{j+1,k})$
vanishes, we will be able to produce a coparabolic bundle which is not
generically globally generated. We will later apply 6.3.6 to this bundle to
obtain Theorem 6.1.1(1).

######  Lemma 6.4.7.

With notation as in 6.4.1, suppose the map $H^{1}(C,T_{C}(-D))\to
H^{1}(C,E_{\star}^{j+1,k})$ (induced by the non-zero map
$\psi_{j+1,k}:T_{C}(-D)\to E_{\star}^{j+1,k}$ of 6.4.3) vanishes. Then the
coparabolic bundle
$\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[\widthof{(E_{\star}^{j+1,k})^{\vee}}]{\kern-0.6pt\bigwedge\kern-0.6pt}{\rule[-505.89pt]{4.30554pt}{505.89pt}}}{}}{0.5ex}}\stackon[1pt]{(E_{\star}^{j+1,k})^{\vee}}{\tmpbox}\otimes\omega_{C}(D)$
is not generically globally generated.

###### Proof.

Since $\psi_{j+1,k}:T_{C}(-D)\to E_{\star}^{j+1,k}$ is nonzero, we obtain a
nonzero Serre dual map

$\displaystyle\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[\widthof{(E_{\star}^{j+1,k})^{\vee}}]{\kern-0.6pt\bigwedge\kern-0.6pt}{\rule[-505.89pt]{4.30554pt}{505.89pt}}}{}}{0.5ex}}\stackon[1pt]{(E_{\star}^{j+1,k})^{\vee}}{\tmpbox}\otimes\omega_{C}(D)\to\omega_{C}\otimes\omega_{C}(D)$

which induces the $0$ map

$\displaystyle
H^{0}(C,\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[\widthof{(E_{\star}^{j+1,k})^{\vee}}]{\kern-0.6pt\bigwedge\kern-0.6pt}{\rule[-505.89pt]{4.30554pt}{505.89pt}}}{}}{0.5ex}}\stackon[1pt]{(E_{\star}^{j+1,k})^{\vee}}{\tmpbox}\otimes\omega_{C}(D))\to
H^{0}(C,\omega_{C}\otimes\omega_{C}(D)$

by Serre duality 2.6.6 and 6.4.4. In particular,
$\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[\widthof{(E_{\star}^{j+1,k})^{\vee}}]{\kern-0.6pt\bigwedge\kern-0.6pt}{\rule[-505.89pt]{4.30554pt}{505.89pt}}}{}}{0.5ex}}\stackon[1pt]{(E_{\star}^{j+1,k})^{\vee}}{\tmpbox}\otimes\omega_{C}(D)$
is not generically globally generated. Indeed, any global section must lie in
the kernel of (6.4), which has corank one in
$\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[\widthof{(E_{0}^{j+1,k})^{\vee}}]{\kern-0.6pt\bigwedge\kern-0.6pt}{\rule[-505.89pt]{4.30554pt}{505.89pt}}}{}}{0.5ex}}\stackon[1pt]{(E_{0}^{j+1,k})^{\vee}}{\tmpbox}\otimes\omega_{C}(D)$.
∎

This concludes our setup for proving Theorem 6.1.1(1). To prove Theorem
6.1.1(2), we will need the following generalization of 6.2.2 to the
coparabolic setting.

######  Lemma 6.4.8.

If $V_{\star}$ is a semistable coparabolic bundle with respect to a divisor
$D=x_{1}+\cdots+x_{n}$, of coparabolic slope $\mu_{\star}(V_{\star})>2g-1+n$,
then $V_{\star}$ is globally generated.

###### Proof.

Let $p\in C$ be a point. Suppose $V_{\star}=\widehat{E}_{\star}$ for
$E_{\star}$ a parabolic bundle. It suffices to show $V_{\star}$ is generated
by global sections at $p$. Indeed, $V_{\star}(-p)$ is a semistable coparabolic
bundle with coparabolic slope $\mu_{\star}(V_{\star}(-p))>2g-2+n$. Therefore,
$\mu_{\star}(E_{\star}^{\vee}(p)\otimes\omega_{C}(D))<0$, implying
$H^{0}(C,E_{\star}^{\vee}(p)\otimes\omega_{C}(D))=0$, as any global section
would be destabilizing. By Serre duality 2.6.6, (taking
$E_{\star}^{\vee}(p)\otimes\omega_{C}(D)$ in place of the parabolic vector
bundle $E_{\star}$ in 2.6.6),
$H^{1}(C,V_{\star}(-p))=H^{0}(C,E_{\star}^{\vee}(p)\otimes\omega_{C}(D))^{\vee}=0$.
As in 6.2.2, the long exact sequence on cohomology associated $V(-p)\to V\to
V|_{p}$ implies $H^{0}(C,V)\otimes\mathscr{O}\to V\to V|_{p}$ is surjective,
as desired. ∎

#### 6.4.9.

We now prove Theorem 6.1.1.

###### Proof of Theorem 6.1.1.

We use notation as in 6.4.1. We aim first to show that if
$(E_{\star}^{\prime},\nabla^{\prime})$ is the isomonodromic deformation of
$(E_{\star},\nabla)$ to an analytically general nearby curve $C^{\prime}$, and
if $E^{\prime}_{*}$ is not semistable, then for every $i$ there are some
$j<i<k$ with
$\operatorname{rk}\operatorname{gr}^{j+1}_{HN}E_{\star}^{\prime}\cdot\operatorname{rk}\operatorname{gr}^{k}_{HN}E_{\star}^{\prime}\geq
g+1.$

By [BHH21, Lemma 5.1], the locus of bundles in a family $\mathscr{E}_{\star}$
on $\mathscr{C}\to\Delta$ which are not semistable form a closed analytic
subset, and if a general member is not semistable, then, after passing to an
open subset of $\Delta$, there is a filtration on $\mathscr{E}_{\star}$
restricting to the Harder-Narasimhan filtration on each fiber. Thus after
replacing $(C,D)$ with an analytically general nearby curve
$(C^{\prime},D^{\prime})$, and replacing $(E_{\star},\nabla)$ with the
restriction $(E_{\star}^{\prime},\nabla^{\prime})$ of the isomonodromic
deformation to $(C^{\prime},D^{\prime})$, we may assume the Harder-Narasimhan
filtration $HN^{\bullet}$ of $E_{\star}^{\prime}$ extends to a filtration of
$\mathscr{E_{\star}}$ on a first-order neighborhood of $C^{\prime}$. We set
$(E^{\prime}_{\star})^{i,j}:=\mathscr{H}\kern-2.0ptom(\operatorname{gr}^{i}_{HN}(E^{\prime}_{\star}),\operatorname{gr}^{j}_{HN}(E^{\prime}_{\star}))_{\star}$;
note that $(E^{\prime}_{\star})^{i,j}$ is semistable by the definition of the
Harder-Narasimhan filtration.

We next verify that for every $0<i<m$, there is some $j<i<k$ for which
$\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[\widthof{((E^{\prime}_{\star})^{j+1,k})^{\vee}}]{\kern-0.6pt\bigwedge\kern-0.6pt}{\rule[-505.89pt]{4.30554pt}{505.89pt}}}{}}{0.5ex}}\stackon[1pt]{((E^{\prime}_{\star})^{j+1,k})^{\vee}}{\tmpbox}\otimes\omega_{C}(D)$
is not generically globally generated. By 6.4.3, for every $0<i<m$, there is
some $j<i$ and $k\geq i+1$ so that the map
$T_{C^{\prime}}(-D^{\prime})\to\mathscr{E}\kern-1.0ptnd(E_{\star}^{\prime})/\mathscr{E}\kern-1.0ptnd_{HN^{\bullet}}(E_{\star}^{\prime})$
induces a nonzero map

$T_{C^{\prime}}(-D^{\prime})\to(E^{\prime}_{\star})^{j+1,k}:=\mathscr{H}\kern-2.0ptom(\operatorname{gr}^{j+1}_{HN}E_{\star}^{\prime},\operatorname{gr}^{k}_{HN}E_{\star}^{\prime})_{\star}.$

By 6.4.4, $H^{1}(C^{\prime},T_{C^{\prime}}(-D^{\prime}))\to
H^{1}(C^{\prime},(E^{\prime}_{\star})^{j+1,k})$ vanishes. Hence, by 6.4.7,
$\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[\widthof{((E^{\prime}_{\star})^{j+1,k})^{\vee}}]{\kern-0.6pt\bigwedge\kern-0.6pt}{\rule[-505.89pt]{4.30554pt}{505.89pt}}}{}}{0.5ex}}\stackon[1pt]{((E^{\prime}_{\star})^{j+1,k})^{\vee}}{\tmpbox}\otimes\omega_{C}(D)$
is not generically globally generated. Note that
$\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[\widthof{((E^{\prime}_{\star})^{j+1,k})^{\vee}}]{\kern-0.6pt\bigwedge\kern-0.6pt}{\rule[-505.89pt]{4.30554pt}{505.89pt}}}{}}{0.5ex}}\stackon[1pt]{((E^{\prime}_{\star})^{j+1,k})^{\vee}}{\tmpbox}\otimes\omega_{C}(D)$
has slope $>2g-2+n$ by 2.6.5.

We are finally in a position to prove Theorem 6.1.1(1). It follows from 6.3.6
that

$\operatorname{rk}\operatorname{gr}^{j+1}_{HN}E_{\star}^{\prime}\cdot\operatorname{rk}\operatorname{gr}^{k}_{HN}E_{\star}^{\prime}=\operatorname{rk}\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[\widthof{((E^{\prime}_{\star})^{j+1,k})^{\vee}}]{\kern-0.6pt\bigwedge\kern-0.6pt}{\rule[-505.89pt]{4.30554pt}{505.89pt}}}{}}{0.5ex}}\stackon[1pt]{((E^{\prime}_{\star})^{j+1,k})^{\vee}}{\tmpbox}\otimes\omega_{C}(D)\geq
g+1.$

Thus Theorem 6.1.1(1) holds.

We now conclude by verifying Theorem 6.1.1(2). By 6.4.8,

$\displaystyle\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[\widthof{((E^{\prime})_{\star}^{j+1,k})^{\vee}}]{\kern-0.6pt\bigwedge\kern-0.6pt}{\rule[-505.89pt]{4.30554pt}{505.89pt}}}{}}{0.5ex}}\stackon[1pt]{((E^{\prime})_{\star}^{j+1,k})^{\vee}}{\tmpbox}\otimes\omega_{C}(D)=\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[\widthof{\mathscr{H}\kern-2.0ptom(\operatorname{gr}^{j+1}_{HN}E_{\star}^{\prime},\operatorname{gr}^{k}_{HN}E_{\star}^{\prime})^{\vee}}]{\kern-0.6pt\bigwedge\kern-0.6pt}{\rule[-505.89pt]{4.30554pt}{505.89pt}}}{}}{0.5ex}}\stackon[1pt]{\mathscr{H}\kern-2.0ptom(\operatorname{gr}^{j+1}_{HN}E_{\star}^{\prime},\operatorname{gr}^{k}_{HN}E_{\star}^{\prime})^{\vee}}{\tmpbox}\otimes\omega_{C}(D$

must have slope at most $2g-1+n$, since it is not generically globally
generated. As
$\mathscr{H}\kern-2.0ptom(\operatorname{gr}^{j+1}_{HN}E_{\star}^{\prime},\operatorname{gr}^{k}_{HN}E_{\star}^{\prime})$
has negative parabolic slope by the definition of the Harder-Narasimhan
filtration and 2.6.5, we find

$\displaystyle-1\leq\mu_{\star}(\mathscr{H}\kern-2.0ptom(\operatorname{gr}^{j+1}_{HN}(E_{\star}^{\prime}),\operatorname{gr}^{k}_{HN}(E_{\star}^{\prime})))<0.$

Using 2.6.5, the parabolic slope of a tensor product of parabolic vector
bundles is the sum of their parabolic slopes, and taking duals negates
parabolic slope. Therefore,

$\displaystyle
0<\mu_{\star}(\operatorname{gr}^{j+1}_{HN}({E}_{\star}^{\prime}))-\mu_{\star}(\operatorname{gr}^{k}_{HN}({E}_{\star}^{\prime}))\leq
1.$

Since

$\displaystyle\mu_{\star}(\operatorname{gr}^{j+1}_{HN}({E}_{\star}^{\prime}))\geq\mu_{\star}(\operatorname{gr}^{i}_{HN}({E}_{\star}^{\prime}))\geq\mu_{\star}(\operatorname{gr}^{i+1}_{HN}({E}_{\star}^{\prime}))\geq\mu_{\star}(\operatorname{gr}^{k}_{HN}({E}_{\star}^{\prime})),$

we also conclude

$0<\mu_{\star}(\operatorname{gr}^{i}_{HN}({E}_{\star}^{\prime}))-\mu_{\star}(\operatorname{gr}^{i+1}_{HN}({E}_{\star}^{\prime}))\leq
1.\qed$

#### 6.4.10.

We now prove 6.1.2, using Theorem 6.1.1 and the AM-GM inequality.

###### Proof of 6.1.2.

It suffices to consider the case where $(E,\nabla)$ has irreducible monodromy,
as an extension of semistable parabolic bundles of the same slope is
semistable.

If $(E_{\star}^{\prime},\nabla^{\prime})$ is an isomonodromic deformation of
$(E_{\star},\nabla)$ to an analytically general nearby curve which is not
semistable, it follows from Theorem 6.1.1(1) that for each $i$, there will be
$j,k$ with $j<i<k$ so that the Harder-Narasimhan filtration $HN$ of
$E_{\star}^{\prime}$ satisfies
$\operatorname{rk}\operatorname{gr}^{j+1}_{HN}E_{\star}^{\prime}\cdot\operatorname{rk}\operatorname{gr}^{k}_{HN}E_{\star}^{\prime}\geq
g+1.$ Since
$\operatorname{rk}\operatorname{gr}^{j+1}_{HN}E_{\star}^{\prime}+\operatorname{rk}\operatorname{gr}^{k}_{HN}E_{\star}^{\prime}\leq\operatorname{rk}E_{\star}^{\prime}=\operatorname{rk}E_{\star}$,
it follows from the arithmetic mean-geometric mean inequality that

$g+1\leq\operatorname{rk}\operatorname{gr}^{j+1}_{HN}E_{\star}^{\prime}\cdot\operatorname{rk}\operatorname{gr}^{k}_{HN}E_{\star}^{\prime}\leq\left(\frac{\operatorname{rk}E_{\star}}{2}\right)^{2}.$

So $\operatorname{rk}E_{\star}\geq 2\sqrt{g+1}$ as desired. ∎

#### 6.4.11.

We conclude by proving Theorem 1.3.4 and 1.3.6.

###### Proof of Theorem 1.3.4 and 1.3.6.

These follow immediately from Theorem 6.1.1 and 6.1.2 respectively, taking
$E_{\star}$ to have the trivial parabolic structure. ∎

##  7\. Variations of Hodge structure on an analytically general curve

We prove Theorem 1.2.12 in § 7.1, Theorem 1.2.5 in § 7.2, 1.2.8 in § 7.3, and
1.2.7 in § 7.4. Finally, we prove an additional application concerning maps
from very general curves to Hilbert modular varieties in § 7.2.

###  7.1. The proof of Theorem 1.2.12

We start with the following lemma, which gives a useful criterion for showing
a monodromy representation is unitary.

######  Lemma 7.1.1.

Suppose $(C,x_{1},\ldots,x_{n})$ is an $n$-pointed hyperbolic curve and
$(E,\nabla)$ is a flat vector bundle on $C\setminus\\{x_{1},\ldots,x_{n}\\}$
underlying a polarizable complex variation of Hodge structure. Let
$(\overline{E}_{\star},\overline{\nabla})$ be the Deligne canonical extension
of $(E,\nabla)$ to a flat vector bundle on $C$ with regular singularities at
the $x_{i}$, with the parabolic structure associated to $\overline{\nabla}$.
If $\overline{E}_{\star}$ is semistable, then the representation of
$\pi_{1}(C\setminus\\{x_{1},\cdots,x_{n}\\})$ associated to $(E,\nabla)$ is
unitary.

###### Proof.

By 4.1.4(2), we may write $\mathbb{V}:=\ker(\nabla)$ as

$\mathbb{V}:=\bigoplus_{i}\mathbb{L}_{i}\otimes W_{i}$

where the $\mathbb{L}_{i}$ each have irreducible monodromy and also underlie
polarizable variations of Hodge structure, and the $W_{i}$ are constant
complex Hodge structures. It suffices to show the representation associated to
each $\mathbb{L}_{i}$ has unitary monodromy. We may therefore reduce to the
case that $\mathbb{V}=\mathbb{L}_{i}$ and assume that $(E,\nabla)$ has
irreducible monodromy.

Let $i$ be maximal such that $F^{i}\overline{E}_{\star}$ is non-zero. Since
$\overline{E}_{\star}$ is semistable, it follows from 4.1.5 that that the
natural map

$\displaystyle F^{i}\overline{E}_{\star}\to
F^{i-1}\overline{E}_{\star}/F^{i}\overline{E}_{\star}\otimes\omega_{C}$

induced by the connection is zero, i.e. the connection preserves
$F^{i}\overline{E}_{\star}$. By irreducibility of the monodromy of
$(E,\nabla)$, we must have that $F^{i}\overline{E}_{\star}$ equals
$\overline{E}_{\star}.$ But in this case $({E},\nabla)$ is unitary, as the
monodromy preserves the polarization, a definite Hermitian form. ∎

#### 7.1.2.

We now recall the setup of Theorem 1.2.12. Let $(C,x_{1},\cdots,x_{n})$ be an
$n$-pointed hyperbolic curve of genus $g$. Let $({E},\nabla)$ be a flat vector
bundle on $C$ with $\operatorname{rk}{E}<2\sqrt{g+1}$ such that $(E,\nabla)$
has regular singularities at the $x_{i}$. Our goal is to show that if an
isomonodromic deformation of ${(E,\nabla)}$ to an analytically general nearby
$n$-pointed curve underlies a polarizable complex variation of Hodge
structure, then $({E},\nabla)$ has unitary monodromy.

###### Proof of Theorem 1.2.12.

As the hypothesis is about the restriction of $(E,\nabla)$ to
$C\setminus\\{x_{1},\cdots,x_{n}\\}$, we may without loss of generality assume
$(E,\nabla)$ is the Deligne canonical extension of
$(E,\nabla)|_{C\setminus\\{x_{1},\cdots,x_{n}\\}}$.

Endow $E$ with the parabolic structure associated to $\nabla$, and denote the
corresponding parabolic bundle by $E_{\star}$. After replacing
$(C,x_{1},\cdots,x_{n})$ with an analytically general nearby curve, and
$(E,\nabla)$ with its isomonodromic deformation to this curve, we may assume
by 6.1.2 that $E_{\star}$ is semistable. Thus $(E,\nabla)$ has unitary
monodromy by 7.1.1. ∎

###  7.2. The proof of Theorem 1.2.5

The proof of Theorem 1.2.5 follows from the integrality assumption and the
following lemma.

######  Lemma 7.2.1.

Suppose $\Gamma$ is a group, $K$ is a number field, and $\rho$ is a
representation

$\rho:\Gamma\to\operatorname{GL}_{m}(\mathscr{O}_{K}).$

If for each embedding $\iota:K\hookrightarrow\mathbb{C}$ the representation
$\rho\otimes_{\mathscr{O}_{K},\iota}\mathbb{C}$ is unitary, then $\rho$ has
finite image.

###### Proof.

Indeed, for $\iota:K\hookrightarrow\mathbb{C}$ an embedding, let

$\rho_{\iota}:\Gamma\to\operatorname{GL}_{m}(\mathbb{C})$

be the corresponding representation
$\rho\otimes_{\mathscr{O}_{K},\iota}\mathbb{C}$. First,

$\prod_{\iota}\rho_{\iota}:\Gamma\to\prod_{\iota}\operatorname{GL}_{m}(\mathbb{C})$

has compact image by the definition of being unitary. Moreover, the image of

$\mathscr{O}_{K}\hookrightarrow\prod_{\iota}\mathbb{C}$

is discrete, since the difference of any two distinct elements of
$\mathscr{O}_{K}$ has norm at least $1$. Hence the image of
$\prod_{\iota}\rho_{\iota}$ is discrete and compact, and therefore finite. ∎

Let $K$ be a number field with ring of integers $\mathscr{O}_{K}$. Let
$(C,x_{1},\cdots,x_{n})$ be an analytically general hyperbolic $n$-pointed
curve of genus $g$, and let $\mathbb{V}$ be a $\mathscr{O}_{K}$-local system
on $C\setminus\\{x_{1},\cdots,x_{n}\\}$ with infinite monodromy. Suppose that
for each embedding $\iota:\mathscr{O}_{K}\hookrightarrow\mathbb{C}$,
$\mathbb{V}\otimes_{\mathscr{O}_{K},\iota}\mathbb{C}$ underlies a polarizable
complex variation of Hodge structure. Our goal is to prove Theorem 1.2.5,
which states that

$\operatorname{rk}_{\mathscr{O}_{K}}(\mathbb{V})\geq 2\sqrt{g+1}.$

###### Proof of Theorem 1.2.5.

We use $\mathscr{T}_{g,n}$ to denote the universal cover of
$\mathscr{M}_{g,n}$. For a fixed representation
$\rho:\pi_{1}(C\setminus\\{x_{1},\cdots,x_{n}\\}))\to\operatorname{GL}_{m}(\mathscr{O}_{K})$
let $T_{\rho}$ denote the set of
$[(C^{\prime},x_{1}^{\prime},\cdots,x_{n}^{\prime})]\in\mathscr{T}_{g,n}$, for
which the associated $\mathscr{O}_{K}$-local system $\mathbb{V}$ has the
following property: for each embedding
$\iota:\mathscr{O}_{K}\hookrightarrow\mathbb{C}$,
$\mathbb{V}\otimes_{\mathscr{O}_{K},\iota}\mathbb{C}$ underlies a polarizable
complex variation of Hodge structure on
$C^{\prime}\setminus\\{x_{1}^{\prime},\ldots,x_{n}^{\prime}\\}$. Let
$M_{\rho}$ denote the image of $T_{\rho}$ under the covering map
$\mathscr{T}_{g,n}\to\mathscr{M}_{g,n}$.

Our goal is to show that an analytically very general point of
$\mathscr{M}_{g,n}$ lies in the complement of the union of the $M_{\rho},$
where $\rho$ ranges over the set of representations of
$\pi_{1}(C\setminus\\{x_{1},\cdots,x_{n}\\})\to\operatorname{GL}_{r}(\mathscr{O}_{K})$,
with infinite image, for $K$ a number field, and $r<2\sqrt{g+1}$. Since there
are only countably many such representations $\rho$, it is enough to show that
an analytically very general point lies in the complement of $M_{\rho}$ for
fixed $\rho$. Since $(C,x_{1},\ldots,x_{n})$ is hyperbolic,
$\mathscr{T}_{g,n}\to\mathscr{M}_{g,n}$ is a covering space of countable
degree, and so the image of a closed analytic set is locally contained in a
countable union of closed analytic subsets. It therefore suffices to show that
for any $\rho$ with infinite monodromy and rank $<2\sqrt{g+1}$, $T_{\rho}$ is
contained in a closed analytic subset of $\mathscr{T}_{g,n}$.

We now show such $T_{\rho}$ as above are contained in a closed analytic subset
of $\mathscr{T}_{g,n}$. Indeed, suppose $\mathbb{V}$ is the local system
associated to $\rho$ on some curve $C\setminus\\{x_{1},\cdots,x_{n}\\}$, and
that for each embedding $\iota:\mathscr{O}_{K}\hookrightarrow\mathbb{C}$,
$\mathbb{V}\otimes_{\mathscr{O}_{K},\iota}\mathbb{C}$, underlies a polarizable
complex variation of Hodge structure. It is enough to show this complex
polarizable variation of Hodge structure does not extend to an analytically
general nearby curve. Indeed, if it did, Theorem 1.2.12 implies
$\prod_{\iota:\mathscr{O}_{K}\to\mathbb{C}}\rho_{\iota}$ has unitary
monodromy, and 7.2.1 implies its monodromy is finite. This contradicts our
assumption that $\rho$ has infinite monodromy. ∎

### 7.3.

###### Proof of 1.2.7.

Let $g\geq 1$ be an integer and let $(C,x_{1},\cdots,x_{n})$ be an
analytically general hyperbolic $n$-pointed curve of genus $g$. Let $U\subset
C\setminus\\{x_{1},\cdots,x_{n}\\}$ be a dense Zariski-open subset. Let
$f:Y\to U$ be a smooth proper morphism, $i\geq 0$ an integer, and suppose
$\mathbb{V}$ is a complex local system on $(C,x_{1},\cdots,x_{n})$ with
infinite monodromy such that $\mathbb{V}|_{U}$ is a summand of
$R^{i}f_{*}\mathbb{C}$. Then we wish to show that
$\dim_{\mathbb{C}}\mathbb{V}\geq 2\sqrt{g+1}$.

It suffices to show that $\mathbb{V}$ satisfies the hypotheses of Theorem
1.2.5. The existence of an $\mathscr{O}_{K}$-structure follows from the fact
that $R^{i}f_{*}\mathbb{C}$ has a $\mathbb{Z}$-structure. Let $\mathbb{W}$ be
the corresponding $\mathscr{O}_{K}$-local system. All that remains is to
verify that for each embedding
$\iota:\mathscr{O}_{K}\hookrightarrow\mathbb{C}$, the corresponding complex
local system
$\mathbb{W}_{\iota}:=\mathbb{W}\otimes_{\mathscr{O}_{K},\iota}\mathbb{C}$
underlies a polarizable complex variation of Hodge structure. But each such
embedding yields a summand $\mathbb{W}_{\iota}|_{U}$ of
$R^{i}f_{*}\mathbb{C}$, Galois-conjugate to the original embedding
$\mathbb{W}|_{U}\subset\mathbb{V}|_{U}\subset R^{i}f_{*}\mathbb{C}.$ Any such
summand underlies a polarizable complex variation of Hodge structure, by
4.1.4(2). Now $\mathbb{W}_{\iota}|_{U}$ extends from $U$ to
$C\setminus\\{x_{1},\cdots,x_{n}\\}$ and so the same is true for the
corresponding complex polarizable variation of Hodge structure by [Sch73,
Corollary 4.11], which completes the proof. ∎

### 7.4.

###### Proof of 1.2.8.

The statement for relative curves $h:S\to C\setminus\\{x_{1},\ldots,x_{n}\\}$
reduces to the statement for abelian varieties upon taking the relative
Jacobian $\operatorname{Pic}^{0}_{h}$, as the Torelli theorem implies that $h$
is isotrivial if $\operatorname{Pic}^{0}_{h}$ is. Hence, it suffices to show
that any abelian scheme $f:A\to C\setminus\\{x_{1},\ldots,x_{n}\\}$ of
relative dimension $r<\sqrt{g+1}$ is isotrivial. If $r<\sqrt{g+1}$ then
$\operatorname{rk}R^{1}f_{*}\mathbb{C}=2r<2\sqrt{g+1}$, so
$R^{1}f_{*}\mathbb{C}$ has finite monodromy by 1.2.7. To show $f$ is
isotrivial, it is enough to show $f$ is trivial after a finite base change.
After passing to a finite étale cover of $C\setminus\\{x_{1},\ldots,x_{n}\\}$,
we may therefore assume $R^{1}f_{*}\mathbb{C}$ has trivial monodromy. In this
case, the theorem of the fixed part [Del71b, Corollaire 4.1.2] implies that
the first part of the Hodge filtration $F^{1}(R^{1}f_{*}\mathbb{C})\subset
R^{1}f_{*}\mathbb{C}$ is also a trivial local system. Since $A$ is a quotient
of $F^{1}(R^{1}f_{*}\mathbb{C})^{\vee}$ by the constant local system
$R^{2g-1}f_{*}\mathbb{Z}\simeq\wedge^{2g-1}R^{1}f_{*}\mathbb{Z}$, (on fibers
this local system is identified with the first homology of $f$ via Poincaré
duality,) $f$ is also trivial. ∎

### 7.5.

We next prove that local systems of geometric origin with low rank are not
Zariski dense in the character variety of an analytically very general genus
$g$ curve. Recall that we use $\mathscr{M}_{B,r}(X)$ for the character variety
parametrizing conjugacy classes of semisimple representations

$\rho:\pi_{1}(C\setminus\\{x_{1},\cdots,x_{n}\\})\to\operatorname{GL}_{r}(\mathbb{C}).$

###### Proof of 1.2.10.

Let $(C,x_{1},\cdots,x_{n})$ be an analytically very general hyperbolic
$n$-pointed curve of genus $g$, and fix an integer $r$ with $1<r<2\sqrt{g+1}$.
Note that we must have $g\neq 0$ because there are no integers $r$ with
$1<r<2\sqrt{0+1}$. By 1.2.7, the points of
$\mathscr{M}_{B,r}(C\setminus\\{x_{1},\cdots,x_{n}\\})$ of geometric origin
correspond to representations $\rho$ with finite image whenever
$r<2\sqrt{g+1}$. We wish to show these finite image representations are not
Zariski dense in $\mathscr{M}_{B,r}(C\setminus\\{x_{1},\cdots,x_{n}\\})$. This
follows from by 7.5.1. upon choosing a topological identification
$C\setminus\\{x_{1},\cdots,x_{n}\\}\simeq\Sigma_{g,n}$. ∎

######  Lemma 7.5.1.

Let $\Sigma_{g,n}$ be a topological $n$-punctured genus $g$ surface with
basepoint $p\in\Sigma_{g,n}$. For $r>1$, the set of representations
$\rho:\pi_{1}(\Sigma_{g,n},p)\to\operatorname{GL}_{r}(\mathbb{C})$ with finite
image are not Zariski dense in the character variety
$\mathscr{M}_{B,r}(\Sigma_{g,n},p)$.

###### Proof.

It suffices to prove non-density of representations with finite image in the
_framed_ character variety

$\mathscr{M}_{B,r}^{\square}(\Sigma_{g,n}):=\operatorname{Hom}(\pi_{1}(\Sigma_{g,n},p),\operatorname{GL}_{r}(\mathbb{C})).$

For $s\in\mathbb{Z}_{>0}$, let
$V_{s}\subset\mathscr{M}_{B,r}^{\square}(\pi_{1}(\Sigma_{g,n},p))$ be the
closed subvariety consisting of those representations $\rho$ such that

$[\rho(x)^{s},\rho(y)^{s}]=1$

for all $x,y\in\pi_{1}(\Sigma_{g,n},p)$.

By Jordan’s theorem ([Jor78, p. 91] or [CR66, Theorem 36.13]) on finite
subgroups of $\operatorname{GL}_{r}(\mathbb{C})$, there is some constant
$m_{r}$ such that for each finite subgroup
$G\subset\operatorname{GL}_{r}(\mathbb{C})$, there exists an abelian normal
subgroup $H\subset G$ of index at most $m_{r}$. Hence $V_{(m_{r}!)}$ contains
all representations with finite image. Thus it suffices to show $V_{s}$ is not
all of $\mathscr{M}_{B,r}^{\square}(\Sigma_{g,n})$ for any $s>0$.

We now write

$\pi_{1}(\Sigma_{g,n},p)=\left\langle
a_{1},\cdots,a_{g},b_{1},\cdots,b_{g},\gamma_{1},\cdots,\gamma_{n}\mid\prod_{i=1}^{g}[a_{i},b_{i}]\cdot\prod_{j=1}^{n}\gamma_{j}\right\rangle$

for the standard presentation of the fundamental group. If $g\geq 2$, let

$\rho:\pi_{1}(\Sigma_{g,n},p)\to\operatorname{GL}_{2}(\mathbb{C})$

be the representation defined by

$a_{1}\mapsto\begin{pmatrix}1&1\\\ 0&1\end{pmatrix}$
$a_{2}\mapsto\begin{pmatrix}1&0\\\ 1&1\end{pmatrix}$

and sending all other generators to the identity. If $g=1$, hyperbolicity of
$\Sigma_{g,n}$ implies $n>0$. Then $\pi_{1}(\Sigma_{g,n},x)$ is free on $n+1$
generators, and we may set $\rho$ to be a representation sending two of the
generators to the matrices above, and all other generators to the identity.
Then $\rho$ does not lie in $V_{s}$ for any $s$ because $\rho(a_{1}^{s})$ does
not commute with $\rho(a_{2}^{s})$ for any integer $s>0$. This completes the
case $r=2$. For the case that $r>2$, the representation
$\rho\oplus\operatorname{triv}^{\oplus r-2}$ lies outside of $V_{s}$ for all
$s$, for the same reason. ∎

###  7.6. An application to Hilbert modular varieties

Let $K$ be a totally real number field of degree $h$ over $\mathbb{Q}$,
$\mathscr{O}_{K}$ its ring of integers, and use $\mathscr{X}_{K}$ to denote
the Hilbert modular stack parameterizing $h$-dimensional principally polarized
abelian varieties $A$ with an injection
$\mathscr{O}_{K}\hookrightarrow\operatorname{End}(A)$. Using 1.2.8, we find
there are no nonconstant maps from an analytically very general $n$-pointed
curve of genus $g$ to the moduli stack of principally polarized abelian
varieties of dimension $h$ whenever $h\leq\sqrt{g+1}$, or equivalently when
$g>h^{2}$. We now show that the analogous statement for Hilbert modular stacks
holds whenever $g\geq 1$. In particular, this improved bound is independent of
$h$. We thank Alexander Petrov for pointing out the following application.

######  Proposition 7.6.1.

Let $\mathscr{X}_{K}$ be the Hilbert modular stack associated to a totally
real field $K$. Let $(C,x_{1},\ldots,x_{n})$ be an analytically very general
hyperbolic $n$-pointed curve of genus $g\geq 1$. Any map
$\phi:C\setminus\\{x_{1},\ldots,x_{n}\\}\to\mathscr{X}_{K}$ is constant.

###### Proof.

We first use the map $\phi$ to produce a rank $2$ integral local system on
$C\setminus\\{x_{1},\ldots,x_{n}\\}$. A map
$\phi:C\setminus\\{x_{1},\ldots,x_{n}\\}\to\mathscr{X}_{K}$ yields an abelian
scheme $f:A\to C\setminus\\{x_{1},\ldots,x_{n}\\}$ with an
$\mathscr{O}_{K}$-action. In particular, $R^{1}f_{*}\mathbb{Z}$ is a complex
PVHS of rank $2h$ with an $\mathscr{O}_{K}$-action. If $K$ has degree $h$,
then this rank $2h$ local system may not be a $\mathscr{O}_{K}$-local system,
as its fibers may only be locally free, as opposed to free. However, after
passing to a finite extension $K^{\prime}$ of $K$, such as the Hilbert class
field of $K$, any such rank $2$ locally free module becomes a free module, and
so $R^{1}f_{*}\mathbb{Z}\otimes_{\mathscr{O}_{K}}\mathscr{O}_{K^{\prime}}$ is
an $\mathscr{O}_{K^{\prime}}$-local system of rank $2$.

We now show the map $\phi$ must be constant. By construction, for any
embedding $\mathscr{O}_{K^{\prime}}\to\mathbb{C}$, the local system
$R^{1}f_{*}\mathbb{Z}\otimes_{\mathscr{O}_{K}}\mathscr{O}_{K^{\prime}}\otimes_{\mathscr{O}_{K^{\prime}}}\mathbb{C}$
underlies a $\mathbb{C}$-PVHS. Hence by Theorem 1.2.5,
$R^{1}f_{*}\mathbb{Z}\otimes_{\mathscr{O}_{K}}\mathscr{O}_{K^{\prime}}$ has
finite monodromy, as

$\displaystyle
2=\operatorname{rk}_{\mathscr{O}_{K}^{\prime}}R^{1}f_{*}\mathbb{Z}\otimes_{\mathscr{O}_{K}}\mathscr{O}_{K^{\prime}}<2\sqrt{g+1}.$

Hence, the same holds for $R^{1}f_{*}\mathbb{Z}$. This implies the map $\phi$
is constant using the theorem of the fixed part [Del71b, Corollaire 4.1.2]
(see the proof of 1.2.8 for a similar argument). ∎

##  8\. Questions

We conclude with some open questions related to our results.

###  8.1. Bounds

######  Question 8.1.1.

Is the bound of $2\sqrt{g+1}$ appearing in 1.3.6, Theorem 1.2.5 and Theorem
1.2.12 sharp? If not, can one explicitly construct low rank geometric
variations of Hodge structure with infinite monodromy on a general curve or
general $n$-pointed curve? Do there exist counterexamples to the above results
if one replaces $2\sqrt{g+1}$ by a linear function of $g$?

We have no reason to believe the bound is sharp. The Kodaira-Parshin trick (as
used in § 5, for example) is one source of variations of Hodge structure on
$\mathscr{M}_{g,n}$ of rank bounded in terms of $g,n$, but it is not the only
one. For example, the representations constructed in [KS16] are
cohomologically rigid and hence underlie integral variations of Hodge
structure by [EG18, Theorem 1.1] and [Sim92, Theorem 3]. Assuming Simpson’s
motivicity conjecture ([Sim92, Conjecture, p. 9]) these constructions are
geometric in nature, though this is not clear from the construction. Of course
it would be extremely interesting to prove that these representations arise
from algebraic geometry.

It the representations constructed in [KS16] have rank growing exponentially
in $g$ [KS16, Corollary 4.5]. It is natural, given our results, to ask if one
can use their methods to produce representations of smaller rank.

We also raise a related question about bounds on maps to the moduli space of
curves.

######  Question 8.1.2.

Fix an integer $g\geq 2$. What is the smallest integer $h\geq 2$ for which the
generic genus $g$ curve, i.e., the generic fiber of
$\mathscr{M}_{g,1}\to\mathscr{M}_{g}$, has a non-constant map to
$\mathscr{M}_{h}$?

######  Remark 8.1.3.

Since a map $C\to\mathscr{M}_{h}$ corresponds to a family of smooth curves of
genus $h$ over $C$, by considering the associated family of Jacobians, it
follows from 1.2.8 that $h\geq\sqrt{g+1}$. The Kodaira-Parshin trick 5.1.1
does not a priori apply to construct maps from the generic curve to
$\mathscr{M}_{h}$, because as written it produces disconnected covers. But one
can apply a variant where one takes a cover defined by a characteristic
quotient of the fundamental group to show there is some (fairly large) value
of $h$ for which the generic genus $g$ curve has a non-constant map to
$\mathscr{M}_{h}$. See [McM00, Theorem 1.4] for more details.

###  8.2. Non-abelian Hodge loci

Let $(C,x_{1},\cdots,x_{n})$ be an $n$-pointed curve, $\mathbb{V}$ a
$\mathbb{Z}$-local system on $C\setminus\\{x_{1},\cdots,x_{n}\\}$ with quasi-
unipotent local monodromy around the $x_{i}$, and let $(\mathscr{E},\nabla)$
be the associated flat vector bundle. We refer to the locus $H_{\mathbb{V}}$
in $\mathscr{T}_{g,n}$ where the corresponding isomonodromic deformation of
$(\mathscr{E},\nabla)$ underlies a polarizable variation of Hodge structure as
an _non-abelian Hodge locus_. By analogy to the famous result on algebraicity
of Hodge loci of Cattani-Deligne-Kaplan [CDK95], it is natural to ask:

######  Question 8.2.1 (Compare to [Sim97, Conjecture 12.3]).

Let $Z$ be an irreducible component of $H_{\mathbb{V}}$. Is the image of $Z$
in $\mathscr{M}_{g,n}$ algebraic?

This would follow if all $\mathbb{Z}$-local systems which underlie polarizable
variations of Hodge structure arise from geometry, which is perhaps a folk
conjecture (and is conjectured explicitly in [Sim97, Conjecture 12.4]). Just
as [CDK95] provides evidence for the Hodge conjecture, a positive answer to
8.2.1 would provide evidence for this conjecture.

When we refer to an analytically very general curve we mean in the sense of
1.2.3. A positive answer to 8.2.1 would allow us to replace this with the
usual algebraic notion of a very general curve in Theorem 1.2.5. It seems
plausible that one can make this replacement in 1.2.7 without requiring input
from 8.2.1, using the main result of [CDK95].

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