# Magnetization profiles at the upper critical dimension
as solutions of the integer Yamabe problem

Alessandro Galvani  SISSA and INFN, Sezione di Trieste, Via Bonomea 265,
I-34136 Trieste, Italy  Giacomo Gori  Institut für Theoretische Physik,
Universität Heidelberg, D-69120 Heidelberg, Germany  CNR-IOM DEMOCRITOS
Simulation Center and SISSA, Via Bonomea 265, I-34136Trieste, Italy  Andrea
Trombettoni  Department of Physics, University of Trieste, Strada Costiera 11,
I-34151 Trieste, Italy  SISSA and INFN, Sezione di Trieste, Via Bonomea 265,
I-34136 Trieste, Italy  CNR-IOM DEMOCRITOS Simulation Center and SISSA, Via
Bonomea 265, I-34136Trieste, Italy

###### Abstract

We study the connection between the magnetization profiles of $O(N)$ models in
a bounded domain at the upper critical dimension and the solutions of the so-
called Yamabe problem in the same domain, which amounts to finding a metric
having constant curvature. Taking the slab as a reference domain, we first
study the magnetization profiles at the upper critical dimensions $d=3$, $4$,
$6$ for different (scale invariant) boundary conditions. By studying the the
saddle-point equations for the magnetization, we find general formulas in
terms of Weierstrass elliptic functions, extending exact results known in
literature and finding new ones for the case of percolation. The zeros and
poles of the Weierstrass elliptic solutions can be put in direct connection
with the boundary conditions. We then show that, for any dimension $d$, the
magnetization profiles are solution of the corresponding integer Yamabe
equation at the same $d$ and with the same boundary conditions. The
magnetization profiles in the specific case of the $4$-dimensional Ising model
with fixed boundary conditions are compared with Monte Carlo simulations,
finding good agreement. These results explicitly confirm at the upper critical
dimension recent results presented in Gori and Trombettoni (2020).

##  I Introduction

Within the field of critical phenomena, a research line that attracted
considerable attention along the years is the study of critical systems in
presence of boundaries Diehl (1997); Binder (2000). Among the motivations for
such a study, the first is to correctly compare theoretical predictions with
experimental results or numerical simulations coming from finite systems. In
general, the introduction of a boundary alters thermodynamic properties and
sub-leading corrections to bulk values have been extensively studied Brankov
et al. (2000). Techniques such as finite size scaling can be used to extract
critical exponents and other important quantities Cardy (1996). In two-
dimensional systems, the coefficient of the next-to-next-to-leading order in
the free energy is related to the central charge of the theory Cardy and
Peschel (1988) and the study of boundary conformal field theory developed in
the last decades as a very active field of research Cardy (2004).

A remarkable property emerging in critical bounded domains is that a given
bulk universality class corresponds to several surface universality classes,
making boundary critical phenomena a rich and well studied playground. In many
cases, boundaries are the source of very interesting effects and phenomena. A
major example is the thermodynamic Casimir effect: it has been studied
thoroughly to obtain universal scaling function and Casimir amplitudes
Vasilyev et al. (2009). The interest in the Casimir effect has shifted the
attention from semi-infinite systems Bray and Moore (1977); Diehl and Dietrich
(1980, 1981); Lubensky and Rubin (1975) to the geometry of a slab Grüneberg
and Diehl (2008); Diehl and Rutkevich (2017); Gambassi and Dietrich (2006a,
b); Vasilyev et al. (2007). Obtaining the critical magnetization profile Krech
(1997); Vassilev et al. (2018) is a crucial piece of information in this line
of research.

The study of critical phenomena with a field theoretical approach Zinn-Justin
(2002) naturally highlights the role of the dimensionality which rules the
relevance of fluctuations at the critical point. For $d$ matching the so-
called upper critical dimension $d_{c}$ Cardy (1996); Mussardo (2020) the
critical theory becomes typically tractable, yielding in a controlled way the
mean-field approximation and the typical logarithmic corrections on top of it
Bellac (1991); Amit and Martin-Mayor (2005). If critical exponents at the
upper critical dimension are found to be just the mean-field ones, in bounded
systems there is the possibility to study inhomogeneous, space-dependent
quantities and corrections due to the presence of boundaries.

Here we are going to focus on magnetization profiles of $O(N)$ models at their
upper critical dimension. They can be a useful starting point for calculation
of critical magnetization profiles at lower dimensions, as they give the
background around which (non-logarithmic) fluctuations can be added. Moreover,
an advantage of being at the upper critical dimension is that one can obtain,
by a saddle-point treatment, analytical expressions to be compared with
numerical results.

Our goal in this paper is three-fold. First, we aim at obtaining magnetization
profiles in a unified framework for different boundary conditions, by
considering the analytic structure of the saddle-point solutions. This will
allow us to recast known results in a more accessible and compact way while
deriving novel predictions, e.g. for percolation at its upper critical
dimension. For the Ising model in $4$ dimension we test the findings obtained
by the discussed approach by comparing them with results of Monte Carlo
simulations.

Another main objective is to provide a geometric interpretation of the results
for the critical magnetization profiles at the upper critical dimension. The
main idea is to write the field theory equations for the magnetization
profiles as a variational solution of a geometric problem. In this way, the
problem of finding the critical profiles is explicitly put in connection with
the solution of a celebrated problem in differential geometry: the Yamabe
problem Yamabe (1960), which amounts to finding a metric, in the same
conformal class of another given metric, that makes the scalar curvature
constant. The Yamabe problem in its various generalizations has been the
object of intense mathematical work in the last decades Lee et al. (1987);
Chang and del Mar González (2011). It is also related to general relativity,
when one looks for solutions of Einstein field equations which are conformally
flat or more generally within a conformal class Akutagawa (1994). A discussion
of the connection between the (mean-field) Ginzburg-Landau equations and the
Hilbert-Einstein action functional for pure gravity in presence of
cosmological term is presented in Kholodenko and Ballard (2007).

As a third motivation, the present study intends to explicitly verify the
validity at the upper critical dimension of results presented in Gori and
Trombettoni (2020). There, a hypothesis – referred to as “uniformization”– is
put forward to relate the critical magnetization profile of a bounded domain
with a metric factor $\gamma(\mathbf{x})$, which is the solution of a
fractional Yamabe equation in the same domain. This metric is then used to
compute distances between points, which are then used to construct spin-spin
correlation functions. In this “critical geometry” approach, the less
correlated points close to the boundary are being put at larger distances by
inflating locally and isotropically the original metric ending up in a Yamabe
problem. In Gori and Trombettoni (2020) the uniformization hypothesis was
supported by numerical simulations of the $3d$ Ising model, given the absence
of analytical results for this model. It is important to notice that below the
upper critical dimension a non-vanishing anomalous dimensions $\eta$ arises:
in order to take this into account, the Yamabe equation has to be modified, by
introducing a fractional power of the Laplacian operator, obtaining the
fractional Yamabe equation del Mar González and Qing (2013). But, at the upper
critical dimension $\eta=0$ and the fractional Yamabe equation reduces to the
standard Yamabe equation, which for clarity we call integer Yamabe equation,
since in it only the usual Laplacian enters. Therefore the study of critical
magnetization profile at $d=d_{c}$ allows us to test, analytically and
numerically, the validity of the “critical geometry” in a solvable case.

We observe that the relation between the critical magnetization profiles and
the integer Yamabe equation holds in any bounded domain. We specialize to the
slab geometry since it is convenient for both analytical calculations and
simulations and for its relevance for the Casimir effect. We will comment on
the general case when useful for our presentation.

The plan of the paper is the following: first we will consider Landau-Ginzburg
actions with potential $\phi^{2d/(d-2)}$, for various $d$: in each case, we
will obtain the saddle-point equation at the corresponding upper critical
dimension; solutions will be given for the possible boundary conditions. These
equations will then be generalized through the introduction of the Yamabe
equation. After solving it for a slab in arbitrary dimension, we recover the
previous solutions. The theoretical magnetization profiles are then compared
to the numerical solution of the Ising mean field equation, and to the results
of a Monte Carlo simulation of the 4 dimensional Ising model.

##  II Magnetization profiles

In the following, we study exact solution for the critical magnetization
profiles for models living at their upper critical dimension. The action at
the upper critical dimension, when expressed in terms of the order parameter,
in general dimension $d$ and with $O(N)$ symmetry has the form:

$S=\int
d^{d}x\left[\frac{1}{2}\phi(-\bigtriangleup)\phi+\frac{1}{2}\mu^{2}\phi^{2}+g\,c_{d}\,\phi^{\frac{2d}{d-2}}\right],$
(1)

where the field $\phi=\phi(\mathbf{x})$ with $\mathbf{x}\in\Omega$ has $N$
components and $\Omega\subset\mathbb{R}^{d}$ is a bounded domain. The operator
$-\bigtriangleup$ is the positive definite Laplacian. The factor

$c_{d}\equiv\frac{(d-2)^{2}}{8}$

in the interaction constant has been chosen for convenience, as it will appear
later. Since in dimension $d$ the field has scaling dimension
$\Delta_{\phi}=\frac{d-2}{2}$, the action (1) is written so that it is at the
upper critical dimension. In (1) the mass term $\propto\phi^{2}$ vanishing at
the critical point is also included Cardy (1996); Mussardo (2020).

The saddle-point equation for the action (1) at the critical point reads

$\bigtriangleup
m(\mathbf{x})=g\frac{d(d-2)}{4}m(\mathbf{x})^{\frac{d+2}{d-2}}$ (2)

with $m(\mathbf{x})=\left\langle\phi(\mathbf{x})\right\rangle$.

We will now specialize the study of the saddle-point approximation for the
action (1) for the physical dimensions taking the values $d=3$, $4$, and $6$.
This is firstly due to their physical relevance, but also for the striking
mathematical properties they display only for these values of $d$. The
peculiarity amounts to the fact that they are the only dimensions where the
magnetization profiles can be expressed as suitable powers of the elliptic
Weierstrass function $\wp$ (see Appendix A and DLMF  for reference).

###  II.1 $\phi^{4}$ theory in four dimensions

In four dimension the action (1) is the well-known $\phi^{4}$ theory:

$S=\int
d^{4}x\left[\frac{1}{2}\phi(-\bigtriangleup)\phi+\frac{1}{2}\mu^{2}\phi^{2}+\frac{1}{2}g\,\phi^{4}\right].$
(3)

Since the theory is at the upper critical dimension, we proceed by writing the
saddle-point equation. The saddle-point equation for the action at the
critical point $\mu=0$ is

$\bigtriangleup m(\mathbf{x})=2\,g\,m(\mathbf{x})^{3}.$ (4)

We are interested in solving equation (4) in the case of a slab domain
$[0,L]\times\mathbb{R}^{3}$. The magnetization $m(\mathbf{x})$ only depends on
the transverse direction $x$, so the Laplacian $\bigtriangleup$ reduces to
$\partial_{x}^{2}$. The different solutions for various boundary conditions
will be analyzed and compared in the rest of the this section. In Section VI
we will compare the analytical results so obtained with Monte Carlo
simulations for the Ising model on a $4d$ slab geometry with fixed boundary
conditions.

Notice that given a solution $m(x)$ of (4) we can generate other solutions by
translation and rescaling ruled by the parameters $x_{0}$, $\beta$, and
$\lambda$, all of them being possibly complex:

$m(x)\rightarrow\beta\,m(\lambda\,(x-x_{0}))\qquad g\rightarrow
g\,\beta^{\frac{d+2}{d-2}}\,\lambda^{2}.$ (5)

This allows us to change the domain/coupling constant as desired; for
convenience we set $g=1$. In general such scaling will alter the boundary
conditions, however scaling invariant conditions, i.e. $m=0$, $\infty$, are
left unchanged. We remark that the scaling (5) applies also to other values of
$d$ and it will thus be useful for the other values of $d$ considered.

The general solution to equation (4) is

$m(x)=\wp_{l}(x)^{1/2},$ (6)

where $\wp_{l}$ is the Weierstrass $\wp$ function with invariants
$(g_{2},g_{3})=(1,0)$. This is the so-called lemniscatic case. When it is
written in terms of the complex variable $z$ as $\wp_{l}(z)$, it corresponds
in the complex plane to a square lattice, with a real half-period
$\omega_{l}=\Gamma(\frac{1}{4})^{2}/(4\sqrt{\pi})$.

The Weierstrass elliptic $\wp$, whose basic properties are refreshed in
Appendix A, is a doubly periodic function. Within one of its domains in the
complex plane, it has a double pole and two zeros possibly coinciding (as
here): taking segments with poles/zeros as their endpoints gives solutions
with various boundary conditions. Along the segments the function has to be
real or have a constant phase (which can removed) in order to be interpreted
as magnetization profiles.

To see the corresponding boundary conditions in the lattice model, let us
focus on the $N=1$ case, the Ising model. We can force the spins on the
boundaries to be aligned ($+$ or $-$ fixed boundary condition, FBC), or have
the spins on one boundary to be free (open boundary condition, OBC). By
combining these conditions on the two slab surfaces we can generate different
critical profiles. In the continuum limit, the possible conformally invariant
boundary conditions are $\left\langle\phi\right\rangle=0$ or
$\left\langle\phi\right\rangle=\pm\infty$.

Of course, on a lattice the magnetization neither vanishes nor diverges. If,
for instance, we impose FBC for an Ising model, the magnetization at the
boundaries will be $+1$. The value at the center of the slab, however, will
scale with the system size as $L^{-\Delta_{\phi}}$, so one can rescale the
magnetization in a way that it is constant at the center: in the thermodynamic
limit, it then diverges at the slab edges. It is this rescaled form which is
accessible via field theory.
Another way to understand it is this: if the lattice magnetization profile
were prolonged a few sites beyond the boundary, these singularities would
appear. This concept is quantified by the extrapolation length $a_{L}$ Diehl
(1997): if the boundaries where we fixed the spins are at $x=0$ and $x=L$,
after we fit the resulting profile $m(x)$, we would see divergences at
$m(-a_{L})$ and at $m(L+a_{L})$. In the $L\rightarrow\infty$ limit,
$a_{L}\rightarrow 0$ and the magnetization diverges at the boundaries.

From the scaling dimension $\Delta_{\phi}$ of the order parameter, we can also
obtain the behavior near the boundary. The possible cases are:

BC | Lattice | $m(x\rightarrow 0)$ | $m(x\rightarrow L)$
---|---|---|---
$++$ | $\uparrow\dots\uparrow$ | $x^{-\Delta_{\phi}}$ | $(L-x)^{-\Delta_{\phi}}$
$+-$ | $\uparrow\dots\downarrow$ | $x^{-\Delta_{\phi}}$ | $-(L-x)^{-\Delta_{\phi}}$
$00$ | $0\dots 0$ | $x$ | $L-x$
Table 1: Labels for the possible boundary conditions, with the corresponding
spin configuration and behavior of the continuous profile near the boundaries.
“0” refers to spins being left free.

The corresponding (unscaled) solutions are

$\displaystyle m_{++}(x)=\wp_{l}(x)^{1/2}$ $\displaystyle
x\in(0,2\,\omega_{l})$ $\displaystyle
m_{+0}(x)=\wp_{l}(x\,e^{i\pi/4})^{1/2}e^{i\pi/4},$ $\displaystyle
x\in(0,\sqrt{2}\,\omega_{l}]$ (7) $\displaystyle
m_{00}(x)=\wp_{l}(\omega_{l}+i\,x)^{1/2},$ $\displaystyle
x\in[-\omega_{l},\omega_{l}].$

Figure 1: $d=4$ \- Above: modulus of the square root of the lemniscatic
elliptic function, poles are white and zeros blue. Below: magnetization
profiles for the different boundary conditions discussed in the text. The
minimum of the OBC profile is a saddle point, so it is also the minimum of the
$++$ profile: its value is $\wp(\omega_{l})^{1/2}=1/\sqrt{2}$.

When considering OBC, we have ignored the trivial solution $m(x)=0$ throughout
the slab. Practically, $m_{00}(x)$ could be obtained by adding a small
magnetic field in the center of the slab, to break the symmetry. By extending
the $m_{+0}(x)$ solution to the interval $x\in(0,2\sqrt{2}\omega_{L})$ we
indeed generate a solution $m_{+-}$ connecting the + and - boundary states.
This reflects indeed the $\mathbb{Z}_{2}$ symmetry of the model as it is
apparent in the saddle-point equation (4) with $m(x)\rightarrow-m(x)$.

$O(N)$ models with $N>1$ have an additional degree of freedom in the choice of
boundary conditions: the angle between the spins on the two boundaries. The
magnetization becomes then a vector in the plane spanned by the boundary
spins, and the saddle-point becomes a system of two equations. So, while the
$+-$ solution is specific to the Ising model, the $++$ one is valid for any
$O(N)$ model with parallel boundary spins.

It is worth pointing out that the $00$ solution is nothing but the reciprocal
of the $++$ solution, up to a constant: shifting the argument of the function
by half a period has the same effect as taking its inverse, since both
operations swap poles and zeros. This is because the fundamental domain of
$\wp_{L}$ is a square, as square lattice tessellation of the plane is self-
dual. This is represented in Fig. 1, together with the plot of the solutions
for the different boundary conditions.

This peculiarity of the elliptic functions must reflect in a property of the
saddle-point equation, valid only for $d=4$ and when the Laplacian reduces to
an ordinary second derivative: if $\phi(x)^{-1}$ is a solution, then so is
${\cal K}\,\phi(x)$, for an appropriate ${\cal K}$. It is

$\frac{d^{2}}{dx^{2}}\phi(x)^{-1}=2\phi(x)^{-3},$ (8)

from which

$\phi\phi^{\prime\prime}-2(\phi^{\prime})^{2}+2=0$ (9)

(where $\phi^{\prime\prime}=\frac{d^{2}}{dx^{2}}\phi$). Taking another
derivative gives

$\frac{\phi^{\prime\prime\prime}}{\phi^{\prime\prime}}=3\frac{\phi^{\prime}}{\phi},$
(10)

and, after integrating and exponentiating, we are left with

$\phi^{\prime\prime}(x)={\cal K}\,\phi(x)^{3},$ (11)

which is the saddle-point equation for $\sqrt{\frac{2}{\cal K}}\,\phi(x)$.

###  II.2 $\phi^{6}$ theory in three dimensions

A $\phi^{6}$ theory has upper critical dimension $d_{c}=3$. The steps of the
previous section can be repeated in this case. We start from the Landau-
Ginzburg action at the critical point, where the couplings of $\phi^{2}$ and
$\phi^{4}$ terms vanish.

$S=\int
d^{3}x\left[-\frac{1}{2}\phi(-\bigtriangleup)\phi+\frac{g}{8}\,\phi^{6}\right].$
(12)

On the slab geometry $[0,L]\times\mathbb{R}^{2}$, we get the saddle-point
equation

$m^{\prime\prime}(x)=\frac{3}{4}\,g\,m^{5},$ (13)

where again due to the scaling property (5) we set $g=1$. The solution is

$m(x)=\wp_{e}(i\,x)^{-1/2}.$ (14)

In the present case, the invariants of the Weierstrass function $\wp_{e}$ are
$(g_{2},g_{3})=(0,1)$, from which its real half-period is
$\omega_{e}=\Gamma(\frac{1}{3})^{3}/(4\pi)$.

The above case $\wp_{e}$ of the Weierstrass function is the so-called
equiharmonic case, where the lattice used to define the elliptic function is
the regular triangular tiling of the plane, see Fig. 2.

Remarkably, the solution for $d=6$, presented in the next subsection, will
turn out to be dual to this one. Again, looking at the poles and zeros of
$m(z)$, we obtain solutions for the possible boundary conditions:

$\displaystyle m_{++}(x)=\wp_{e}(\omega_{e}+i\,x)^{-1/2}$ $\displaystyle
x\in(-\frac{\omega_{e}}{\sqrt{3}},\frac{\omega_{e}}{\sqrt{3}})$ $\displaystyle
m_{+0}(x)=\wp_{e}(x\,e^{i\pi/6})^{-1/2}e^{-i\pi/6}$ $\displaystyle
x\in(-\frac{2\omega_{e}}{\sqrt{3}},0]$ (15) $\displaystyle
m_{00}(x)=\wp_{e}(x)^{-1/2}$ $\displaystyle x\in[0,2\omega_{e}].$

These solutions could also be expressed through Jacobi elliptic functions, as
was done for the $++$ solution in (Borjan and Upton, 1998), through Fisher-De
Gennes theory (Fisher and Upton, 1990). Again due to $\mathbb{Z}_{2}$
invariance, reflected in the $m(x)\rightarrow-m(x)$ symmetry in the saddle-
point equation, the $m_{+0}(x)$ can be extended further in the interval
$x\in(-\frac{2}{\sqrt{3}}\omega_{e},\frac{2}{\sqrt{3}}\omega_{e})$ yielding a
$m_{+-}(x)$ profile.

Figure 2: As in Fig. 1, but here for $d=6$, the modulus of $m(z)$ in the
complex plane, with the solutions for the different boundary conditions.

###  II.3 $\phi^{3}$ theory in six dimensions

Finally we turn to the $\phi^{3}$ theory in $d=6$, whose action,

$S=\int
d^{6}x\left[\frac{1}{2}\phi(-\bigtriangleup)\phi+2\,g\,\phi^{3}\right],$ (16)

is used to describe percolation at the upper critical dimension Amit et al.
(1977); de Alcantara Bonfirm et al. (1981). This case clearly differs from the
previously considered $d=3$, $4$ cases cases since it lacks $\mathbb{Z}_{2}$
symmetry. This implies the absence of opposing ($+-$) boundary conditions.
This becomes clear when one thinks of the possible boundary conditions for
percolation: one can either force a boundary to belong to the percolating
cluster ($+$), or leave it free, but opposing boundaries no longer make sense.
However, we can still find a solution connecting the $+$ and $0$ boundary
states. The saddle-point equation now is

$m^{\prime\prime}(x)=6\,g\,m(x)^{2}.$ (17)

The general solution is actually (taking $g=1$ as in the previous cases) the
function $\wp_{e}$:

$m(x)=\wp_{e}(x),$ (18)

meaning that the profiles are just the square of the reciprocal of the
$\phi^{6}$ results, as evidenced in Fig. 3. Now the double pole in the origin
directly yields the expected divergence for the $m_{++}$ order parameter
profile, since the dimension of the field is $\frac{d-2}{2}=2$. The three
profiles are now $m_{++}(x)$, $m_{+0}(x)$ and $m_{00}(x)$ and are the
following:

$\displaystyle m_{++}(x)=\wp_{e}(x)$ $\displaystyle x\in(0,2\omega_{e}),$
$\displaystyle m_{+0}(x)=\wp_{e}(x\,e^{i\pi/6})e^{i\pi/3}$ $\displaystyle
x\in(0,\frac{2\omega_{e}}{\sqrt{3}}],$ (19) $\displaystyle
m_{00}(x)=\wp_{e}(\omega_{e}+i\,x)$ $\displaystyle
x\in[-\frac{\omega_{e}}{\sqrt{3}},\frac{\omega_{e}}{\sqrt{3}}].$

Figure 3: The zero and pole structure for $d=6$ is inverted with respected to
Fig. 2: this means that the segment representing $++$ BCs now represents $00$
BCs. What would be the $+-$ profile now ends at $0$, giving the mixed boundary
condition $+0$.

##  III The Yamabe equation

In the following we provide the solution to the saddle-point equation for
$\phi^{2m}$ theories, valid for critical dimensions $d_{c}=\frac{2m}{m-1}$ (in
this general case, they will not possess the nice analytic properties
displayed by the $d=3$, $4$, $6$ cases discussed above).

In order to introduce a geometric interpretation of the critical magnetization
profile, we notice that it is possible to re-write the saddle-point equation
in a convenient form. Since the scaling dimension of the order parameter field
is $\Delta_{\phi}=\frac{d-2}{2}$, we introduce a function $\gamma(\mathbf{x})$
which acts as a point-dependent scale. We then write

$m(\mathbf{x})=\alpha\gamma(\mathbf{x})^{-\Delta_{\phi}}=\alpha\gamma(\mathbf{x})^{-\frac{d-2}{2}},$
(20)

where $\alpha$ is a constant, to be fixed.

The main point of this Section, as well as the main result of the paper, is
that the solution $m(\mathbf{x})$ of the saddle-point equation (2), once
rescaled as in (20), can be written as

$(-\bigtriangleup)\gamma(\mathbf{x})^{-\frac{d-2}{2}}=-\frac{d(d-2)}{4}\gamma(\mathbf{x})^{-\frac{d+2}{2}},$
(21)

where $\alpha$ in (20) is chosen to be $\alpha=g^{(2-d)/4}$ so that the r.h.s.
has the correct coefficient to obtain what is known as the Yamabe equation.
Eq. (21) can be also recast in the form

$1-|\vv{\nabla}\gamma(\mathbf{x})|^{2}+\frac{2}{d}\gamma(\mathbf{x})\bigtriangleup\\!\gamma(\mathbf{x})=0.$
(22)

The Yamabe equation has the geometric interpretation of a constraint on the
curvature of a manifold. This equation can be viewed as a special case of the
more general fractional Yamabe equation, which contains a fractional
Laplacian, allowing it to describe theories with nonvanishing anomalous
dimension as done in Gori and Trombettoni (2020). In the following, whenever
we mention the Yamabe equation, we are referring to the integer one, with
standard Laplacian, as first introduced in Yamabe (1960) .

To understand how a geometric problem emerges from one-point functions of a
bounded system at criticality, we discuss the connection of the obtained
results with the “uniformization” hypothesis put forward in Gori and
Trombettoni (2020).

###  III.1 Uniformization hypothesis and the Yamabe equation

The main property a system typically gains at its critical points is conformal
invariance Polyakov (1970, 1974). Heuristically, this means that every point
and every region of the system look the same. Introducing a boundary clearly
breaks this property. The question addressed in Gori and Trombettoni (2020) is
then: is there a way to recover it? If a metric is introduced which sets the
boundary at an infinite distance, then we would no longer have a distinction
between points close to the boundary and points deep in the bulk. The only
changes of the euclidean metric that we can allow are conformal
transformations, since the system must still be locally euclidean. This means
that the choice of metric reduces to the choice of a function
$\gamma(\mathbf{x})$ which sets a local scale:

$\delta_{ij}\rightarrow g_{ij}=\frac{\delta_{ij}}{\gamma(\mathbf{x})^{2}},$
(23)

$\delta_{ij}$ being the flat metric, i.e. the identity matrix. As usual,
summation over repeated indices is implied and $i,j=1,\cdots,d$.

Constraints on the function $\gamma$ have to be imposed. Since we have a
curved space, we should look into the various quantities that describe its
curvature, the most obvious one being the Ricci scalar curvature. The main
hypothesis in Gori and Trombettoni (2020) is that the metric must make a
bounded critical system as uniform as possible: this means making the scalar
curvature constant. This curvature would have to be negative, since spaces
with positive curvature, like spheres, lack boundaries. The simplest examples
of space with constant negative curvature are the Poincaré half plane and disk
models.

Starting from the metric with an unknown $\gamma(x)$, one can compute the
Christoffel symbols, from which one gets the Ricci tensor, and finally the
Ricci scalar:

$R=R_{ij}g^{ij}=\kappa,\qquad\kappa<0.$ (24)

Without losing generality, we can set $\kappa=-1$. We can now write this
condition as an equation for the factor $\gamma(\mathbf{x})$, getting (21),
supplemented by the condition $\gamma(\mathbf{x})=0$ at the boundaries of the
domain $\Omega$. This is enough to get solutions which, close to the boundary,
are proportional to the euclidean distance from it. The distance from any
point to the boundary, computed with this metric, is therefore infinite, as
desired.

To frame it more generally, the Yamabe problem consists in finding a metric in
the conformal class of another given metric for a smooth Riemann space, that
makes the scalar curvature constant Lee et al. (1987). A solution has been
proven to exist for any such space, provided it is compact.

The two-dimensional limit of (21) gives the Liouville equation

$(-\bigtriangleup)\log\gamma(\mathbf{x})=-\gamma(\mathbf{x})^{-2}.$ (25)

The Yamabe equation is a nonlinear differential equation.There are a few cases
where the Yamabe equation has simple solutions: e.g., for a ball of radius $R$
in any dimension, one finds $\gamma=\frac{R^{2}-|\mathbf{x}|^{2}}{2R}$ Chang
and del Mar González (2011); Gori and Trombettoni (2020); the case of the slab
geometry will be treated in the next section.

One-point functions and the scale factor transform similarly under a dilation
of the system $\Omega\rightarrow\lambda\Omega$:

$\left\langle\phi_{\lambda\Omega}(\lambda\mathbf{x})\right\rangle=\lambda^{-\Delta_{\phi}}\left\langle\phi_{\Omega}(\mathbf{x})\right\rangle,\qquad\gamma_{\lambda\Omega}(\lambda\mathbf{x})=\lambda\gamma_{\Omega}(\mathbf{x}),$
(26)

where $\Delta_{\phi}$ is the scaling dimension of the field. Therefore, once
$\gamma(\mathbf{x})$ is known, all one-point functions are determined up to a
constant ${\cal C}$:

$\left\langle\phi(\mathbf{x})\right\rangle=\frac{{\cal
C}}{\gamma(\mathbf{x})^{\Delta_{\phi}}}.$ (27)

As an example, for a half space in any dimension, $x_{1}>0$, with
$\mathbf{x}=\\{x_{1},x_{2},\ldots,x_{d}\\}\in\mathbb{R}^{d}$, we have
$\gamma(\mathbf{x})=x_{1}$, so

$\left\langle\phi(\mathbf{x})\right\rangle=\frac{{\cal
C}}{x_{1}^{\Delta_{\phi}}},$ (28)

reproducing a standard result of boundary conformal field theory Cardy (2004).

The introduction of $\gamma(\mathbf{x})$ in (20) should now be clearer: the
mean-field equation for a (multi-)critical of the $O(N)$ model is equivalent
to the Yamabe equation at the corresponding upper critical dimension. Once the
scale factor $\gamma(\mathbf{x})$ is obtained by solving the Yamabe equation,
the magnetization is recovered through (27).

##  IV Analytical results for the slab geometry

For the slab geometry we can actually obtain the solution for general
dimension $d$ in implicit form. We denote the solution of the Yamabe equation
(21) with $\gamma_{d}(x)$ to emphasize the dependence on the dimension $d$.
For convenience we assume $x\in[0,2]$ (i.e., $L=2$), so that the center of the
slab is in $x=1$.

For $x\in[0,1]$ and FBC, the relation between $x$ and $\gamma_{d}$ is given in
terms of the ${}_{2}F_{1}$ hypergeometric function for FBC. From the Yamabe
equation one straightforwardly finds

$x={}_{2}F_{1}\left(\frac{1}{d},\frac{1}{2};1+\frac{1}{d};\left(\frac{\gamma_{d}}{\gamma_{m}}\right)^{d}\right)\,\gamma_{d}.$
(29)

By reflecting $x$ around $1$: $x\rightarrow 2-x$ the other branch is obtained.
The constant

$\gamma_{m}=\frac{\Gamma\left(\frac{1}{2}+\frac{1}{d}\right)}{\sqrt{\pi}\,\Gamma\left(1+\frac{1}{d}\right)}$
(30)

is the (maximum) value acquired by the conformal factor at the center of the
slab. Equation (29) is valid for any $d\geq 2$, including the cases
$d=4,6,3,2$ and also real values of $d$. Notice that a slab configuration
cannot be defined for $d<2$. The formulas for other boundary conditions are
written below.

We pause here to comment that one sees a (minor) advantage of using the Yamabe
equation in the form (22) instead of using directly the saddle-point equation
(2). The latter, when written in the slab geometry, gives rise to the so
called Emden-Fowler equation Polyanin and Zaitsev (1995). The latter, for a
function $m(x)$, reads in its canonical form $m^{\prime\prime}(x)=A\,x^{\cal
N}m(x)^{\cal M}$ Polyanin and Zaitsev (1995), with our case corresponding
evidently to ${\cal N}=0$ and ${\cal M}=(d+2)/(d-2)$. The case ${\cal N}=0$
can be solved by quadrature Polyanin and Zaitsev (1995), writing the solution
in term of an integral and for certain values of ${\cal M}$ the corresponding
analytical expressions are tabulated Polyanin and Zaitsev (1995). However,
solving the Yamabe equation (22) and using the procedure prescribed in
ordinary differential equations textbooks (or directly `Mathematica`) one
finds that the solution is given by the inverse of the hypergeometric function
${}_{2}F_{1}$ for any $d$. Therefore, rewriting $m$ in terms of $\gamma$ via
the equation (20) may also help to find easier analytical solutions, as the
case of the slab geometry shows.

Let us now discuss how to relate the result (29) to different boundary
conditions. We write the magnetization via (20) as
$m_{d}(x)=\gamma_{d}(x)^{-\Delta_{\phi}}=\gamma_{d}(x)^{-\frac{d-2}{2}}$ where
$\gamma_{d}(x)$ is given in (29). If in $m_{d}(x)$ we set $d=3$, $4$, and $6$
and aptly rescale the $x$ domain we recover the $++$ boundary magnetization
profiles given in II. This can be achieved by using the identities reported in
Appendix B. The other profiles, referring to different boundary conditions,
can be found by taking $x$ in (29) to be complex according to the following
scheme:

$\displaystyle m_{++}(x)=m_{d}(x)$ $\displaystyle x\in(0,2),$ $\displaystyle
m_{+0}(x)=m_{d}(x\,e^{i\,\pi/d})e^{-i\,\pi\Delta_{\phi}/d}$ $\displaystyle
x\in(0,\sec(\pi/d)],$ $\displaystyle m_{00}(x)=m_{d}(1+i\,x)$ $\displaystyle
x\in[-\tan(\pi/d),\tan(\pi/d)].$

It is worth observing that the three BCs can be found by evaluating $\gamma$
on the right triangle $\mathbf{T}=(0,1,1+i\tan(\pi/d))$. Indeed the function
$z(w)=w^{1/d}{}_{2}F_{1}\left(\frac{1}{d},\frac{1}{2};1+\frac{1}{d};w\right)$
appearing in (29) is nothing but the Schwarz function mapping the upper-half-
plane ($w$ variable) onto $\mathbf{T}$ ($z$ variable, also $x$ when real)
keeping $z=0,\,1$ fixed. In the convenient variable $z$ we have that
$\gamma=w^{1/d}$ and $m=w^{-\Delta_{\phi}/d}=w^{-\frac{d-2}{2d}}$. The above
consideration should shed some light on the appearance of the Weierstrass
functions $\wp_{e}$, $\wp_{l}$ in Section II and on the peculiarity of
$d=3,4,6$: for those values, copies of the triangle $\mathbf{T}$ create a
regular tessellation of the complex plane.

Indeed, we can see what the general result (29) simplifies to when we
substitute $d=4,3,6,2$:

  * $d=4$:

the inverse of (29) is $\wp_{l}(x)^{-1/2}$, the lemniscatic Weierstrass
function used in (4), see Appendix B.

  * $d=6$:

the inverse is the equiharmonic elliptic function $\wp_{e}(x)$ as seen in
(18).

  * $d=3$:

this case is dual to the $d=6$ case, since the solution here is simply the
square root of the reciprocal of the previous solution.

  * $d=2$:

the value $d=2$ cannot be directly chosen in the Yamabe equation (21): a limit
has to be performed, yielding the Liouville equation (25). One may then assume
that plugging $d=2$ into (29) will give an incorrect result. Surprisingly,
that is not the case: for $d=2$, the function
${}_{2}F_{1}(\frac{1}{2},\frac{1}{2};\frac{3}{2};\gamma^{2}\frac{\pi^{2}}{4})\gamma$
reduces to the inverse sine: indeed, $\gamma_{2}(x)=\frac{2}{\pi}\sin(\pi
x/2)$.

They agree with previous results obtained in particular cases: profiles for
$++$ and $+-$ boundary conditions were found for $d=4$ in Krech (1997), and
for $d=2$ in Carlon and Iglói (1998); the $++$ profiles for $d=2,3,4$ are also
found in Borjan and Upton (1998). In both Krech (1997) and Borjan and Upton
(1998) Jacobi functions were used. Switching to Weierstrass functions allowed
us in the present paper to write profiles for various dimensions in a compact
way, retrieving the previously listed known results, finding new results for
all the conformally invariant boundary conditions. We also obtain new results
for all the critical magnetization profiles in $d=6$.

##  V Solution of lattice mean-field equations

In this section we discuss the solution of the mean-field equations for the
Ising model in a slab geometry directly on the lattice. The goal is to see how
lattice profiles can be accurately described by a continuous function from
field theory in the thermodynamic limit. For this reason in this section we
take the slab coordinate $x$ to be an integer variable, going from $1$ to $L$,
and then take $L\to\infty$. In this calculation it is not necessary to specify
the number of sites in the directions perpendicular to $x$, since we can
assume that the probability distribution function $P$ of any spin depends only
on its transverse coordinate $x$. For this reason we put

$P(s_{i})=\frac{1+m_{x}}{2}\delta_{s_{i},1}+\frac{1-m_{x}}{2}\delta_{s_{i},-1},$
(31)

where $s_{i}=\pm 1$ is the Ising variable in the site $i$ having $x$ as
coordinate along the direction of the slab.

We can find the discrete magnetization profile $m_{x}=\langle s_{i}\rangle$
minimizing the free energy. The mean-field equation Bellac (1991) is

$m_{x}=\tanh\left(\beta\sum_{j}\left\langle s_{j}\right\rangle\right).$ (32)

This is the same equation used to find the inverse critical temperature
$\beta_{c}$, in the bulk, i.e. in the case of absence of boundaries. With the
Ising coupling $J$ set to $1$ for convenience, it is $\beta_{c}=1/2d$. Of
course, it differs from the critical temperature of the actual Ising lattice,
see the next Section.

In a four dimensional slab with $d=4$, this becomes

$m_{x}=\tanh\left[\beta(6m_{x}+m_{x+1}+m_{x-1})\right].$ (33)

For the case of FBC $m_{1}=m_{L}=1$, this equation can be solved iteratively
until the difference between the two sides is smaller than a fixed threshold.

The value of the magnetization at the center of the slab decays, as the system
size increases, as $1/L$. Since profiles for different sizes must have the
same functional shape at the critical point, we rescale them by multiplying
each profile by its corresponding size $L$. This also clarifies the
correspondence between fixing the boundary spins to $+1$ in the lattice model
and diverging boundary conditions in the field theory. As the size increases,
the rescaled boundary magnetization grows proportional to $L$.

At the same time, we can comment about the extrapolation length $a_{L}$ Diehl
(1997). the latter decreases, meaning that the point where the profile
diverges gets closer to the lattice boundaries. In the limit
$L\rightarrow\infty$, $a_{L}\rightarrow 0$ and $m(0)=m(L)\rightarrow\infty$.

Figure 4: Collapse of the mean-field magnetization profiles. Since they are
symmetric around the center of the slab, we only plot the left half. Each set
of magnetization values has been rescaled by a factor $L$, while the $x$
coordinate has been rescaled through its extrapolation length.

Once the rescaling is done, we clearly see a collapse of the various profiles,
showing that we are at the critical point. The corresponding results are in
Fig. 4. For large $L$, the extrapolation length vanishes and the mean-field
profile coincides with the saddle-point solution.

##  VI Monte Carlo simulation of the $4d$ Ising model on a slab

The previous predictions have been obtained by performing the saddle-point
approximation on the action (1). It is very well known that in four dimensions
for the $\phi^{4}$ theory, and in general at the upper critical dimension, the
critical exponents are the mean-field ones Cardy (1996); Mussardo (2020); Amit
and Martin-Mayor (2005); Bellac (1991). This is routinely exploited in
conformal bootstrap calculations for the bulk geometry, where in $d=4$ the
critical theory is Gaussian Rychkov (2017). However, in a bounded domain – to
the best of our knowledge – we do not know a proof that the critical
magnetization profile in the thermodynamic limit given by the saddle-point
approximation is exact in $d=4$ for the $\phi^{4}$ theory, although it is
expected. For this reason we decided to numerically test the saddle-point
findings by Monte Carlo simulations and validate our predictions via numerical
experiments. In order to obtain a numerical check of the magnetization
profiles, we will concentrate on the Ising model on the slab geometry with
FBC. Notice that explicit numerical investigations for high dimensions of
phenomena arising from inhomogeneities are rather sparse especially if
compared to two-dimensional systems. Three dimensional models did received of
course attention, but validation of basic predictions from scale/conformal
invariance have recently been tested Gori and Trombettoni (2015); Cosme et al.
(2015).

We performed Monte Carlo simulations of the Ising model at its upper critical
dimension, $4$, in a slab of sizes $L$ in the transverse direction, and $4L$
in the other three directions; $L$ ranges from $16$ to $56$. The FBC are
implemented by fixing the spins to the same value ($+1$) on the two faces in
the transverse direction, while the other directions have periodic boundary
conditions. We can then expect the $++$ magnetization profile from (II.1). The
critical inverse temperature $\beta_{c}$ is taken to be $\beta_{c}=0.1496927$
from Luijten (1997), see also Bittner et al. (2002); Lundow and Markström
(2009).

The simulation uses the standard Metropolis algorithm, whose moves are single
spin flips. After a thermalization time, we measured the average magnetization
and local energy along hyperplanes parallel to the boundaries. As explained in
Sections II and V, since the spins are fixed at the edge faces and one is at
the bulk critical temperature, the magnetization profile will start from the
value $1$ at one boundary, will decrease as one approaches the center, should
reach a value of order of $1/L$, and then rise again up to $1$ in the other
slab boundary. One can rescale the numerical data in a way that the value of
the magnetization at the center is constant, meaning that the boundary
magnetization increases with $L$.

Figure 5: Collapse of the critical magnetization profiles for slabs of different sizes. Again, we only plot half the profile. Each set of points has been rescaled with a multiplicative constant and its extrapolation length, obtained from the fit 34. $L$ | $a_{L}$
---|---
16 | 0.2465
24 | 0.1724
32 | 0.1314
40 | 0.1070
48 | 0.0884
56 | 0.0404
Table 2: Decreasing extrapolation lengths as the size increases.

The magnetization data is then fitted with

$m(x)=k\left[\gamma\left(\frac{x}{1+a_{L}}\right)\right]^{-1},$ (34)

where $k$ is a multiplicative constant roughly proportional to the system
size, and $a_{L}$ is the extrapolation length, with the index L denoting the
dependence on the size. The extrapolation length shrinks the lattice profile
so that the divergence of the order parameter would appear a few sites beyond
the edges Cardy (1996); Diehl (1997). We see that it becomes smaller as the
size grows, meaning that larger sizes describe a larger portion of the saddle-
point profile. In Fig. 5, we plot the magnetization profiles obtained for
different sizes, compared with the prediction from (27). The magnetization
$m(x)$ for different sizes are plotted as functions of the respective rescaled
variable $\xi=x/(1+a_{L})$, which highlights the collapse.

Despite qualitative agreement between the prediction and the numerical data,
the distance between raw data points and the theoretical curve is larger than
the estimated numerical error (in Fig. 5, smaller than the point sizes). This
is to be expected: while (27) gives the correct mean-field behavior,
observables at the upper critical dimension also include some logarithmic
corrections. To account for them, we proceed as in Luijten (1997). At the
critical temperature, the main finite size corrections to the bulk
magnetization take the form

$m=c\frac{(\ln L)^{1/4}}{L}\sqrt{b_{0}+\frac{b_{1}}{\ln L}},$ (35)

where terms with higher powers of $1/\ln L$ have been neglected. In our case,
the magnetization is a function of $x$ (or rather of the rescaled coordinate
$\xi$), and therefore so are $b_{0}$ and $b_{1}$. For every $\xi$ we find the
values of $b_{0}(\xi)$ and $b_{1}(\xi)$ which best fit $m(\xi,L)$, seen as a
function of $L=\\{16,24,32,40,48,56\\}$ 111Notice that in Luijten (1997) the
(bulk) $4D$ Ising model at criticality is studied for linear sizes
$L^{\prime}$ from $2$ up to $48$ (to match with our notations, the hypercube
in Luijten (1997) is $L^{\prime}\times L^{\prime}\times L^{\prime}\times
L^{\prime}$, while our slab is $4L\times 4L\times 4L\times L$)..

In order to compare with the analytical prediction (27), the constant $c$ in
(35) is chosen so that $\sqrt{b_{0}(0)}=m(0)$. We then plot, in Fig. 6, the
ratio $\sqrt{b_{0}(\xi)}/m(\xi)$, and we see that it remains close to 1. As we
approach the boundaries, the value of the first logarithmic correction $b_{1}$
grows, explaining the deviation of the ratio from 1. Larger values of $b_{1}$
simply imply that it is numerically harder to measure the values predicted by
(27) near the boundaries, as it requires simulating even larger systems.

Figure 6: Ratio of the square root of the fit parameter $b_{0}$ and the
expected magnetization. The boundary $x=-1$ correspond to $\xi=-1/(1+a_{L})$,
and the first few points close to the boundary must be discarded, so the plot
starts from $\xi=-0.39$.

From $\xi=-1$ to $\xi=-0.4$ we decided not to plot the ratio
$\sqrt{b_{0}(\xi)}/m(\xi)$ since, due to the values of the extrapolation
length, for these $\xi$’s we do not have points for each of the considered
values of $L$. We remind that we are in the left part of the slab geometry,
$\xi=0$ corresponding to the center and $\xi=-1$ to the left edge (we
discretize $\xi$ in steps of $0.01$). We notice that by fitting the points
from the magnetization using (35) without $b_{1}$ one obtains a clearly worse
estimate of the $\xi^{2}$, confirming the validity of the fitting function
(35). Moreover, for the points in Fig. 6 the value
$b_{1}/\ln{L_{\mathrm{m}ax}}$ is always 3 to 4 times smaller than $b_{0}$,
where $L_{\mathrm{m}ax}=56$ is the maximum value of $L$ we were able to
simulate. This shows that the term proportional to $b_{1}$ is actually a
relatively small perturbation of the $b_{0}$ term. The standard deviations
$\sigma$ in Fig. 6 are obtained in an unweighted way, i.e. fitting without
weighting the magnetization data with their respective errors. The point for
$\xi=0$ has the ratio $\sqrt{b_{0}(\xi=0)}/m(0)$ equals $1$ by construction,
and with standard deviation – determined as explained above – $\sigma=0.04$.
The final result, depicted in Figure (6), shows that $\sqrt{b_{0}(\xi)}$ is
compatible with $m(\xi)$ within $\sigma$ for $-0.32\leq\xi\leq 0$ and within
$2\sigma$ for $-0.39\leq\xi\leq-0.33$. To obtain more data in the range
$-1<\xi\leq-0.4$ one should have larger sizes.

The conclusion is that for the data we have, the analytical predictions is in
agreement with the Monte Carlo numerical results within $2\sigma$.

##  VII Conclusions

In this paper we studied the magnetization profiles of $O(N)$ models at their
upper critical dimensions for different (scale invariant) boundary conditions.
Taking the slab as a reference domain, we first studied the magnetization
profiles at the upper critical dimensions $d=3$, $4$, $6$. We put the zeros
and poles of the Weierstrass elliptic solutions (written as a function of a
complex variable) in connection with the different boundary conditions. We
found general formulas in terms of Weierstrass elliptic functions, extending
known results and finding new ones for the percolation.

We then studied the connection between the critical magnetization profiles in
a general domain and the solutions of the Yamabe problem in the same domain
and with the same boundary conditions.

The solutions of the Yamabe equation solve the so-called Yamabe problem, which
amounts to finding a metric having constant curvature. This shows that the
saddle-point equation for the critical magnetization at the upper critical
dimension is equivalent to a purely geometric description, where the euclidean
metric is conformally altered to obtain a space with uniform negative
curvature. In the slab geometry, using the Yamabe equations we derived
analytical expressions for critical magnetization profile as inverse
hypergeometric functions valid for any dimension $d\geq 2$. The expressions
are valid for fixed boundary conditions, but with suitable shifts in the
argument of the slab coordinate variable we can obtain the corresponding
solutions for the other considered boundary conditions. Lattice mean-field
results for the slab geometry in $d=4$ have been also presented.

The magnetization profiles in the specific case of the $4$-dimensional Ising
model with fixed boundary conditions in the slab are compared with Monte Carlo
simulations, finding good agreement.

The presented results explicitly confirm at the upper critical dimension
recent results presented in Gori and Trombettoni (2020). There, general
correlation functions of the magnetization field in a bounded domain have been
conjectured to be related to the solution of a fractional Yamabe problem,
which with vanishing anomalous dimension becomes the integer problem here
considered.

As next step at the upper critical dimension, a worthwhile subject of
investigation would be to study spin-spin correlation functions in the four-
dimensional case for bounded domains, and check how/if they depend on the
geometric distance defined by the metric obtained from the solution of the
integer Yamabe problem in the considered domain. Finally, one would like to
study the case of small anomalous dimensions, which would lead to the study of
a fractional Yamabe problem with the Laplacian entering with a fractional,
near to an integer value, exponent. This would be interesting both to compare
with other perturbative approaches and for the challenging task to develop a
suitable perturbative schemes to solve the fractional Yamabe equation relevant
for critical models below the upper critical dimension.
We also wonder if the analytic structure found can be extended beyond the slab
geometry at the upper critical dimension treated here. When the domain is not
of the form $[0,L]\times\mathbb{R}^{d-1}$, is it still possible, given a
magnetization profile, to obtain profiles for other boundary conditions by
some process of continuation?
If we instead consider the fractional Yamabe problem (in the slab to begin
with) can we use similar techniques to the ones described here to provide a
solution? The solution to this problem was already found in Gori and
Trombettoni (2020) via generalization of AdS/CFT-borne scattering techniques
Graham and Zworski (2003) providing a beautiful mathematical framework. When
searching for explicit solutions to the fractional Yamabe problem within this
framework, however, one is faced with great mathematical challenges, as it
requires to both solve the vacuum Einstein equations for a metric in $d+1$
dimensions, and then find the solution of a nonlinear eigenvalue problem in
the obtained metric space. It would be very appealing to have simpler schemes
available.

More concretely, the next step will be to study a three dimensional system, a
case which is also numerically more accessible, to test the predictions of the
fractional Yamabe equation against data below $d_{c}$. It is then worth asking
whether the profiles obtained from $\gamma(\mathbf{x})$ and the unknown
function that describes two-point correlations can be approximately recovered
in a perturbative manner, as an expansion around the four dimensional
solutions.

Acknowledgements: We thank J. Cardy, N. Defenu, S. Dietrich, T. Enss, A.
Gambassi and A. Squarcini for useful discussions at various stages of this
work. GG is supported by the Deutsche Forschungsgemeinschaft (DFG, German
Research Foundation) under Germany’s Excellence Strategy EXC 2181/1 –
390900948 (the Heidelberg STRUCTURES Excellence Cluster).

##  Appendix A Reminders on Weierstrass elliptic functions

Elliptic functions appear in numerous areas of physics. They get their name
from their property of being the inverse of elliptic integrals.
A complex function of one complex variable $f(z)$ is called elliptic if it is
meromorphic (its only singularities are poles) and is doubly periodic, i.e.,

$f(z+2\omega_{1})=f(z),\qquad f(z+2\omega_{2})=f(z),$ (36)

with $\omega_{1}/\omega_{2}\notin\mathbb{R}$. $\omega_{1}$ and $\omega_{2}$
are called half-periods. The double periodicity induces a tessellation of the
complex plane in parallelograms. It is therefore sufficient to know the values
of the function within one of these parallelograms, say the one with vertices
$0$, $2\omega_{1}$, $2\omega_{2}$ and $2\omega_{1}+2\omega_{2}$.

The Weierstrass $\wp$ function is probably the most intuitive elliptic
function to construct, starting from the definition and the requirement of
having only a double pole within each cell:

$\begin{split}\wp(z)=\frac{1}{z^{2}}+\sum_{(m,n)\neq(0,0)}\left(\frac{1}{(z+2\,m\,\omega_{1}+2\,n\,\omega_{2})^{2}}\right.-\\\
-\left.\frac{1}{(2\,m\,\omega_{1}+2\,n\,\omega_{2})^{2}}\right).\qquad\end{split}$
(37)

Instead of the half periods, the Weierstrass function can be identified with
another pair of numbers, $g_{2}$ and $g_{3}$, called invariants. They are the
lowest order coefficients in the Laurent expansion of $\wp$ around $0$:

$\wp(z)=\frac{1}{z^{2}}+\frac{g_{2}}{20}z^{2}+\frac{g_{3}}{28}z^{4}+O(z^{6}).$
(38)

The invariants can be obtained from the half periods as

$\begin{split}g_{2}=60\sum_{(m,n)\neq(0,0)}\frac{1}{(2m\omega_{1}+2n\omega_{2})^{4}}\\\
g_{3}=140\sum_{(m,n)\neq(0,0)}\frac{1}{(2m\omega_{1}+2n\omega_{2})^{6}}.\end{split}$
(39)

The importance of the invariants comes from the fact that they appear in the
differential equation that the Weierstrass function obeys:

$\wp^{\prime 2}(z)=4\wp^{3}(z)-g_{2}\wp(z)-g_{3}=0.$ (40)

The particular cases encountered in the text are:

  * •

$(g_{2},g_{3})=(1,0)$, called lemniscatic elliptic function $\wp_{l}$ here,
which gives orthogonal semiperiods
$\omega_{1}=\omega_{l}=\Gamma(\frac{1}{4})^{2}/(4\sqrt{\pi})$,
$\omega_{2}=i\omega_{1}$, that tessellate the complex plane with squares.

  * •

$(g_{2},g_{3})=(0,1)$, the equiharmonic case $\wp_{e}$ here, with
$\omega_{1}=\omega_{e}=\Gamma(\frac{1}{3})^{3}/(4\pi)$ and
$\omega_{2}=\frac{1}{2}(\sqrt{3}i-1)\omega_{1}$, which produces a tessellation
of parallelograms each made of two equilateral triangles.

##  Appendix B Inverse functions for the lemniscatic and equiharmonic
Weierstrass elliptic functions

In the main text explicit solutions for the nonlinear ODE
$m^{\prime\prime}(x)\propto m^{\frac{d+2}{d-2}}$ have been derived for special
values of $d$ in terms of the lemniscatic and equiharmonic Weierstrass
elliptic functions $\wp_{l}$ and $\wp_{e}$. When solving the equation for
generic $d$ a solution has been presented in implicit form. This allows us to
write the following inversion formulas for $\wp_{l}$ and $\wp_{e}$:

${\wp_{l}}^{-1}(x)=\frac{1}{\sqrt{x}}{\,\,}_{2}F_{1}\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\frac{1}{4x^{2}}\right)$
(41)

${\wp_{e}}^{-1}(x)=\frac{1}{\sqrt{x}}{\,\,}_{2}F_{1}\left(\frac{1}{6},\frac{1}{2};\frac{7}{6};\frac{1}{4x^{3}}\right).$
(42)

These two results can be obtained by considering different representation of
the elliptic integrals $\int\mathrm{d}x\frac{1}{\sqrt{4x^{3}-1}}$ and
$\int\mathrm{d}x\frac{1}{\sqrt{4x^{3}-x}}$.

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A simple proof of the Baillon–Haddad theorem on open subsets of Hilbert spaces
This research was supported by the German Research Foundation (DFG) under grant numbers WA 3626/3-2 and WA 3636/4-2
within the priority program “Non-smooth and Complementarity-based Distributed Parameter
Systems: Simulation and Hierarchical Optimization” (SPP 1962).]

Daniel Wachsmuth
Institut für Mathematik,
Universität Würzburg,
97074 Würzburg, Germany<EMAIL_ADDRESS>Gerd Wachsmuth
Brandenburgische Technische Universität Cottbus-Senftenberg,
Institute of Mathematics,
03046 Cottbus, Germany,

We give a simple proof of the Baillon–Haddad theorem
for convex functions defined
on open and convex subsets of Hilbert spaces.
We also state some generalizations and limitations.
In particular, we discuss equivalent characterizations of the
Lipschitz continuity of the derivative of convex functions
on open and convex subsets of Banach spaces.

Baillon–Haddad theorem,
strong smoothness

§ INTRODUCTION

A very important result in convex analysis is the
Baillon–Haddad theorem
which states
that the derivative $f'$ is $\frac1L$-cocoercive
whenever $f \colon X \to \R$
is convex and differentiable with $L$-Lipschitz continuous derivative,
Here, $X$ is a (real) Banach space.
In <cit.>
it was shown that this remains true
if $f$ is defined on an open and convex subset of a (real) Hilbert space.
The corresponding proof is quite involved,
since it uses generalized second-order derivatives
and a reduction to the finite-dimensional situation.
We also refer to <cit.> for further comments and references.
We give a short and direct proof,
see <ref>.

It is well known that the $L$-Lipschitz continuity of the derivative of $f$ is
equivalent to a number of important properties of $f$ and its convex conjugate.
In <ref>, we investigate which of these equivalences remain valid
if $f$ is defined on an open and convex subset of a (real) Banach or Hilbert space.

§ THE BAILLON–HADDAD THEOREM ON OPEN SUBSETS OF HILBERT SPACES

We start by a characterization of differentiable functions with Lipschitz derivatives.

Let $O \subset H$ be an open and convex subset of the Hilbert space $H$.
Suppose that $f \colon O \to \R$ is (Gâteaux) differentiable.
For $L \ge 0$, the following are equivalent.

The derivative $f' \colon O \to H\dualspace$ is $L$-Lipschitz,

\begin{equation}
\label{eq:L_lipschitz}
\norm{
f'(y) - f'(x)
\le
L \norm{
y - x
\qquad\forall x,y \in O
\end{equation}

Both $f'$ and $-f'$ satisfy a one-sided Lipschitz estimate with constant $L$,

\begin{equation}
\label{eq:two_sided_lipschitz}
\abs{
\dual{f'(y) - f'(x)}{y - x}_H
\le
L \norm{y - x}_H^2
\qquad\forall x,y \in O.
\end{equation}

The function $f$
has a first order Taylor expansion with remainder $\frac L2 \norm{\cdot}_H^2$, i.e.,

\begin{equation}
\label{eq:taylor}
\abs[\big]{
f(y) - f(x) - \dual{f'(x)}{y - x}_H
\le
\frac L2 \norm{y - x}_H^2
\qquad\forall x,y \in O.
\end{equation}

The implication “<ref>$\Rightarrow$<ref>”
is a simple application of the Cauchy–Schwarz inequality.

To prove “<ref>$\Rightarrow$<ref>”,
we employ the fundamental theorem of calculus
and get

\begin{align*}
\abs[\big]{
f(y) - f(x) - \dual{f'(x)}{y - x}_H
\abs*{
\int_0^1 \dual{f'(x + t (y -x)) - f'(x)}{y - x}_H \, \d t
\\&
\le
\int_0^1 \frac1t \abs[\big]{
\dual[\big]{f'(x + t (y -x)) - f'(x)}{t(y - x)}_H
} \, \d t
\\&
\le
\int_0^1 \frac1t
\norm{t(y - x)}_H^2
\d t
\frac L2 \norm{y - x}_H^2
\end{align*}

In order to check “<ref>$\Rightarrow$<ref>”,
we take an arbitrary $\rho > 0$ and set

\begin{equation}\label{eq_O_rho}
O_\rho :=
\set{
x \in O
\given
\forall h \in H, \norm{h}_H \le \rho : x +  h \in O
\end{equation}

It is clear that $O_\rho$ is again convex.
Let us choose some $x,y \in O_\rho$
with $\norm{y - x}_H \le \rho$
and $d \in H$ with $y+d, x-d \in O$.
From (<ref>),
we get the inequalities

\begin{align*}
f(x-d) - f(y) - \dual{f'(y)}{x-d-y}_H &\le \frac L2 \norm{y-x+d}_H^2,\\
f(y+d) - f(x) - \dual{f'(x)}{y+d-x}_H &\le \frac L2 \norm{y-x+d}_H^2,\\
-\parens[\big]{f(y+d) - f(y) - \dual{f'(y)}{\mrep[r]{d}{-d}}_H} &\le \frac L2 \norm{d}_H^2,\\
-\parens[\big]{f(x-d) - f(x) - \dual{f'(x)}{-d}_H} &\le \frac L2 \norm{d}_H^2.
\end{align*}

Adding these inequalities leads to

\begin{equation*}
\dual{f'(y) - f'(x)}{y-x+2 d}_H
\le
L \norm{y-x+d}_H^2 + L \norm{d}_H^2.
\end{equation*}

Next, we specialize to
$d = (x - y + g) / 2$ for some $g \in H$ with $\norm{g}_H \le \rho$.
Note that
$\norm{d}_H \le (\norm{y-x}_H + \norm{g}_H)/2 \le \rho$.
Together with $x,y \in O_\rho$ we find
$x - d, y + d \in O$.
For arbitrary $g \in H$ with $\norm{g}_H \le \rho$, this leads to

\begin{equation}
\label{eq:ineq_with_g}
\dual{f'(y) - f'(x)}{g}_H
\le
\frac L4 \norm{y - x + g}_H^2 + \frac L4 \norm{x - y + g}_H^2
\frac L2 \norm{y - x}_H^2 + \frac L2 \norm{g}_H^2
\end{equation}

where we used the parallelogram identity.
We denote by $G \in H$ the Riesz representative of $f'(y) - f'(x) \in H\dualspace$.
In case
$\norm{G}_H = \norm{f'(y) - f'(x)}_{H\dualspace} \ge \rho L$,
we take $g = \rho G / \norm{G}_H$
in (<ref>)
and obtain

\begin{align*}
\rho \norm{f'(y) - f'(x)}_{H\dualspace}
\dual{f'(y) - f'(x)}{g}_H
\le
\frac L2 \norm{y - x}_H^2 + \frac L2 \norm{g}_H^2
\le
L \rho^2
\end{align*}

This yields $\norm{G}_H \le \rho L$.
Thus, we can insert $g = G / L$
(in case $L > 0$)
in (<ref>)
and obtain

\begin{equation*}
\frac 1 L \norm{f'(y) - f'(x)}^2_{H\dualspace}
\le
\frac L2 \norm{y - x}_H^2 + \frac 1 {2 L} \norm{f'(y) - f'(x)}_{H\dualspace}^2
\end{equation*}

this shows

\begin{equation*}
\norm{f'(y) - f'(x)}_{H\dualspace}
\le
L \norm{y - x}_H
\qquad
\forall x,y \in O_\rho, \norm{y - x}_H \le \rho
\end{equation*}

For arbitrary $x,y \in O_\rho$,
we can choose $n \in \N$, $n \ge \norm{y - x}_H / \rho$
and set
$x_i := x + i (y - x) / n$, $i = 0,\ldots, n$.
Due to $\norm{x_i - x_{i-1}}_H = \norm{y - x}_H / n \le \rho$,
we can use the above estimate to achieve

\begin{align*}
\norm{f'(y) - f'(x)}_{H\dualspace}
\norm{f'(x_n) - f'(x_0)}_{H\dualspace}
\le
\sum_{i = 1}^n \norm{f'(x_i) - f'(x_{i-1})}_{H\dualspace}
\\
\sum_{i = 1}^n L \norm{x_i - x_{i-1}}_H
\sum_{i = 1}^n L \frac{\norm{y - x}_H}{n}
= L \norm{y - x}_H
\qquad\forall x,y \in O_\rho
\end{align*}

$O = \bigcup_{\rho > 0} O_\rho$

The implications
remain to hold in the Banach space setting.
However, in the proof of
, we have utilized the parallelogram identity in (<ref>).
Thus, the proof does not generalize to Banach spaces.
Instead, we could employ the triangle inequality,
which leads to

\begin{equation*}
\frac L4 \norm{y - x + g}_H^2 + \frac L4 \norm{x - y + g}_H^2
\le
L \norm{y - x}_H^2 + L \norm{g}_H^2
\end{equation*}

By adapting the remaining part of the proof,
we still arrive at (<ref>)
but with $2 L$ instead of $L$.

By means of an example, we demonstrate that the assertion of <ref>
indeed fails in Banach spaces.

We choose $X = (\R^2, \norm{\cdot}_\infty)$,
thus, $X\dualspace = (\R^2, \norm{\cdot}_1)$.
We define $f \colon \R^2 \to \R$ via

\begin{equation*}
f(x_1, x_2) := \frac12 (x_1^2 - x_2^2)
\end{equation*}

\begin{equation*}
\abs{\dual{ f'(y) - f'(x)}{ y - x}_{\R^2}}
=  \abs{(y_1 - x_1)^2 - (y_2 - x_2)^2}
\le
% \max\{ \abs{x_1-y_1}, \abs{x_2 - y_2} \}^2
% =
\norm{ y - x }_\infty^2
\end{equation*}

for all $x,y \in \R^2$,
i.e., (<ref>)
is satisfied with $L = 1$.

\begin{equation*}
\norm{ f'(x) - f'(0) }_1
% =  \norm{ x }_1
= \abs{x_1} + \abs{-x_2}
\le \hat L \norm{ x - 0 }_\infty
\quad\forall x \in \R^2
\end{equation*}

only holds
for $\hat L \ge 2$.
Thus, $f' \colon X \to X\dualspace$ is only Lipschitz continuous with constant $2$.

Further, we need a characterization of cocoercive operators
on Hilbert spaces.

Let $O \subset H$ be a subset of the Hilbert space $H$.
Then, for $T \colon O \to H\dualspace$ and $L > 0$
the following are equivalent.

* $T$ is $1/L$-cocoercive, i.e.,

\begin{equation*}
\dual{T(y) - T(x)}{y - x}_H
\ge
\frac1L \norm{T(y) - T(x)}_{H\dualspace}^2
\qquad\forall x,y \in O.
\end{equation*}

* $2 T/L - R$ is nonexpansive,

\begin{equation*}
\norm{
2 T(y)/L - R(y)
\parens*{
2 T(x)/L - R(x)
\le
\norm{y - x}_H
\qquad\forall x,y \in O.
\end{equation*}

Here, $R \colon H \to H\dualspace$ is the Riesz isomorphism of $H$.

We note that this result follows from some simple and straightforward calculations.

As a last prerequisite,
we show that the so-called strong smoothness of $f$ implies
Gâteaux differentiability.

Let $O \subset X$ be an open and convex subset of the Banach space $X$.
Suppose that $f \colon O \to \R$ is convex, lower semicontinuous
and strongly smooth, i.e.,

\begin{equation*}
f\parens[\big]{ \lambda x + (1-\lambda) y}
\frac{L}{2} \lambda (1-\lambda) \norm{y - x}_X^2
\ge
\lambda f(x) + (1-\lambda) f(y)
\quad\forall x,y \in O, \lambda \in (0,1)
\end{equation*}

holds for some $L > 0$.
Then, $f$ is Gâteaux differentiable on $O$.

For $x \in O$, $h \in X$ and $t > 0$ small enough,
we apply the smoothness inequality to $x \pm t h$
and $\lambda = 1/2$.
This yields

\begin{equation*}
f(x) + \frac{L}{2} \norm{t h}_X^2
\ge
\frac12 f(x + t h) + \frac12 f(x - t h).
\end{equation*}

Sorting terms and dividing by $t/2$ gives

\begin{equation*}
\ge
\lim_{t \searrow 0}
\parens*{
\frac{f(x + t h) - f(x)}{t} + \frac{f(x - t h) - f(x)}{t}
L t \norm{h}_X^2
f'(x; h) + f'(x; -h)
\end{equation*}

Recall that the existence of the directional derivatives
follows from $x \in O = \interior(\dom f)$,
see <cit.>.
This result also gives the sublinearity $f'(x; h) + f'(x; -h) \ge 0$,
thus $f'(x; \cdot)$ is linear.
Since $f$ is locally Lipschitz continuous on $O$ by
the functional $f'(x; \cdot)$
is Lipschitz continuous as well.
Thus, $f'(x; \cdot) \in X\dualspace$
and this shows the Gâteaux differentiability of $f$.

Now we are in position to prove the main result.

Let $O \subset H$ be an open and convex subset of the Hilbert space $H$.
Suppose that $f \colon O \to \R$
is convex.
Then, for $L > 0$, the following are equivalent.

$f$ is (Gâteaux) differentiable
and $f'$ is $L$-Lipschitz.

$\frac L2 \norm{\cdot}_H^2 - f$ is convex
and $f$ is lower semicontinuous.

$f$ is (Gâteaux) differentiable
and $f'$ is $1/L$-cocoercive.

Since $H$ is assumed to be a Hilbert space,
it is straightforward to check that
implies the strong smoothness of $f$,
$f$ is also Gâteaux differentiable in case

We set $h(x) := \frac L2 \norm{x}_H^2 - f(x)$ for $x \in O$.
For the auxiliary function $h$
we have $h'(x) = L R(x) - f'(x)$.
This directly yields

\begin{align*}
\ref{thm:main:1}
\dual{f'(y) - f'(x)}{y - x}_H \le L \norm{y - x}_H^2
\quad\forall x,y \in O
\\
\dual{h'(y) - h'(x)}{y - x}_H \ge \mrep{0}{L \norm{y - x}_H^2}
\quad\forall x,y \in O
\\
\ref{thm:main:2}
\end{align*}

For the other equivalence,
we employ <ref>.
To this end, we apply <ref>
to the function $g \colon O \to \R$
defined via $g(x) := 2 f(x) / L - \frac12 \norm{x}_H^2$.
Note that $g'(x) = 2 f'(x) / L - R$.
This yields

\begin{align*}
\ref{thm:main:1}
\mrep[r]{0}{-\norm{y - x}_H^2} \le \dual{f'(y) - f'(x)}{y - x}_H \le L \norm{y - x}_H^2
\quad\forall x,y \in O
\\
-\norm{y - x}_H^2 \le \dual{\mrep[r]{g}{f}'(y) - \mrep[r]{g}{f}'(x)}{y - x}_H \le \norm{y - x}_H^2\mrep{}{L}
\quad\forall x,y \in O
\\
\text{$g'$ is nonexpansive}
% \\&
\quad\Leftrightarrow\quad
\ref{thm:main:3}
\qedhere
\end{align*}

The above prove is an adaption of the proof of
to the situation without second-order differentiability
and, thus, gives a simple answer to <cit.>.
It is currently not clear whether the equivalence
of <ref> and <ref> remains to hold
if $H$ is only assumed to be a Banach space.
Note that both main ingredients of the proof (<ref>)
cannot be transferred directly to Banach spaces, see also the discussion in the next section.
The next example shows
that the implication “<ref>$\Rightarrow$<ref>” fails in Banach spaces.

We choose $X = (\R^2, \norm{\cdot}_\infty)$,
thus, $X\dualspace = (\R^2, \norm{\cdot}_1)$.
We define $f \colon \R^2 \to \R$ via

\begin{equation*}
f(x_1, x_2) := \frac12 (x_1^2 + x_2^2)
\end{equation*}

\begin{equation*}
\norm{f'(y) - f'(x)}_1
=  \abs{y_1 - x_1} + \abs{y_2 - x_2}
\le
2\norm{ y - x }_\infty
\end{equation*}

for all $x,y \in \R^2$.
this inequality is satisfied with equality if $x,y\in \operatorname{span}\set{(1,1)}$.
Hence, <ref> is satisfied if and only if $L \ge 2$.

Define $h(x):=\frac L2 \norm{x}_X^2 - f(x) = \max(x_1^2,x_2^2)- \frac12 (x_1^2 + x_2^2)$.
Then $h( (1,0)) >0$ but $h( (1,\pm1))=0$. Consequently, $h$ is not convex, and  <ref> is violated.
In fact, $h$ is not convex for all $L>0$.

in Hilbert spaces,
is equivalent to the
strong smoothness of $f$.
Then, the equivalence between Lipschitzness of $f'$
and strong smoothness of $f$
continues to hold in Banach spaces,
see <ref> below.

§ CONVEX FUNCTIONS ON OPEN, CONVEX SUBSETS OF BANACH SPACES

In this section, we address generalizations and limitations
of <ref>.
In particular, we are interested in convex functions defined on an open subset of a Banach space.
We investigate, which of the claims of <ref> and conditions well-known to be equivalent to $L$-Lipschitz continuity of
the derivative remain true in this general situation.

For a convex and lower semicontinuous function
$f \colon O \to \R$,
where $O$ is an open and convex subset of a Banach space $X$,
we consider the following assertions
with some fixed $L > 0$.

The function $f$ is strongly smooth

\begin{equation*}
f\parens[\big]{ \lambda x + (1-\lambda) y}
\frac{L}{2} \lambda (1-\lambda) \norm{y - x}_X^2
\ge
\lambda f(x) + (1-\lambda) f(y)
\quad\forall x,y \in O, \lambda \in (0,1)
\end{equation*}

The descent lemma holds

\begin{equation*}
f(y) \le f(x) + \dual{x\dualspace}{y - x}_X + \frac L2 \norm{y - x}_X^2
\qquad\forall x,y \in O, x\dualspace\in \partial f(x)
\end{equation*}

We have

\begin{equation*}
\dual{y\dualspace - x\dualspace}{y - x}_X \le L \norm{y - x}_{X}^2
\qquad \forall (x,x\dualspace), (y,y\dualspace) \in \graph \partial f
\end{equation*}

The subdifferential is Lipschitz continuous
(thus single-valued)

\begin{equation*}
\norm{y\dualspace - x\dualspace}_{X\dualspace} \le L \norm{y - x}_X
\qquad \forall (x,x\dualspace), (y,y\dualspace) \in \graph \partial f
\end{equation*}

The subdifferential is cocoercive

\begin{equation*}
\dual{y\dualspace - x\dualspace}{y - x}_X \ge \frac1L \norm{y\dualspace - x\dualspace}_{X\dualspace}^2
\qquad \forall (x,x\dualspace), (y,y\dualspace) \in \graph \partial f
\end{equation*}

We have

\begin{equation*}
f(y) \ge f(x) + \dual{x\dualspace}{y - x}_X + \frac 1{2 L} \norm{y\dualspace - x\dualspace}_{X\dualspace}^2
\qquad \forall (x,x\dualspace), (y,y\dualspace) \in \graph \partial f
\end{equation*}

Any of these conditions imply the Gâteaux differentiability of $f$ on $O$.
Moreover, the following relations hold.

We have

In case that $X$ is a Hilbert space,
we have additionally
This results in
and these conditions are implied by

In case that $O = X$, all assertions are equivalent.

The case <ref>
follows from
and case <ref>
is implied by <ref>.
shows that
implies the Gâteaux differentiability of $f$.
Thus, it remains to check the implications from <ref>.
For later reuse we recall that $\partial f(x) \ne \emptyset$
for all $x \in O$,
since $f$ is continuous on $O$,
see <cit.>.

We already know that $f$ is Gâteaux differentiable.
Moreover, <ref> gives

\begin{equation*}
\frac{f(y + \lambda (x - y)) - f(y)}{\lambda}
\frac L2 (1- \lambda) \norm{y - x}_X^2
\ge
f(x) - f(y).
\end{equation*}

for arbitrary $\lambda \in (0,1)$
and $\lambda \searrow 0$ results in

\begin{equation*}
\dual{f'(y)}{x - y}_X
\frac L2 \norm{y - x}_X^2
\ge
f(x) - f(y).
\end{equation*}

Since $\partial f(y) = \set{f'(y)}$,
this gives
with exchanged roles of $x$ and $y$.

We set $x_\lambda := \lambda x + (1-\lambda) y \in O$
and choose an arbitrary
$x_\lambda\dualspace \in \partial f(x_\lambda)$.

\[
\begin{aligned}
f( x ) &\le f(x_\lambda) + \dual{x_\lambda\dualspace}{x - x_\lambda}_X + \frac L2 \norm{x - x_\lambda}_X^2, \\
f( y ) &\le f(x_\lambda) + \dual{x_\lambda\dualspace}{y - x_\lambda}_X + \frac L2 \norm{y - x_\lambda}_X^2.
\end{aligned}
\]

We multiply the first inequality by $\lambda$
and the second one by $(1-\lambda)$.
Adding the resulting inequalities
and using
$x - x_\lambda = (1-\lambda) (x - y)$, $y - x_\lambda = \lambda (y - x)$,
we get

This follows from adding the inequality
with $(x,x\dualspace)$ and $(y,y\dualspace)$ exchanged.

We choose an arbitrary $n \in \N$ and define

\begin{equation*}
x_0 := x,
\quad
x_n := y,
\quad
x_i := x + \frac{i}{n} (y - x),
\quad
x_0\dualspace := x\dualspace,
\quad
x_n\dualspace := y\dualspace
\end{equation*}

Further we choose some arbitrary
$x_i\dualspace \in \partial f(x_i)$ for $i = 1,\ldots, n-1$.

\begin{align*}
&f(y) - f(x) - \dual{x\dualspace}{y - x}_X
f(x_n) - f(x_0) - \dual{x_0\dualspace}{x_n - x_0}_X
\\&\qquad
\sum_{i = 1}^n f(x_i) - f(x_{i-1}) - \dual{x_0\dualspace}{x_i - x_{i-1}}_X
\\&\qquad
\le
\sum_{i = 1}^n \dual{x_i\dualspace - x_0\dualspace}{x_i - x_{i-1}}_X
% \\&\qquad
\sum_{i = 1}^n \frac{1}{i} \dual{x_i\dualspace - x_0\dualspace}{x_i - x_0}_X
\\&\qquad
\le
L \sum_{i = 1}^n \frac{1}{i} \norm{x_i - x_0}_X^2
% \\&\qquad
L \sum_{i = 1}^n \frac{i}{n^2} \norm{x_1 - x_0}_X^2
L \frac{n+1}{2 n} \norm{y - x}_X^2
\end{align*}

Now, the claim follows from $n \to \infty$.

Since <ref> implies <ref>,
$f$ is differentiable on $O$.
For an arbitrary $x \in O$, there exists $\rho > 0$
with $B_\rho(x) := \set{y \in X \given \norm{y - x}_X \le \rho } \subset O$.
Now, let additionally $y \in B_\rho(x)$ with $x \ne y$ and $\varepsilon > 0$
be given.
By the Hahn-Banach theorem we get $z \in X$ with $\norm{z}_X = \norm{y - x}_X \le \rho$
$(1-\varepsilon) \norm{f'(y) - f'(x)}_{X\dualspace} \norm{z}_X \le \dual{f'(y) - f'(x)}{z}_X$.
Thus, $x + z \in B_\rho(x) \subset O$
and we have

\begin{align*}
&(1-\varepsilon) \norm{f'(y) - f'(x)}_{X\dualspace} \norm{z}_X
\\
&\qquad\le 0 + \dual{f'(y) - f'(x)}{z}_X
\\
\bracks*{f(x + z) - f(y) - \dual{f'(y)}{x+z-y}_X} + \dual{f'(y) - f'(x)}{z}_X
\\
\bracks{f(x + z) - f(x) - \dual{f'(x)}{z}_X}
\bracks{f(x) - f(y) - \dual{f'(y)}{x - y}_X}
\\
\frac L2 \norm{z}_X^2
\frac L2 \norm{ y - x }_X^2
L \norm{ y - x }_X^2
\end{align*}

Dividing by $\norm{z}_X = \norm{y-x}_X$
passing to the limit $\varepsilon \searrow 0$ yields

\begin{equation*}
\norm{f'(y) - f'(x)}_{X\dualspace}
\le
L\norm{y - x}_X
\end{equation*}

for all $x,y \in O$
such that $y \in B_\rho(x) \subset O$
with $\rho > 0$.

Now, if $x,y \in O$ are arbitrary,
there exists $\rho > 0$ with
$B_\rho(x),B_\rho(y) \subset O$.
We pick $n \in \N$ with $\norm{y - x}_X / n < \rho$
and define
$x_i := x + \frac{i}{n}(y - x)$, $i = 0,\ldots, n$.
By convexity we have $B_\rho(x_i) \subset O$
for all $i = 0,\ldots, n$.

\begin{equation*}
\norm{f'(y) - f'(x)}_{X\dualspace}
\le
\sum_{i = 1}^n \norm{f'(x_i) - f'(x_{i-1})}_{X\dualspace}
\le
\sum_{i = 1}^n L \norm{x_i - x_{i-1}}_{X}
L \norm{y - x}_X
\end{equation*}

and this shows <ref>.

This follows from adding the inequality
with $(x,x\dualspace)$ and $(y,y\dualspace)$ exchanged.

are straightforward.

* In the case
the missing implication
cannot hold due to the counterexample by <cit.>,
which is for the Euclidean case $X=\R^n$.

* In the general case,
it is currently not clear whether any of
the (equivalent) conditions
implies condition

The next lemma shows that the failure of
in case <ref>
is due to the missing convexity of the range of $f'$.
Since any of the properties in <ref> imply Gâteaux differentiability,
we will formulate all the following results for a Gâteaux differentiable function to simplify the presentation.

Let $O \subset X$ be an open and convex subset of the Banach space $X$.
Suppose that
the convex, lower semicontinuous and
Gâteaux differentiable function
$f \colon O \to \R$
<ref> <ref>,
$f'$ is cocoercive with constant $L > 0$.
Further, let $x,y \in O$ be given, such that

\begin{equation*}
[f'(x), f'(y)]
\set{
(1-\lambda) f'(x) + \lambda f'(y)
\given
\lambda \in [0,1]
\subset
\set{
\given
z \in O
\end{equation*}

\begin{equation*}
f(y) \ge f(x) + \dual{f'(x)}{y - x}_X + \frac 1{2 L} \norm{f'(y) - f'(x)}_{X\dualspace}^2
\end{equation*}

That is,
<ref> <ref>
holds on a subset of $O$.

Let $n \in \N$ be arbitrary
and define

\begin{equation}
\label{eq:xip}
x_i\dualspace :=
\parens*{
1 - \frac{i}{n}
} f'(x)
\frac{i}{n} f'(y)
\end{equation}

Due to our assumption, there exists $x_i \in O$, $i = 1,\ldots, n-1$,
such that $f'(x_i) = x_i\dualspace$ for all $i = 1, \ldots, n-1$.
We further set $x_0 := x$ and $x_n := y$.
Now, we have

\begin{align*}
&f(y) - f(x) - \dual{f'(x)}{y - x}_X
\sum_{i = 1}^n f(x_i) - f(x_{i-1}) - \dual{f'(x_0)}{x_i - x_{i-1}}_X
\\&\qquad
\ge
\sum_{i = 1}^n \dual{f'(x_{i-1}) - f'(x_0)}{x_i - x_{i-1}}_X
% \\&
\sum_{i = 1}^n (i-1) \dual{f'(x_i) - f'(x_{i-1})}{x_i - x_{i-1}}_X
\\&\qquad
\ge
\frac1L \sum_{i = 1}^n (i-1) \norm{f'(x_i) - f'(x_{i-1})}_{X\dualspace}^2
% \\&
\frac1{L n^2} \sum_{i = 1}^n (i-1) \norm{f'(y) - f'(x)}_{X\dualspace}^2
\\&\qquad
\frac{n (n-1)}{2 L n^2} \norm{f'(y) - f'(x)}_{X\dualspace}^2
\end{align*}

Passing to the limit $n \to \infty$ yields the claim.

Under slightly stronger assumptions on $f$ than in the previous lemma,  condition <ref> implies <ref>.

Let $O \subset X$ be an open and convex subset of the Banach space $X$.
Suppose that $f \colon O \to \R$ is convex,  lower semicontinuous, and Gâteaux differentiable.
Let $L>0$ be given such that <ref> <ref>, which is

\begin{equation*}
f(y) \le f(x) + \dual{f'(x)}{y - x}_X + \frac L2 \norm{y - x}_X^2
\qquad\forall x,y \in O
\end{equation*}

is satisfied.
Let $\rho>0$ and $x,y\in O_\rho$ be given with $O_\rho$ defined in (<ref>).

\[
[f'(x), f'(y)] \subset f'(O_\rho),
\]

\begin{equation}
\label{eq:yet_another_inequality}
\dual{f'(y) - f'(x)}{y - x}_X \ge \frac1L \norm{f'(y) - f'(x)}_{X\dualspace}^2
\end{equation}

which shows that <ref> <ref> holds on a subset of $O$.

First, we consider the case that
$x,y\in O_\rho$ are given
such that
$\norm{f'(y) - f'(x)}_{X\dualspace} < L\rho$.
Take $d\in X$ with $\norm{d}_X<\rho$. Then using the convexity and property <ref>,
we get

\begin{align*}
\dual{f'(y) - f'(x)}{d}_X
&= \dual{f'(y)-f'(x)}{y-x}_X + \dual{f'(y)-f'(x)}{d-y+x}_X \\
&\le \dual{f'(y)-f'(x)}{y-x}_X + f(x+d)-f(y) + f(y-d)-f(x) \\
& \le  \dual{f'(y)-f'(x)}{y-x}_X + \dual{f'(y)-f'(x)}{-d}_X + L \norm{d}_X^2.
\end{align*}

This implies

\[
\dual{f'(y)-f'(x)}{d}_X - \frac L2\norm{d}_X^2  \le \frac12\dual{f'(y)-f'(x)}{y-x}_X.
\]

For arbitrary $\varepsilon>0$, we can choose $d \in X$ with $\norm{d}_X = \frac1L\norm{f'(y) - f'(x)}_{X\dualspace}$ such that $\dual{f'(y)-f'(x)}{d}_X \ge \frac1L\norm{f'(y) - f'(x)}_{X\dualspace}^2 -\varepsilon$.
This implies

\[
\frac 1{2L}\norm{f'(y) - f'(x)}_{X\dualspace}^2 -\varepsilon  \le \frac12\dual{f'(y)-f'(x)}{y-x}_X.
\]

Since $\varepsilon>0$ was arbitrary, this shows (<ref>) in case that $\norm{f'(y) - f'(x)}_{X\dualspace}$ is small.

Now, let $x,y \in O_\rho$ be given such that
$[f'(x), f'(x)] \subset f'(O_\rho)$.
We use the same construction as in the proof of <ref>. Let a number $n\in \N$ be given such that
$\norm{f'(y) - f'(x)}_{X\dualspace} < L \rho n$.
Let $x_i\dualspace$ be given as in (<ref>)
and choose $x_i \in O_\rho$ with
$f'(x_i) = x_i\dualspace$ for all $i=1, \dots, n-1$.
We set $x_0=x$ and $x_n:=y$.
By construction, we have $\norm{f'(x_{i+1})-f'(x_i)}_{X\dualspace} = \frac1n \norm{f'(y) - f'(x)}_{X\dualspace} < L\rho$.
Then, the first part of the proof gives

\[
\frac1L \norm{f'(x_i) - f'(x_{i-1})}_{X\dualspace}^2 \le \dual{f'(x_i)-f'(x_{i-1})}{x_i-x_{i-1}}_X \quad \forall i=1,\dots, n.
\]

By construction, this implies

\[
\frac 1{Ln^2} \norm{f'(y) - f'(x)}_{X\dualspace}^2 \le \frac1{n}\dual{f'(y)-f'(x)}{x_i-x_{i-1}}_X \quad \forall i=1,\dots, n.
\]

Summation over $i=1,\dots, n$ yields the claim.

Missing 'biblatex' package
The bibliography requires the 'biblatex' package.

Springer Science
Business Media LLC

journaltitleIsrael Journal of Mathematics
titleQuelques propriétés des opérateurs angle-bornés et $n$-cycliquement monotones

journaltitleJournal of Convex Analysis
titleThe Baillon–Haddad Theorem Revisited

Berlin: Springer

isbn978-1-4419-9466-0/hbk; 978-1-4419-9467-7/ebook
journaltitleCMS Books in Mathematics/Ouvrages de Mathématiques de la SMC
titleConvex analysis and monotone operator theory in Hilbert spaces

journaltitleJournal of Convex Analysis
titleOn the Properties of Convex Functions over Open Sets

Springer Science
Business Media LLC

journaltitleApplied Mathematics & Optimization
titleAn Enhanced Baillon–Haddad Theorem for Convex Functions Defined on Convex Sets

World Scientific

titleConvex Analysis in General Vector Spaces

Missing 'biblatex' package
The bibliography requires the 'biblatex' package.

Springer Science
Business Media LLC

journaltitleIsrael Journal of Mathematics
titleQuelques propriétés des opérateurs angle-bornés et $n$-cycliquement monotones

journaltitleJournal of Convex Analysis
titleThe Baillon–Haddad Theorem Revisited

Berlin: Springer

isbn978-1-4419-9466-0/hbk; 978-1-4419-9467-7/ebook
journaltitleCMS Books in Mathematics/Ouvrages de Mathématiques de la SMC
titleConvex analysis and monotone operator theory in Hilbert spaces

journaltitleJournal of Convex Analysis
titleOn the Properties of Convex Functions over Open Sets

Springer Science
Business Media LLC

journaltitleApplied Mathematics & Optimization
titleAn Enhanced Baillon–Haddad Theorem for Convex Functions Defined on Convex Sets

World Scientific

titleConvex Analysis in General Vector Spaces