where we used $\widehat{\xi}_{N}(p)\leq\widehat{\xi}_{N}(0)$, $\ell\geq
2\mathfrak{a}_{N}$, $\sum_{p\in\Lambda_{+}^{*}\setminus
P_{\mathrm{B}}}\gamma(p)\lesssim N$ and (2.36). Similarly, we estimate the
term related to the second-to-last term on the right-hand side of (3.38) by

$2\widehat{\xi}_{N}(0)\operatorname{Tr}_{\mathfrak{F}}[\mathcal{N}_{+}\widetilde{\Gamma}_{0}]\leq
8\pi\mathfrak{a}_{N}\widetilde{N}_{0}\sum_{p\in\Lambda_{+}^{*}}\gamma_{p}+C\ell^{-1}.$
(3.46)

Finally, the last term on the right-hand side of (3.38) satisfies

$\widehat{\xi}_{N}(0)\int_{\mathbb{C}}\zeta(z)|z|^{4}\,\mathrm{d}z\leq
4\pi\mathfrak{a}_{N}\int_{\mathbb{C}}\zeta(z)|z|^{4}\,\mathrm{d}z+C\ell^{-1}.$
(3.47)

Combining (3.39), (3.40), (3.44), (3.45), (3.46) and (3.47), we find the
claim. ∎

We can now conclude the proof of Proposition 3.6. Inserting (3.36) into
(3.24), we find (3.19), with
$\mathcal{E}_{\mathcal{V}}=V_{1}+V_{2}+V_{3}+\mathcal{E}_{\Xi}$. The result
now follows from (3.25), (3.26) and (3.37). ∎

###  3.3 Final upper bound

We are now prepared to prove Proposition 3.1. Observing that
$\mathcal{K}+\mathcal{Q}_{\mathrm{B}}=\mathcal{H}_{\mathrm{B}}(z)+\mu_{0}\mathcal{N}_{+}$
and using (2.23), we find

$\begin{split}\operatorname{Tr}_{\mathfrak{F}}[\mathcal{K}\widetilde{\Gamma}_{0}]+\int_{\mathbb{C}}\zeta(z)\operatorname{Tr}_{\mathfrak{F}_{+}}[\mathcal{Q}_{\mathrm{B}}\widetilde{G}(z)]\,\mathrm{d}z=&\operatorname{Tr}_{\mathfrak{F}_{+}}[\mathcal{H}^{\mathrm{diag}}\widetilde{G}^{\mathrm{diag}}]+\mu_{0}\operatorname{Tr}_{\mathfrak{F}_{+}}[\mathcal{N}_{+}\widetilde{G}(z)]\\\
\leq&\operatorname{Tr}_{\mathfrak{F}_{+}}[\mathcal{H}^{\mathrm{diag}}G^{\mathrm{diag}}]+\mu_{0}\sum_{p\in\Lambda_{+}^{*}}\gamma_{p}+\mathcal{E}_{\mathrm{Bog}},\end{split}$
(3.48)

with

$\mathcal{E}_{\mathrm{Bog}}=(\kappa_{0}^{-1}-1)\operatorname{Tr}[(\mathcal{H}^{\mathrm{diag}}-E_{0})G^{\mathrm{diag}}]+\mu_{0}\big{(}\operatorname{Tr}_{\mathfrak{F}_{+}}[\mathcal{N}_{+}\widetilde{G}(z)]-\operatorname{Tr}_{\mathfrak{F}_{+}}[\mathcal{N}_{+}G(z)]\big{)}.$

The lowest eigenvalue $E_{0}<0$ of $\mathcal{H}_{\mathrm{B}}(z)$ has been
defined in (2.23). The bounds
$|p|^{2}-\mu_{0}\leq\varepsilon(p)\lesssim|p|^{2}-\mu_{0}$ and an application
of Lemma 2.4 show

$\operatorname{Tr}[(\mathcal{H}^{\mathrm{diag}}-E_{0})G^{\mathrm{diag}}]=\sum_{p\in\Lambda_{+}^{*}}\frac{\varepsilon(p)}{\exp(\beta\varepsilon(p))-1}\lesssim\sum_{p\in\Lambda_{+}^{*}}\frac{(|p|^{2}-\mu_{0})}{\exp(\beta(|p|^{2}-\mu_{0}))-1}\lesssim\beta^{-5/2}.$

In combination with Lemma 2.7, Lemma 2.10, and
$\mu_{0}=-\beta^{-1}\log(1+N_{0}^{-1})$, this implies

$\mathcal{E}_{\mathrm{Bog}}\lesssim\exp(-cN^{\delta_{\mathrm{Bog}}}),$ (3.49)

for some $c>0$. The claim thus follows by combining (3.48), (3.49), and
Propositions 3.2 and 3.6. ∎

##  4 Estimate for the entropy

The goal of this section is to prove the lower bound in Proposition 4.1 for
the entropy of the trial state $\Gamma$ defined in (2.14). It should be
compared to [13, Proposition 4.1] and to [52, Lemma 2]. We also refer to the
discussion in Remark 2.3 above. The main improvement is that we can estimate
the influence of correlations added to the eigenfunctions of the state
$|z\rangle\langle z|\otimes\widetilde{G}(z)$, while this has been possible in
[13] only for correlations added to those of $\widetilde{\Gamma}_{0}$. The
additional freedom we obtain by doing this is a crucial ingredient for our
analysis in Section 3. We recall that the assumptions stated at the beginning
of Section 3 also apply in this section.

######  Proposition 4.1.

The entropy of the state $\Gamma$ in (2.14) satisfies

$S(\Gamma)=S(\widetilde{G}^{\mathrm{diag}})+S^{\mathrm{cl}}(\zeta)+\mathcal{E}_{\mathrm{S}},$
(4.1)

with $\widetilde{G}^{\mathrm{diag}}$ in (2.4), $\zeta$ in (2.9),
$S^{\mathrm{cl}}$ defined below (1.21) and

$\mathcal{E}_{\mathrm{S}}\gtrsim-N\ell^{2}.$ (4.2)

###### Proof.

We define the function $\varphi(x)=-x\log(x)$ for $x\geq 0$ and we assume that
$\\{\xi_{k}\\}_{k}$ is an orthonormal basis of eigenfunctions of $\Gamma$.
This allows us to write

$\begin{split}S(\Gamma)=&\sum_{k}\varphi\big{(}\langle\xi_{k},\Gamma\xi_{k}\rangle\big{)}=\sum_{k}\varphi\Big{(}\int_{\mathbb{C}}\zeta(z)\sum_{\alpha\in\widetilde{\mathcal{A}}}\frac{\widetilde{\lambda}_{\alpha}}{\|F\phi_{z,\alpha}\|^{2}}\left|\langle\xi_{k},F\phi_{z,\alpha}\rangle\right|^{2}\,\mathrm{d}z\Big{)}.\end{split}$

Next, we define

$Z_{k}=\int_{\mathbb{C}}\sum_{\alpha\in\widetilde{\mathcal{A}}}\frac{\left|\langle\xi_{k},F\phi_{z,\alpha}\rangle\right|^{2}}{\|F\phi_{z,\alpha}\|^{2}}\,\mathrm{d}z$

and observe that (2.67) implies the bound

$\begin{split}Z_{k}\leq&(1+CN\ell^{2})\int_{\mathbb{C}}\sum_{\alpha\in\mathcal{A}}\left|\langle\xi_{k},F\phi_{z,\alpha}\rangle\right|^{2}\,\mathrm{d}z\\\
=&(1+CN\ell^{2})\int_{\mathbb{C}}\sum_{\alpha\in\mathcal{A}}\langle z\otimes
T_{z}\Psi_{\alpha},F\xi_{k}\rangle\langle F\xi_{k},z\otimes
T_{z}\Psi_{\alpha}\rangle\,\mathrm{d}z\\\
=&(1+CN\ell^{2})\int_{\mathbb{C}}\langle
z,\operatorname{Tr}_{\mathfrak{F}_{+}}[F|\xi_{k}\rangle\langle\xi_{k}|F]z\rangle=(1+CN\ell^{2})\operatorname{Tr}[F|\xi_{k}\rangle\langle\xi_{k}|F]\leq(1+CN\ell^{2}),\end{split}$
(4.3)

as long as $N\ell^{2}$ is sufficiently small. To come to the last line, we
used the fact that the set $\\{T_{z}\Psi_{\alpha}\\}_{\alpha}$ is an
orthonormal basis of $\mathfrak{F}_{+}$ for fixed $z\in\mathbb{C}$, the
completeness relation $\int_{\mathbb{C}}|z\rangle\langle
z|\,\mathrm{d}z=\mathds{1}_{\mathfrak{F}_{0}}$, and the bound $F\leq 1$ for
the multiplication operator $F$ on $\mathfrak{F}$.

Using the identity $\varphi(xy)=x\varphi(y)+y\varphi(x)$ for $x,y\geq 0$, we
can thus write

$\begin{split}S(\Gamma)=\sum_{k}Z_{k}\varphi\Big{(}\int_{\mathbb{C}}\sum_{\alpha\in\mathcal{A}}\zeta(z)\widetilde{\lambda}_{\alpha}\frac{\left|\langle\xi_{k},F\phi_{z,\alpha}\rangle\right|^{2}}{Z_{k}\|F\phi_{z,\alpha}\|^{2}}\,\mathrm{d}z\Big{)}+\sum_{k}\varphi\left(Z_{k}\right)\int_{\mathbb{C}}\sum_{\alpha\in\mathcal{A}}\zeta(z)\widetilde{\lambda}_{\alpha}\frac{\left|\langle\xi_{k},F\phi_{z,\alpha}\rangle\right|^{2}}{Z_{k}\|F\phi_{z,\alpha}\|^{2}}\,\mathrm{d}z.\end{split}$
(4.4)

With (4.3) we see that the second term on the right-hand side can be bounded
by

$\begin{split}-\sum_{k}\log(Z_{k})\int_{\mathbb{C}}\zeta(z)\sum_{\alpha\in\widetilde{\mathcal{A}}}&\widetilde{\lambda}_{\alpha}\frac{\left|\langle\psi_{k},F\phi_{z,\alpha}\rangle\right|^{2}}{\|F\phi_{z,\alpha}\|^{2}}\\\
\gtrsim&-N\ell^{2}\sum_{k}\int_{\mathbb{C}}\zeta(z)\sum_{\alpha\in\widetilde{\mathcal{A}}}\widetilde{\lambda}_{\alpha}\frac{\left|\langle\psi_{k},F\phi_{z,\alpha}\rangle\right|^{2}}{\|F\phi_{z,\alpha}\|^{2}}=-N\ell^{2},\end{split}$
(4.5)

where the equality in the second line follows from the fact that
$\\{\xi_{k}\\}_{k}$ is an orthonormal basis. Moreover, an application of
Jensen’s inequality to the strictly concave function $\varphi$ shows

$\begin{split}\sum_{k}Z_{k}&\varphi\Big{(}\int_{\mathbb{C}}\sum_{\alpha\in\widetilde{\mathcal{A}}}\zeta(z)\widetilde{\lambda}_{\alpha}\frac{\left|\langle\xi_{k},F\phi_{z,\alpha}\rangle\right|^{2}}{Z_{k}\|F\phi_{z,\alpha}\|^{2}}\,\mathrm{d}z\Big{)}\geq\sum_{k}Z_{k}\int_{\mathbb{C}}\sum_{\alpha\in\widetilde{\mathcal{A}}}\varphi(\zeta(z)\widetilde{\lambda}_{\alpha})\frac{\left|\langle\xi_{k},F\phi_{z,\alpha}\rangle\right|^{2}}{Z_{k}\|F\phi_{z,\alpha}\|^{2}}\,\mathrm{d}z\\\
=&\int_{\mathbb{C}}\sum_{\alpha\in\widetilde{\mathcal{A}}}\varphi(\widetilde{\lambda}_{\alpha}\zeta(z))\,\mathrm{d}z=\sum_{\alpha\in\widetilde{\mathcal{A}}}\varphi(\widetilde{\lambda}_{\alpha})+\int_{\mathbb{C}}\varphi(\zeta(z))\,\mathrm{d}z=S(\widetilde{G}^{\mathrm{diag}})+S^{\mathrm{cl}}(\zeta).\end{split}$
(4.6)

We insert (4.5) and (4.6) into (4.4) to conclude the proof. ∎

##  5 Proof of Theorem 1.1

Let us fix some $0<\varepsilon_{0}<1/12$. To prove (1.11) we show two distinct
upper bounds, corresponding to the two terms in the minimum appearing on the
right-hand side. The first upper bound is obtained with the trial state
$\Gamma$ in (2.14) and yields the term
$F^{\mathrm{BEC}}-8\pi\mathfrak{a}_{N}N_{0}^{2}$ in the minimum. It is only
valid in the parameter regime in which $N_{0}\gtrsim N^{2/3+\varepsilon_{0}}$
holds. A second bound that is valid for all $N_{0}\leq N$ (but useful only if
$N_{0}\ll N^{5/6}$) and contributes the term $F_{0}^{\mathrm{BEC}}$ in the
minimum will be obtained at the end of this section with a much simpler trial
state. Before we discuss these issues in more detail, we provide the final
upper bound for the free energy of $\Gamma$. We recall that for this part of
the analysis the assumptions stated at the beginning of Section 3 hold.

In combination, Proposition 3.1 and 4.1 imply the upper bound

$\begin{split}\mathcal{F}(\Gamma)=&\operatorname{Tr}[\mathcal{H}_{N}\Gamma]-\beta^{-1}S(\Gamma)\\\
\leq&\operatorname{Tr}[\mathcal{H}^{\mathrm{diag}}G^{\mathrm{diag}}]-\beta^{-1}S(\widetilde{G}^{\mathrm{diag}})+4\pi\mathfrak{a}_{N}\int_{\mathbb{C}}\zeta(z)|z|^{4}\,\mathrm{d}z+\beta^{-1}S^{\mathrm{cl}}(\zeta)+\mu_{0}\sum_{p\in\Lambda_{+}^{*}}\gamma_{p}\\\
&+8\pi\mathfrak{a}_{N}\Bigg{[}\Big{(}\sum_{p\in\Lambda_{+}^{*}}\gamma_{p}\Big{)}^{2}+\widetilde{N}_{0}\sum_{p\in\Lambda_{+}^{*}\setminus
P_{\mathrm{B}}}\gamma_{p}+\widetilde{N}_{0}\sum_{p\in\Lambda_{+}^{*}}\gamma_{p}\Bigg{]}+\mathcal{E}_{\mathcal{H}}-\beta^{-1}\mathcal{E}_{\mathrm{S}},\end{split}$
(5.1)

where $\mathcal{E}_{\mathcal{H}}$ and $\mathcal{E}_{\mathrm{S}}$ satisfy (3.2)
and (4.2), respectively.

We first remove the particle number cutoff from the entropy of the Bogoliubov
Gibbs state. As in Section 4 we denote $\varphi(x)=-x\log(x)$ for $x\geq 0$.
Using $\kappa_{0}=\sum_{\alpha\in\widetilde{\mathcal{A}}}\lambda_{\alpha}\leq
1$, we find

$\begin{split}S(\widetilde{G}^{\mathrm{diag}})=\sum_{\alpha\in\widetilde{\mathcal{A}}}\varphi(\widetilde{\lambda}_{\alpha})\geq&\sum_{\alpha\in\widetilde{\mathcal{A}}}\varphi(\lambda_{\alpha})+\varphi(\kappa_{0}^{-1})\sum_{\alpha\in\widetilde{\mathcal{A}}}\lambda_{\alpha}=S(G^{\mathrm{diag}})+\log(\kappa_{0})-\sum_{\alpha\in\mathcal{A}\setminus\widetilde{\mathcal{A}}}\varphi(\lambda_{\alpha}).\end{split}$
(5.2)

An application of (2.36) shows

$\log(\kappa_{0})\gtrsim-\exp(-cN^{\delta_{\mathrm{Bog}}}).$ (5.3)

Using the definition of $\lambda_{\alpha}$ above (2.10) and (2.37), we find

$\begin{split}\sum_{\alpha\in\mathcal{A}\setminus\widetilde{\mathcal{A}}}\varphi(\lambda_{\alpha})=&\sum_{\alpha\in\mathcal{A}\setminus\widetilde{\mathcal{A}}}\lambda_{\alpha}\log\Big{(}\sum_{\alpha^{\prime}\in\mathcal{A}}e^{-\beta
E_{\alpha^{\prime}}}\Big{)}+\beta\sum_{\alpha\in\mathcal{A}\setminus\widetilde{\mathcal{A}}}\lambda_{\alpha}E_{\alpha}\\\
\leq&\sum_{\alpha\in\mathcal{A}\setminus\widetilde{\mathcal{A}}}\lambda_{\alpha}\log\Big{(}\sum_{\alpha^{\prime}\in\mathcal{A}}e^{-\beta
E_{\alpha^{\prime}}}\Big{)}+C\exp(-cN^{\delta_{\mathrm{Bog}}}),\end{split}$

with $E_{\alpha}$ in (2.10). A standard computation shows that

$\begin{split}\log\Big{(}\sum_{\alpha\in\mathcal{A}}e^{-\beta
E_{\alpha}}\Big{)}=&-\sum_{p\in\Lambda^{*}_{+}}\log\left(1-\exp(-\beta\varepsilon(p))\right)\leq-\sum_{p\in\Lambda^{*}_{+}}\log\left(1-\exp(-\beta|p|^{2})\right)\\\
\lesssim&-\beta^{-3/2}\int_{0}^{\infty}(\beta+x^{2})\log(1-\exp(-x^{2}))\,\mathrm{d}x\lesssim
N.\end{split}$

The inequalities above follow from $\varepsilon(p)\geq|p|^{2}$, the fact that
$x\mapsto-\log(1-\exp(-x))$ is monotone decreasing for $x\geq 0$, and Lemma
2.4 with $\lambda=0$. In combination with (2.36), this implies

$\sum_{\alpha\in\mathcal{A}\setminus\widetilde{\mathcal{A}}}\lambda_{\alpha}\log\Big{(}\sum_{\alpha\in\mathcal{A}}e^{-\beta
E_{\alpha}}\Big{)}\lesssim\exp(-cN^{\delta_{\mathrm{Bog}}}).$ (5.4)

Inserting (5.3) and (5.4) into (5.2) yields

$S(\widetilde{G}^{\mathrm{diag}})\geq
S(G^{\mathrm{diag}})-C\exp(-cN^{\delta_{\mathrm{Bog}}}).$ (5.5)

We also have

$\operatorname{Tr}[\mathcal{H}^{\mathrm{diag}}G^{\mathrm{diag}}]-\beta^{-1}S(G^{\mathrm{diag}})=\beta^{-1}\sum_{p\in\Lambda_{+}^{*}}\log\big{(}1-e^{-\beta\varepsilon(p)}\big{)}.$
(5.6)

The following Lemma, which is proved in [13, Appendix B], provides us with an
asymptotic expansion of the term on the right-hand side of (5.6).

######  Lemma 5.1.

Let $\beta=\kappa\beta_{\mathrm{c}}$, with $\beta_{\mathrm{c}}$ defined in
(1.8) and $\kappa\in(0,\infty)$. There exists a constant $C>0$ such that, for
every $N$,

$\begin{split}\frac{1}{\beta}\sum_{p\in
P_{\mathrm{B}}}&\log\big{(}1-e^{-\beta\varepsilon(p)}\big{)}\\\
\leq&\frac{1}{\beta}\sum_{p\in
P_{\mathrm{B}}}\log\big{(}1-e^{-\beta(|p|^{2}-\mu_{0})}\big{)}+8\pi\mathfrak{a}_{N}N_{0}\sum_{p\in
P_{\mathrm{B}}}\frac{1}{e^{\beta(|p|^{2}-\mu_{0})}-1}\\\
&-\frac{1}{2\beta}\sum_{p\in\Lambda_{+}^{*}}\Bigg{[}\frac{16\pi\mathfrak{a}_{N}N_{0}}{|p|^{2}}-\log\Big{(}1+\frac{16\pi\mathfrak{a}_{N}N_{0}}{|p|^{2}}\Big{)}\Bigg{]}+\frac{CN_{0}^{2}}{N^{2}}\Big{(}N^{\delta_{\mathrm{Bog}}}+\frac{1}{\beta
N^{\delta_{\mathrm{Bog}}}}+\frac{1}{\beta^{2}N_{0}}\Big{)}.\end{split}$

An application of Lemma 5.1 shows

$\begin{split}\frac{1}{\beta}\sum_{p\in\Lambda_{+}^{*}}&\log\big{(}1-e^{-\beta\varepsilon(p)}\big{)}\leq\frac{1}{\beta}\sum_{p\in\Lambda_{+}^{*}}\log\big{(}1-e^{-\beta(|p|^{2}-\mu_{0})}\big{)}+8\pi\mathfrak{a}_{N}N_{0}\sum_{p\in
P_{\mathrm{B}}}\gamma^{\mathrm{id}}_{p}\\\
&-\frac{1}{2\beta}\sum_{p\in\Lambda_{+}^{*}}\Bigg{[}\frac{16\pi\mathfrak{a}_{N}N_{0}}{|p|^{2}}-\log\Big{(}1+\frac{16\pi\mathfrak{a}_{N}N_{0}}{|p|^{2}}\Big{)}\Bigg{]}+CN^{2/3-\delta_{\mathrm{Bog}}},\end{split}$
(5.7)

where we introduced the notation
$\gamma^{\mathrm{id}}_{p}=\big{(}\exp(\beta(|p|^{2}-\mu_{0}))-1\big{)}^{-1}$
for $p\in\Lambda_{+}^{*}$.

The sum of the third and fourth term on the right-hand side of (5.1) equals

$4\pi\mathfrak{a}_{N}\int_{\mathbb{C}}\zeta(z)|z|^{4}\,\mathrm{d}z+\beta^{-1}S^{\mathrm{cl}}(\zeta)=-\beta^{-1}\log\Big{(}\int_{|z|^{2}\leq\widetilde{N}}e^{-\beta(4\pi\mathfrak{a}_{N}|z|^{4}-\widetilde{\mu}|z|^{2})}\,\mathrm{d}z\Big{)}+\widetilde{\mu}\widetilde{N}_{0},$
(5.8)

with $\widetilde{N}_{0}$ defined in (2.74), and where we used the shorthand
notation $\widetilde{N}=\widetilde{c}N$. Extending the integral inside the
logarithm to the whole complex plane, we generate the error term

$-\beta^{-1}\log\Big{(}1-\frac{\int_{|z|^{2}>\widetilde{N}}e^{-\beta(4\pi\mathfrak{a}_{N}|z|^{4}-\widetilde{\mu}|z|^{2})}\,\mathrm{d}z}{\int_{\mathbb{C}}e^{-\beta(4\pi\mathfrak{a}_{N}|z|^{4}-\widetilde{\mu}|z|^{2})}\,\mathrm{d}z}\Big{)}.$

Setting $\eta=\widetilde{\mu}\sqrt{\beta/(4h)}$ and $A=\sqrt{\beta
h}\widetilde{c}N$, and computing the integrals in polar coordinates with the
change of variables $t=\sqrt{\beta h}|z|^{2}-\eta$, we find

$\frac{\int_{|z|^{2}>\widetilde{N}}e^{-\beta(4\pi\mathfrak{a}_{N}|z|^{4}-\widetilde{\mu}|z|^{2})}\,\mathrm{d}z}{\int_{\mathbb{C}}e^{-\beta(4\pi\mathfrak{a}_{N}|z|^{4}-\widetilde{\mu}|z|^{2})}\,\mathrm{d}z}=\frac{\int_{A-\eta}^{\infty}e^{-t^{2}}\,\mathrm{d}t}{\int_{-\eta}^{\infty}e^{-t^{2}}\,\mathrm{d}t}.$

We know from the proof of Lemma B.1 that $\eta<A/2$. If $0\leq\eta<A/2$, we
estimate the denominator from below by a constant and apply (B.5) to obtain a
bound for the numerator. This shows

$\frac{\int_{A-\eta}^{\infty}e^{-t^{2}}\,\mathrm{d}t}{\int_{-\eta}^{\infty}e^{-t^{2}}\,\mathrm{d}t}\lesssim\int_{A/2}^{\infty}e^{-t^{2}}\,\mathrm{d}t\lesssim\exp(-cN^{1/3}).$

In contrast, if $\eta<0$ we apply (B.5) in both the numerator and the
denominator, and find

$\frac{\int_{A-\eta}^{\infty}e^{-t^{2}}\,\mathrm{d}t}{\int_{-\eta}^{\infty}e^{-t^{2}}\,\mathrm{d}t}\lesssim
e^{-A^{2}+2A\eta}\lesssim\exp(-cN^{1/3}).$

Hence,

$\begin{split}-\beta^{-1}\log\Big{(}\int_{|z|^{2}\leq\widetilde{N}}&e^{-\beta(4\pi\mathfrak{a}_{N}|z|^{4}-\widetilde{\mu}|z|^{2})}\,\mathrm{d}z\Big{)}\\\
\leq&-\beta^{-1}\log\Big{(}\int_{\mathbb{C}}e^{-\beta(4\pi\mathfrak{a}_{N}|z|^{4}-\widetilde{\mu}|z|^{2})}\,\mathrm{d}z\Big{)}+Ce^{-cN^{1/3}}\\\
\leq&-\beta^{-1}\log\Big{(}\int_{\mathbb{C}}e^{-\beta(4\pi\mathfrak{a}_{N}|z|^{4}-\mu|z|^{2})}\,\mathrm{d}z\Big{)}-N_{0}(\widetilde{\mu}-\mu)+Ce^{-cN^{1/3}},\end{split}$
(5.9)

with $\mu$ in (1.13). In the last step, we used the fact that the first term
on the second line of (5.9) is a concave function of $\widetilde{\mu}$ and
that

$\frac{1}{\beta}\frac{\partial}{\partial\widetilde{\mu}}\log\Big{(}\int_{\mathbb{C}}e^{-\beta(4\pi\mathfrak{a}_{N}|z|^{4}-\widetilde{\mu}|z|^{2})}\,\mathrm{d}z\Big{)}\Big{|}_{\widetilde{\mu}=\mu}=N_{0}$

holds.

Inserting (5.5)–(5.9) into (5.1), we get

$\begin{split}\mathcal{F}(\Gamma)\leq&\frac{1}{\beta}\sum_{|p|\in\Lambda_{+}^{*}}\log(1-e^{-\beta(|p|^{2}-\mu_{0})})+\mu_{0}\sum_{p\in\Lambda_{+}^{*}}\gamma_{p}-\beta^{-1}\log\left(\int_{\mathbb{C}}e^{-\beta(4\pi\mathfrak{a}_{N}|z|^{4}-\mu|z|^{2})}\right)\\\
&+8\pi\mathfrak{a}_{N}\Bigg{[}N_{0}\sum_{p\in
P_{\mathrm{B}}}\gamma_{p}^{\mathrm{id}}+\Big{(}\sum_{p\in\Lambda_{+}^{*}}\gamma_{p}\Big{)}^{2}+\widetilde{N}_{0}\sum_{p\in\Lambda_{+}^{*}\setminus
P_{\mathrm{B}}}\gamma_{p}+\widetilde{N}_{0}\sum_{p\in\Lambda_{+}^{*}}\gamma_{p}\Bigg{]}\\\
&+\mu
N_{0}+\widetilde{\mu}(\widetilde{N}_{0}-N_{0})-\frac{1}{2\beta}\sum_{p\in\Lambda_{+}^{*}}\Bigg{[}\frac{16\pi
a_{N}N_{0}}{|p|^{2}}-\log\Big{(}1+\frac{16\pi
a_{N}N_{0}}{|p|^{2}}\Big{)}\Bigg{]}\\\
&+CN^{2/3-\delta_{\mathrm{Bog}}}+\mathcal{E}_{\mathcal{H}}-\beta^{-1}\mathcal{E}_{\mathrm{S}}.\end{split}$
(5.10)

With Lemma 2.1, Lemma 2.7 and Lemma 2.10, we infer that

$\sum_{p\in\Lambda_{+}^{*}}\gamma_{p}=N-\widetilde{N}_{0}+E,$

with $E\lesssim N^{5/3+\delta_{\mathrm{Bog}}}\ell^{2}+N^{3}\ell^{4}\lesssim
N$. Using this, we see that the terms inside the bracket on the second line of
(5.10) are bounded from above by

$\begin{split}2\widetilde{N}_{0}&(N-\widetilde{N}_{0})+(N-\widetilde{N}_{0})^{2}+N_{0}\sum_{p\in
P_{\mathrm{B}}}\gamma^{\mathrm{id}}_{p}-\widetilde{N}_{0}\sum_{p\in
P_{\mathrm{B}}}\gamma_{p}+CNE\\\
=&2\widetilde{N}_{0}(N-\widetilde{N}_{0})+(N-\widetilde{N}_{0})^{2}+\widetilde{N}_{0}(\widetilde{N}_{0}-N_{0})+(N_{0}-\widetilde{N}_{0})\sum_{p\in
P_{\mathrm{B}}}\gamma^{\mathrm{id}}_{p}+CNE\\\
=&N^{2}-N_{0}^{2}+\widetilde{N}_{0}(N_{0}-\widetilde{N}_{0})+(N_{0}-\widetilde{N}_{0})^{2}+(N_{0}-\widetilde{N}_{0})\sum_{p\in
P_{\mathrm{B}}}\gamma^{\mathrm{id}}_{p}+CNE.\end{split}$

Moreover, an application of Lemma 2.16 shows

$\begin{split}8\pi\mathfrak{a}_{N}\Big{[}(N_{0}-\widetilde{N}_{0})^{2}+(N_{0}-\widetilde{N}_{0})\sum_{p\in
P_{\mathrm{B}}}\gamma^{\mathrm{id}}_{p}+CNE\Big{]}\lesssim&N^{1/3+\delta_{\mathrm{Bog}}}+N^{5/3+\delta_{\mathrm{Bog}}}\ell^{2}+N^{3}\ell^{4}.\end{split}$

The above considerations show that the second line of (5.10) is bounded from
above by

$8\pi\mathfrak{a}_{N}(N^{2}-N_{0}^{2})+8\pi\mathfrak{a}_{N}\widetilde{N}_{0}(N_{0}-\widetilde{N}_{0})+C[N^{1/3+\delta_{\mathrm{Bog}}}+N^{5/3+\delta_{\mathrm{Bog}}}\ell^{2}+N^{3}\ell^{4}].$

Next, we consider the second term in the above equation and the second term on
the third line of (5.10). Let $\varepsilon\in(0,\varepsilon_{0})$. If
$N_{0}\gtrsim N^{5/6+\varepsilon}$, we infer from (B.1) in Appendix C that

$(8\pi\mathfrak{a}_{N}\widetilde{N}_{0}-\widetilde{\mu})(N_{0}-\widetilde{N}_{0})\lesssim
e^{-cN^{\varepsilon}}.$

On the other hand, if $N_{0}\lesssim N^{5/6+\varepsilon}$, Lemma 2.16, Lemma
B.1 and the lower bound $N_{0}\gtrsim N^{2/3+\varepsilon_{0}}$ imply

$|8\pi\mathfrak{a}_{N}\widetilde{N}_{0}(N_{0}-\widetilde{N}_{0})|\lesssim
N^{1/3+2\varepsilon}+N^{3/2+\delta_{\mathrm{Bog}}+\varepsilon}\ell^{2}+N^{17/6+\varepsilon}\ell^{4}$

and

$\begin{split}|\widetilde{\mu}(\widetilde{N}_{0}-N_{0})|\lesssim&\Big{(}\frac{1}{\widetilde{N}_{0}\beta}+\frac{1}{\sqrt{\beta
N}}+\frac{\widetilde{N}_{0}}{N}\Big{)}\Big{(}\frac{N_{0}}{N\beta}+\frac{N_{0}^{2}}{N^{2}}+N^{3}\ell^{4}+N^{5/3+\delta_{\mathrm{Bog}}}\ell^{2}\Big{)}\\\
\lesssim&N^{1/3+2\varepsilon}+N^{3}\ell^{4}+N^{5/3+\delta_{\mathrm{Bog}}}\ell^{2}.\end{split}$

With the bound $|\mu_{0}|=\beta^{-1}\log(1+N_{0}^{-1})\leq 1/(N_{0}\beta)$, we
see that

$\begin{split}\mu_{0}\sum_{p\in\Lambda_{+}^{*}}\gamma_{p}=&\mu_{0}(N-N_{0})+\mu_{0}(N_{0}-\widetilde{N}_{0}+E)\\\
\leq&\mu_{0}(N-N_{0})+C(N^{1/3}+N^{5/3+\delta_{\mathrm{Bog}}}\ell^{2}+N^{3}\ell^{4}).\end{split}$

We collect the above estimates, insert the bounds for
$\mathcal{E}_{\mathcal{H}}$ and $\mathcal{E}_{\mathrm{S}}$ in (3.2) and (4.2),
respectively, choose $\varepsilon$ sufficiently small, and find

$\begin{split}\mathcal{F}(\Gamma)\leq&\frac{1}{\beta}\sum_{|p|\in\Lambda_{+}^{*}}\log(1-e^{-\beta(|p|^{2}-\mu_{0})})+\mu_{0}(N-N_{0})-\frac{1}{\beta}\log\left(\int_{\mathbb{C}}e^{-\beta(4\pi
a_{N}|z|^{4}-\mu|z|^{2})}\right)+\mu N_{0}\\\
&+8\pi\mathfrak{a}_{N}(N^{2}-N_{0}^{2})-\frac{1}{2\beta}\sum_{p\in\Lambda_{+}^{*}}\Bigg{[}\frac{16\pi
a_{N}N_{0}}{|p|^{2}}-\log\Bigg{(}1+\frac{16\pi
a_{N}N_{0}}{|p|^{2}}\Bigg{)}\Bigg{]}\\\
&+C\Big{(}N^{2/3-\delta_{\mathrm{Bog}}}+N^{2/3+3\delta_{\mathrm{Bog}}}\ell^{1/2}+N^{5/3+3\delta_{\mathrm{Bog}}}\ell^{2}+N^{3}\ell^{4}+\ell^{-1}\Big{)}.\end{split}$
(5.11)

The error size is minimized by choosing $\delta_{\mathrm{Bog}}=1/18$,
$\ell=N^{-11/18}$, which yields the first upper bound.

To prove an upper bound that is valid in the non-condensed phase, we need a
different trial state. In this regime, the Bogoliubov corrections become
negligible compared to the size of the error term in Theorem 1.1. As undressed
trial state, we can thus take the Gibbs state of the ideal gas with an
appropriate cutoff for the number of particles:

$\widetilde{G}^{\mathrm{id}}=\frac{\mathds{1}_{\\{\mathcal{N}\leq\widetilde{c}N\\}}\exp(-\beta(\mathrm{d}\Gamma(-\Delta-\widetilde{\mu}_{0})))}{\operatorname{Tr}_{\mathfrak{F}}[\mathds{1}_{\\{\mathcal{N}\leq\widetilde{c}N\\}}\exp(-\beta(\mathrm{d}\Gamma(-\Delta-\widetilde{\mu}_{0})))]},$

where $\widetilde{c}>1$ is chosen independent of $N$. The chemical potential
$\widetilde{\mu}_{0}$ is uniquely determined by the condition
$\operatorname{Tr}_{\mathfrak{F}}[\mathcal{N}\widetilde{G}^{\mathrm{id}}]=N$.
The final trial state $\widetilde{\Gamma}$ is obtained by dressing
$\widetilde{G}^{\mathrm{id}}$ with the correlation structure $F$ in the same
way as we did in (2.14). Since the eigenfunctions of
$\widetilde{G}^{\mathrm{id}}$ can be chosen to be eigenfunctions of
$\mathcal{N}$, it is easy to check that, in contrast to the previous regime,
the correlation structure does not alter the expected number of particles in
the trial state. Computing the free energy of $\widetilde{\Gamma}$ is a
straightforward replication of Sections 3 and 4 in a simplified setting. We
therefore leave the details to the reader and only state the final result:

$\mathcal{F}(\widetilde{\Gamma})\leq
F_{0}(\beta,N)+8\pi\mathfrak{a}_{N}N^{2}+CN^{11/18}.$ (5.12)

Theorem 1.1 is a direct consequence of (5.11) and (5.12). To see this, let us
make the following remarks. Using Proposition 1.2 one easily checks that the
minimum in (1.11) is attained by the first term if $N_{0}\gtrsim
N^{5/6+\varepsilon}$ with $\varepsilon>0$. Since the term on the second line
of (1.11) is negative, it can be pulled out of the minimum in this parameter
regime. If $N_{0}\lesssim N^{5/6+\varepsilon}$ this term can also be pulled
out of the minimum, because it is bounded in absolute value by a constant
times $N^{1/3+2\varepsilon}$, which is smaller than our remainder term if
$\varepsilon$ is chosen sufficiently small. Moreover, a short computation that
uses Proposition 1.2 and the identity $\mu_{0}=-\ln(1+N_{0}^{-1})/\beta$ show
that the condition $N_{0}\lesssim N^{2/3+\varepsilon_{0}}$ with
$\varepsilon_{0}<1/12$ implies

$F^{\mathrm{BEC}}-8\pi\mathfrak{a}_{N}N_{0}^{2}\geq
F^{\mathrm{BEC}}_{0}-\frac{\mu_{0}}{2}+\mathcal{O}(N^{1/2}).$ (5.13)

Using additionally $\mu_{0}<0$, we thus find

$\min\\{F^{\mathrm{BEC}}-8\pi\mathfrak{a}_{N}N_{0}^{2},F^{\mathrm{BEC}}_{0}\\}=F^{\mathrm{BEC}}_{0}+\mathcal{O}(N^{1/2}).$

This explains why it is sufficient to prove the bound for the trial state
$\Gamma$ only in the parameter regime $N_{0}\gtrsim N^{2/3+\varepsilon_{0}}$,
concluding our proof of Theorem 1.1. ∎

Acknowledgments. A. D. gratefully acknowledges funding from the Swiss National
Science Foundation (SNSF) through the Ambizione grant PZ00P2 185851. M. C.
gratefully acknowledges support from the European Research Council through the
ERC–AdG CLaQS. It is our pleasure to thank Chiara Boccato, Phan Thành Nam,
Marcin Napiórkowski, Alessandro Olgiati and Benjamin Schlein for inspiring
discussions.

— Appendix —

##  Appendix A Properties of the solution to the scattering equation

In this appendix we recall some well-known properties of the solution $f(|x|)$
to the zero energy scattering equation and of $f_{\ell}(x)$ defined above and
in (2.12), respectively. In the whole section we assume that $V$ is a
nonnegative measurable and radial function that vanishes outside the ball with
radius $R>0$. All results that we state without proof can be found in [40,
Appendix C]333For general interaction potentials, the scattering equation in
[40, Theorem C.1] has to be understood in the sense of quadratic forms and not
in the sense of distributions as claimed. That is, functions used to test the
equation should be elements of the form domain of the energy functional
$\mathcal{E}_{R}$ (notation from the reference). All proofs in the reference
apply, with minor adjustments..

Let us introduce the energy functional

$\mathcal{E}_{\ell}[\phi]=\int_{B_{\ell}}\left(|\nabla\phi|^{2}+\frac{1}{2}V_{N}(x)|\phi(x)|^{2}\right)\mathrm{d}x$

with $\ell>R$. The function $f_{\ell}$ is the unique minimizer of
$\mathcal{E}_{\ell}$ among all $H^{1}$ functions $\phi$ satisfying the
boundary condition $\phi(x)=1$ for $|x|=\ell$, and its energy is given by

$\mathcal{E}_{\ell}[f_{\ell}]=\min_{\begin{subarray}{c}\phi\in
H^{1}(B_{\ell})\\\ \phi=1\text{ on }\partial
B_{\ell}\end{subarray}}\mathcal{E}_{\ell}[\phi]=\frac{4\pi\mathfrak{a}_{N}}{1-\mathfrak{a}_{N}/\ell}.$
(A.1)

Here $\mathfrak{a}_{N}<\ell$ is a positive number called the scattering length
of the potential $V_{N}$. It is easy to see that, by scaling,
$\mathfrak{a}_{N}=N^{-1}\mathfrak{a}$, where $\mathfrak{a}$ is the scattering
length of the unscaled potential $V$.

The function $f(|x|)$ is monotonically non-decreasing in $|x|$, and it is
bounded from above and from below by

$1\geq f(r)\geq 1-\frac{\mathfrak{a}_{N}}{r},$

with equality in the lower bound for $r\geq R$. This, in particular, implies

$1\geq
f_{\ell}(x)\geq\frac{(1-\mathfrak{a}_{N}/|x|)_{+}}{1-\mathfrak{a}_{N}/\ell}\geq(1-\mathfrak{a}_{N}/|x|)_{+}.$
(A.2)

We also need the following integral bounds on $f_{\ell}$.

######  Lemma A.1.

Let $u_{\ell}=1-f_{\ell}$. We have

$\int_{\mathbb{R}^{3}}u_{\ell}(x)\mathrm{d}x\lesssim\mathfrak{a}_{N}\ell^{2},\qquad\int_{\mathbb{R}^{3}}u_{\ell}(x)^{2}\mathrm{d}x\lesssim\mathfrak{a}_{N}^{2}\ell,\quad\text{
and }\quad\int_{\mathbb{R}^{3}}|\nabla
f_{\ell}(x)|\,\mathrm{d}x\lesssim\mathfrak{a}_{N}\ell.$ (A.3)

###### Proof.

With (A.2), we see that

$0\leq u_{\ell}(x)\leq\begin{cases}0&|x|\geq\ell,\\\
2\mathfrak{a}_{N}/|x|&\mathfrak{a}_{N}\leq|x|<\ell,\\\
1&|x|<\mathfrak{a}_{N},\end{cases}$

which implies the first two bounds in (A.3). The last bound follows from
$f^{\prime}\geq 0$ and one integration by parts:

$\begin{split}\int|\nabla
f_{\ell}|=&\frac{4\pi}{f(\ell)}\int_{0}^{\ell}f^{\prime}(r)r^{2}\mathrm{d}r=\frac{4\pi}{f(\ell)}\left[f(\ell)\ell^{2}-2\int_{0}^{\ell}f(r)r\mathrm{d}r\right]\\\
\leq&4\pi\ell^{2}-8\pi\int_{0}^{\ell}(1-\mathfrak{a}_{N}/r)_{+}r\mathrm{d}r=8\pi\mathfrak{a}_{N}\ell-4\pi\mathfrak{a}_{N}^{2}\leq
8\pi\mathfrak{a}_{N}\ell.\end{split}$

To come to the second line, we additionally used (A.2). ∎

##  Appendix B Properties of the effective functional for the condensate

We recall the definition of $\zeta$ in (2.9) with the cutoff parameter
$\widetilde{c}$. The following is an adaptation of [13, Lemma C.1].

######  Lemma B.1.

We assume that $\widetilde{c}>2$ and consider the combined limit $N\to\infty$,
$\beta=\kappa\beta_{\mathrm{c}}$ with $\kappa\in(0,\infty)$ and
$\beta_{\mathrm{c}}$ in (1.8). Let $1<\widetilde{b}<\widetilde{c}/2$ and
choose a sequence $M=M(N)\in\mathbb{R}$ such that $0<M<\widetilde{b}N$ for
every $N\in\mathbb{N}$. Then there exists a unique
$\widetilde{\mu}\in\mathbb{R}$ such that
$\int_{\mathbb{C}}|z|^{2}\zeta(z)\,\mathrm{d}z=M$, with $\zeta$ in (2.9).
Moreover, for every $\varepsilon>0$, there exists $c>0$ such that:

  1. 1.

If $M\gtrsim N^{5/6+\varepsilon}$, then

$\left|\widetilde{\mu}-8\pi\mathfrak{a}_{N}M\right|\lesssim
e^{-cN^{\varepsilon}}.$ (B.1)

  2. 2.

If $M\lesssim N^{5/6-\varepsilon}$, then

$\left|\widetilde{\mu}+\frac{1}{\beta
M}\right|\lesssim\frac{N^{-2\varepsilon}}{\beta M}.$ (B.2)

  3. 3.

For any $M=M(N)$, we have

$|\widetilde{\mu}|\lesssim\left(\frac{1}{M\beta}+\frac{1}{\sqrt{\beta
N}}+\frac{M}{N}\right).$ (B.3)

###### Proof.

Proceeding as in the proof of [13, Lemma C.1], we compute

$\sqrt{\beta
h}\int_{\mathbb{C}}\zeta(z)|z|^{2}\,\mathrm{d}z=\frac{1-\exp(-A^{2}+2A\eta)+\sqrt{\pi}\eta\exp(\eta^{2})\operatorname{erf}[-\eta,A-\eta]}{\sqrt{\pi}\exp(\eta^{2})\operatorname{erf}[-\eta,A-\eta]}\eqqcolon\Upsilon(\eta),$
(B.4)

where $h=4\pi\mathfrak{a}_{N}$, $\eta=\widetilde{\mu}\sqrt{\beta/(4h)}$,
$A=\widetilde{c}\sqrt{\beta h}N$ and

$\operatorname{erf}[a,b]=\frac{2}{\sqrt{\pi}}\int_{a}^{b}e^{-t^{2}}\,\mathrm{d}t$

with $a,b\in\mathbb{R}$. From [1, Eq. 7.1.21] we know that

$2\exp(x^{2})\int_{x}^{\infty}e^{-t^{2}}\,\mathrm{d}t=\frac{1}{x}-\frac{1}{2x^{3}}+Q(x),\qquad|Q(x)|\leq\frac{3}{4x^{5}},$
(B.5)

for $x>0$.

Using (B.5) we find

$\Upsilon(\eta)=\frac{\frac{1}{2\eta^{2}}+\eta
Q(-\eta)-\eta\exp(-A^{2}+2A\eta)\big{(}\frac{1}{\eta}+\frac{1}{A-\eta}-\frac{1}{2(A-\eta)^{3}}+Q(A-\eta)\big{)}}{-\frac{1}{\eta}+\frac{1}{2\eta^{3}}+Q(-\eta)-\exp(-A^{2}+2A\eta)\big{(}\frac{1}{A-\eta}-\frac{1}{2(A-\eta)^{3}}+Q(A-\eta)\big{)}}$
(B.6)

for $\eta<0$. This, in particular, implies $\Upsilon(\eta)\to 0$ in the limit
$\eta\to-\infty$, for fixed $A$. Moreover, a direct computation shows
$\Upsilon(A/2)=A/2$. With

$\frac{\partial}{\partial\widetilde{\mu}}\int_{\mathbb{C}}\zeta(z)|z|^{2}\,\mathrm{d}z=\beta\int_{\mathbb{C}}\zeta(z)\left(|z|^{2}-\left(\int_{\mathbb{C}}\zeta(w)|w|^{2}\,\mathrm{d}w\right)\right)^{2}\,\mathrm{d}z>0,$

we conclude that for $0<M<\widetilde{b}N$ there exists a unique solution
$\eta$ to the equation $\sqrt{\beta h}M=\Upsilon(\eta)$. In the following we
derive the asymptotic behavior of this solution for large $N$.

Let us first consider the case $M\lesssim N^{5/6-\varepsilon}$, where we have
$\Upsilon(\eta)\lesssim N^{-\varepsilon}$. By comparing this to

$\Upsilon(0)=\frac{1-\exp(-A^{2})}{2\int_{0}^{A}e^{-t^{2}}\,\mathrm{d}t}\to\pi^{-1/2}\quad\text{
as }A\to\infty,$ (B.7)

and using the monotonicity of $\Upsilon(\eta)$, we see that $\eta<0$, for
sufficiently large $N$. Thus, it follows from (B.6) that $\eta\to-\infty$ as
$N\to\infty$, and moreover

$\sqrt{\beta h}M=-\frac{1}{2\eta}+\mathcal{O}(|\eta|^{-3}),$

which immediately implies (B.2).

Let us now assume that $M\gtrsim N^{5/6+\varepsilon}$, in which case we have

$\Upsilon(\eta)=\sqrt{\beta h}M\gtrsim N^{\varepsilon}.$ (B.8)

From the assumption $\widetilde{b}<\widetilde{c}/2$ we know that

$\Upsilon(\eta)=\sqrt{\beta h}M<A/2=\Upsilon(A/2).$

In combination with (B.7) and the monotonicity of $\Upsilon(\cdot)$, this
allows us to conclude that $0<\eta<A/2$, for large $N$. An application of
(B.5) shows

$\Upsilon(\eta)=\eta+\frac{1-e^{-A(A-2\eta)}}{2e^{\eta^{2}}\sqrt{\pi}-\frac{1}{\eta}+\frac{1}{2\eta^{3}}-Q(\eta)-e^{-A(A-2\eta)}\big{(}\frac{1}{A-\eta}-\frac{1}{2(A-\eta)^{3}}+Q(A-\eta)\big{)}},$
(B.9)

which together with (B.8) implies $\eta\to+\infty$ as $N\to\infty$, provided
$M\gtrsim N^{5/6+\varepsilon}$. Using again $\eta<A/2$ and (B.9), we see that

$|\eta-\sqrt{\beta h}M|\lesssim\exp(-N^{2\varepsilon}),$

which is (B.1).

Finally, the bound (B.3) follows from the asymptotics of $\Upsilon$ for
$\eta\to\pm\infty$ and the fact that if $\eta$ is bounded, then
$\widetilde{\mu}\lesssim(\beta N)^{-1/2}$. ∎

##  Appendix C Estimate of the expected number of particles in $\Gamma$

This section is devoted to the proof of Lemma 2.1. We start by showing (2.15)
uniformly in $\widetilde{\mu}\in\mathbb{R}$. We will often need to remove
cutoffs from the expectation of observables on the Gibbs state
$\widetilde{G}^{\mathrm{diag}}$, producing errors that are exponentially small
in $N$. We will omit such errors because they can be absorbed in the remaining
error bounds. Since the proof is analogous to that of Lemma 3.7 we only sketch
it.

Using (2.69) and (3.23) without the restriction $(k,h)\neq(i,j)$ and expanding
the relevant numerator and denominator, we find

$\begin{split}\operatorname{Tr}[\mathcal{N}\Gamma]=&\operatorname{Tr}[\mathcal{N}\widetilde{\Gamma}_{0}]+\frac{1}{2}\int_{\mathbb{C}}\zeta(z)\sum_{\alpha\in\widetilde{\mathcal{A}}}\widetilde{\lambda}_{\alpha}\int_{\Lambda}\varrho_{z,\alpha}^{(1)}(x)\,\mathrm{d}x\int_{\Lambda^{2}}u_{12}\varrho_{z,\alpha}^{(2)}(x_{1},x_{2})\,\mathrm{d}x_{1}\,\mathrm{d}x_{2}\,\mathrm{d}z\\\
&-\frac{1}{2}\int_{\Lambda^{3}}u_{12}\varrho^{(3)}_{\Gamma_{0}}(x_{1},x_{2},x_{3})\,\mathrm{d}x_{1}\,\mathrm{d}x_{2}\,\mathrm{d}x_{3}+\mathcal{O}(N\ell^{2}+N^{3}\ell^{4}).\end{split}$
(C.1)

From (2.27) and (2.60) we know that

$\begin{split}\int_{\Lambda}\varrho_{z,\alpha}^{(1)}(x)\,\mathrm{d}x=&\langle\phi_{z,\alpha},\mathcal{N}\phi_{z,\alpha}\rangle=|z|^{2}+N_{\alpha}+\sum_{p\in
P_{\mathrm{B}}}v_{p}^{2}\Big{(}1+\langle\Psi_{\alpha},a_{p}^{*}a_{p}\Psi_{\alpha}\rangle\Big{)}.\end{split}$
(C.2)

Using (2.57), we see that the second term on the right-hand side of (C.1)
equals

$\frac{1}{2}\Big{(}\int
u_{\ell}\Big{)}\int_{\mathbb{C}}\zeta(z)\sum_{\alpha\in\widetilde{\mathcal{A}}}\widetilde{\lambda}_{\alpha}\int_{\Lambda}\varrho_{z,\alpha}^{(1)}(x)\,\mathrm{d}x\int_{\Lambda}\varrho_{z,\alpha}^{(2)}(x,x)\,\mathrm{d}x\,\mathrm{d}z+\mathcal{E}_{\mathrm{C}}^{(1)},$
(C.3)

with

$\begin{split}\mathcal{E}_{\mathrm{C}}^{(1)}=&\frac{1}{2}\int_{\mathbb{C}}\zeta(z)\sum_{\alpha\in\widetilde{\mathcal{A}}}\widetilde{\lambda}_{\alpha}\int_{\Lambda}\varrho_{z,\alpha}^{(1)}(x)\,\mathrm{d}x\int_{\Lambda^{2}}u_{\ell}(x_{2}-x_{1})\\\
&\times\int_{0}^{1}(x_{2}-x_{1})\cdot\nabla_{2}\varrho_{z,\alpha}^{(2)}(x_{1},x_{1}+t(x_{2}-x_{1}))\,\mathrm{d}t\,\mathrm{d}x_{1}\,\mathrm{d}x_{2}\,\mathrm{d}z.\end{split}$
(C.4)

It follows from (2.57), (A.3), (C.2) and the cutoff in
$\widetilde{\lambda}_{\alpha}$ that

$\begin{split}|\mathcal{E}_{\mathrm{C}}^{(1)}|\lesssim&\mathfrak{a}_{N}N\ell^{3}\int_{\mathbb{C}}\zeta(z)\sum_{\alpha\in\widetilde{\mathcal{A}}}\widetilde{\lambda}_{\alpha}\sup_{x_{1},x_{2}\in\Lambda}\big{|}\nabla_{2}\varrho_{z,\alpha}^{(2)}(x_{1},x_{2})\big{|}\mathrm{d}z\\\
\lesssim&\ell^{3}\Big{(}N^{3/2}\big{(}\operatorname{Tr}_{\mathfrak{F}_{+}}[\mathcal{K}\widetilde{G}^{\mathrm{diag}}]\big{)}^{1/2}+N^{4\delta_{\mathrm{Bog}}}(\beta^{-2}N^{2\delta_{\mathrm{Bog}}}+1)+N^{1+2\delta_{\mathrm{Bog}}}(1+\beta^{-1})\Big{)}\\\
\lesssim&\ell^{3}N^{2}(\beta^{-1/2}+1).\end{split}$ (C.5)

To come to the last line we also used the second bound in (2.32) and the
assumption $\delta_{\mathrm{Bog}}<1/12$.

The same arguments used to prove (2.56) show

$\varrho_{z,\alpha}^{(2)}(x,x)=|z|^{4}+4|z|^{2}N_{\alpha}+2N_{\alpha}(N_{\alpha}-1)+\mathcal{O}\Big{(}(|z|^{2}+N_{\alpha})(N_{\alpha}^{<}+N^{\delta_{\mathrm{Bog}}})+N^{3\delta_{\mathrm{Bog}}}(N_{\alpha}^{<}+1)^{2}\Big{)},$
(C.6)

uniformly in $x\in\Lambda$. This follows from the observation that, thanks to
the translation invariance of the eigenstates $\Psi_{\alpha}$, the phases in
the expansion (2.59) drop out when $x_{1}=x_{2}$. Inserting (C.6) into (C.3)
and using (A.3), (C.2), (C.4), (C.5) and Corollary 2.9, we see that the second
term on the right-hand side of (C.1) equals

$\begin{split}\frac{1}{2}\Big{(}\int
u_{\ell}\Big{)}&\int_{\mathbb{C}}\zeta(z)\Big{\\{}|z|^{6}+5|z|^{4}\operatorname{Tr}_{\mathfrak{F}_{+}}[\mathcal{N}_{+}G^{\mathrm{diag}}]+6|z|^{2}(\operatorname{Tr}_{\mathfrak{F}_{+}}[\mathcal{N}_{+}G^{\mathrm{diag}}])^{2}\\\
&+2(\operatorname{Tr}_{\mathfrak{F}_{+}}[\mathcal{N}_{+}G^{\mathrm{diag}}])^{3}\Big{\\}}\,\mathrm{d}z+\mathcal{O}\Big{(}\ell^{2}N^{1+\delta_{\mathrm{Bog}}}(\beta^{-1}+1)+\ell^{3}N^{2}(\beta^{-1/2}+1)+\ell^{4}N^{3}\Big{)}.\end{split}$
(C.7)

As for the term on the second line of (C.1), it is easy to see, using Lemma
2.15, that

$\int_{\Lambda^{3}}u_{12}\varrho^{(3)}_{\Gamma_{0}}(x_{1},x_{2},x_{3})\,\mathrm{d}x_{1}\,\mathrm{d}x_{2}\,\mathrm{d}x_{3}=\Big{(}\int
u_{\ell}\Big{)}\int_{\Lambda^{2}}\varrho^{(3)}_{\Gamma_{0}}(x_{1},x_{1},x_{2})\,\mathrm{d}x_{1}\,\mathrm{d}x_{2}+\mathcal{O}(N^{2}\ell^{3}\beta^{-1/2}).$

From here, using Wick’s theorem as in the proof of (3.35) we arrive at

$\begin{split}-\frac{1}{2}\int_{\Lambda^{3}}&u_{12}\varrho^{(3)}_{\Gamma_{0}}(x_{1},x_{2},x_{3})\,\mathrm{d}x_{1}\,\mathrm{d}x_{2}\,\mathrm{d}x_{3}\\\
=&-\frac{1}{2}\Big{(}\int
u_{\ell}\Big{)}\int_{\mathbb{C}}\zeta(z)\Big{\\{}|z|^{6}+5|z|^{4}\operatorname{Tr}_{\mathfrak{F}_{+}}[\mathcal{N}_{+}G^{\mathrm{diag}}]\\\
&+6|z|^{2}(\operatorname{Tr}_{\mathfrak{F}_{+}}[\mathcal{N}_{+}G^{\mathrm{diag}}])^{2}+2(\operatorname{Tr}_{\mathfrak{F}_{+}}[\mathcal{N}_{+}G^{\mathrm{diag}}])^{3}\Big{\\}}\,\mathrm{d}z+\mathcal{O}\Big{(}N^{2}\ell^{3}\beta^{-1/2}+N\ell^{2}\beta^{-1}\Big{)}.\end{split}$
(C.8)

Inserting (C.7) and (C.8) into (C.1), we get (2.15). It remains to prove the
existence statement of the lemma.

We denote

$M(\widetilde{\mu})=\int_{\mathbb{C}}|z|^{2}\zeta(z)\,\mathrm{d}z$

and observe that
$\operatorname{Tr}_{\mathfrak{F}_{+}}[\mathcal{N}_{+}\widetilde{G}(z)]$ is
independent of $z\in\mathbb{C}$, and that

$\operatorname{Tr}[\mathcal{N}\widetilde{\Gamma}_{0}]=M(\widetilde{\mu})+\operatorname{Tr}_{\mathfrak{F}_{+}}[\mathcal{N}_{+}\widetilde{G}(z)]$

holds. Using Lemmas 2.7, 2.10, 2.12 and Equation (2.54), it is easy to see
that

$\left|\operatorname{Tr}_{\mathfrak{F}_{+}}[\mathcal{N}_{+}\widetilde{G}(z)]-N+N_{0}\right|\lesssim\frac{N_{0}}{N\beta}+\frac{N_{0}^{2}}{N^{2}}.$

We can thus write

$\operatorname{Tr}[\mathcal{N}\Gamma]-N=M(\widetilde{\mu})-N_{0}+\left(\operatorname{Tr}[\mathcal{N}\widetilde{\Gamma}_{0}]-\operatorname{Tr}[\mathcal{N}\Gamma]\right)+\mathcal{O}\Big{(}\frac{N_{0}}{N\beta}+\frac{N_{0}^{2}}{N^{2}}\Big{)},$

which, for $\beta\gtrsim\beta_{\mathrm{c}}$, $\delta_{\mathrm{Bog}}<1/12$ and
$\ell\leq cN^{-7/12}$, for some sufficiently small $c>0$, yields

$M(\widetilde{\mu})-N_{0}-\Big{(}\frac{1}{2}+C\frac{N_{0}}{N}\Big{)}N^{2/3}\leq\operatorname{Tr}[\mathcal{N}\Gamma]-N\leq
M(\widetilde{\mu})-N_{0}+\Big{(}\frac{1}{2}+C\frac{N_{0}}{N}\Big{)}N^{2/3},$
(C.9)

for some constant $C>0$ independent of $\widetilde{\mu}$ and $N$. From the
proof of Lemma B.1 we know that

$\lim_{\widetilde{\mu}\to-\infty}M(\widetilde{\mu})=0,\qquad\lim_{\widetilde{\mu}\to+\infty}M(\widetilde{\mu})=\widetilde{c}N,$
(C.10)

for fixed $N\in\mathbb{N}$. Equations (C.9), (C.10) and the assumption
$N_{0}\geq N^{2/3}$ imply that $\operatorname{Tr}[\mathcal{N}\Gamma]-N$ takes
positive and negative values as a function of $\widetilde{\mu}$, for fixed $N$
large enough. The claim now follows from the continuity of the map

$\widetilde{\mu}\mapsto\operatorname{Tr}[\mathcal{N}\Gamma]=\int_{\mathbb{C}}\zeta(z)\sum_{\alpha\in\widetilde{\mathcal{A}}}\frac{\widetilde{\lambda}_{\alpha}}{\|F\phi_{z,\alpha}\|^{2}}\sum_{n=1}^{\infty}n\int_{\Lambda^{n}}\left|\big{(}F_{n}\phi_{z,\alpha}^{n}\big{)}\right|^{2}\,\mathrm{d}X^{n}\,\mathrm{d}z,$

which is a consequence of the dominated convergence theorem. ∎

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(Marco Caporaletti) Institute of Mathematics, University of Zurich
Winterthurerstrasse 190, 8057 Zurich, Switzerland
E-mail address<EMAIL_ADDRESS>

(Andreas Deuchert) Institute of Mathematics, University of Zurich
Winterthurerstrasse 190, 8057 Zurich, Switzerland
E-mail address<EMAIL_ADDRESS>


# A construction of minimal coherent filling pairs

Hong Chang and William W. Menasco  Hong Chang
Department of Mathematics
University at Buffalo–SUNY
Buffalo, NY 14260-2900, USA
<EMAIL_ADDRESS>William W. Menasco
Department of Mathematics
University at Buffalo–SUNY
Buffalo, NY 14260-2900, USA
<EMAIL_ADDRESS>
###### Abstract.

Let $S_{g}$ denote the genus $g$ closed orientable surface. A _coherent
filling pair_ of simple closed curves, $(\alpha,\beta)$ in $S_{g}$, is a
filling pair that has its geometric intersection number equal to the absolute
value of its algebraic intersection number. A _minimally intersecting_ filling
pair, $(\alpha,\beta)$ in $S_{g}$, is one whose intersection number is the
minimal among all filling pairs of $S_{g}$. In this paper, we give a simple
geometric procedure for constructing minimal intersecting coherent filling
pairs on $S_{g},\ g\geq 3,$ from the starting point of a coherent filling pair
of curves on a torus. Coherent filling pairs have a natural correspondence to
square-tiled surfaces, or origamis, and we discuss the origami obtained from
the construction.

##  1\. Introduction

A simple closed curve on a compact closed surface, $S_{g}$, of genus $g\geq 2$
is called _essential_ if it does not bound a disc. As such, going forward a
“curve in $S_{g}$” will mean an essential simple closed curve in $S_{g}$. Two
curves in $S_{g}$ intersect coherently if all the intersection points have the
same orientation provided that the two curves are oriented. Note that it does
not depend on the choice of the orientation of the curves. Two curves are in
_minimal position_ if the number of intersections of these curves is the
minimal within the curves isotopy classes. It is a simple observation that a
coherently intersecting pair are already intersecting minimally within their
isotopy classes. Thus, for coherently intersecting curves this convenience
allows us to drop the distinction between working with a curve pair and their
isotopy classes. A pair of curves in $S_{g}$ is _filling_ if over all
representatives from their isotopy classes their complement in $S_{g}$ is a
collection of discs.

Let $\alpha,\beta\subset S_{g}$ be a filling pair. We call $(\alpha,\beta)$ a
minimally intersecting filling pairs if the intersecting number,
$i(\alpha,\beta)$, is minimal among all filling pairs on $S_{g}$. For Euler
characteristics reasons, $i(\alpha,\beta)\geq 2g-1$. Additionally, for Euler
characteristic reasons a minimally intersecting filling pair would have the
property that $S_{g}\setminus(\alpha\cap\beta)$ is a single open disc. For
$g=1$ this low bound is geometrical realizable with a meridian-longitude pair.
For $g=2$, an exhaustive search the finite possibilities for a filling pair
intersecting $3$ times establishes that none exist and that at least $4$
intersections is needed. However, Aougab and Huang showed in [1] that for all
$g\geq 3$ there exists filling pairs of curves whose intersection achieves the
$2g-1$ minima. (More recently, also see [7, 8].) Moreover, the minima can be
obtained with $\alpha$ and $\beta$ intersecting coherently shown in [2], so
“minimally intersecting filling coherent pair” is not an empty set for $g\geq
3$.

The construction of minimally intersecting filling coherent pairs in [2]
largely utilizes the algebraic techniques coming from the symmetric groups. In
this paper we give an alternate geometric construction coming from simple cut-
and-paste techniques. Our construction allows one to rapidly construct by hand
such filling pairs for any genus.

###  1.1. Coherent filling pairs and origamis.

For a filling pair of curves, $\alpha,\beta\subset S_{g}$, positioned in their
isotopy classes so as to have $|\alpha\cap\beta|$ being minimal, the
$4$-valent graph, $\alpha\cup\beta=C^{1}\subset S_{g}$, can be thought of as
the $1$-skeleton of a $2$-dimensional CW structure on $S_{g}$. The dual graph,
$\widehat{C}^{1}\subset S_{g}$, is also the $1$-skeleton of a CW structure
that has every $2$-cell being a quadrilateral containing a single $0$-cell, an
intersection point from $\alpha\cap\beta(\subset C^{1})$. Giving each such
quadrilateral $2$-cell a $[0,1]\times[0,1]$ Euclidean square structure, we
obtain a square tiling of $S_{g}$. If we have $\alpha$ and $\beta$
intersecting coherently then an orientation assignment to these two curves
gives a natural way to assign the “right-side, left-side, bottom, top”
categories to the four boundary $1$-cells of each square tile. It follows that
each $1$-cell of $\widehat{C}^{1}$ is either a left/right side gluing or a
bottom/top gluing of two square tiles. That is, we have an origami structure
for $S_{g}$. Specifically, we have the following result from the literature.

######  Theorem 1 ([5, 7]).

A coherent filling pair of curves naturally corresponds to an origami on
$S_{g}$.

The Euclidean square tiles of an origami gives us a flat geometry except at
finitely many branched points—one for each $2$-cell of the CW structure—which
correspond to the points where the corners of the tiles are adjoined. An
origami coming from an intersecting coherent filling pair is said to have a
single _horizontal cylinder_ and a single _vertical cylinder_. Fig. 1 (Fig. 1
of [2]) illustrates such an origami for $S_{3}$ which happens to be the unique
origami up to labeling for genus three [4, 2].

Figure 1. The left illustrates a filling pair (which is $C^{1}$), the
associated square tiles, and the left/right bottom/top gluing assignment for
the origami structure. The right illustrates its geometric realization on
$S_{3}$. The left’s numeric labeling of red/blue edges of $C^{1}$ correspond
to the numeric labeling of the right. Note that there is exactly one branched
point since there is a single component of $S_{3}\setminus(\alpha\cup\beta)$.

Flat metrics on $S_{g}$ coming from the Euclidean square tilings on $S_{g}$
induces a horizontal foliation on each individual square tile that can be
extended to a measured singular foliation $\mathcal{F}$ on $S_{g}$. A
fundamental theorem of Hubbard-Masur associates a unique quadratic
differential to this horizontal foliation and, therefore, to the square-tiling
[6]. A origami of $S$ can thus be interpreted as a point in the space of unit-
area quadratic differentials. When $\mathcal{F}$ is orientable, the origami
supports the structure of a translation surface, and the associated quadratic
differential will be the square of an abelian differential.

###  1.2. Outline

In § 2, we introduce the cut-and-paste surgery operations that we will be
utilizing. These surgery operations have the property that, starting with a
coherent filling pair of curves on a torus, we will trade an increase in the
genus of the surface for a reduction in the number of discs in the complement
of the filling pair. A minimal intersecting filling pair will be realized when
the number of disc components is reduced to one.

Previous work on constructing and understanding minimal intersecting filling
pairs has focused on the growth of the number of nonequivalent pairs [1, 2, 3,
4]. As such, understanding how one might create nonequivalent minimal coherent
filling pairs via our surgeries is of interest. In § 3, we will use a “tree-
graph” analysis to understand how our surgery construction can yield
nonequivalent filling pairs.

Finally, in § 4 we generalize the construction to $S_{g,p}$, oriented surfaces
of genus $g$ with $p$ punctures. In the setting of punctures surfaces the
complement of a filling pair of curves is a collection of discs and once
punctured discs.

### Notation

Throughout this note we will use $S_{g}$ to denote a closed orientable surface
of genus, $g$. $S_{g,p}$ will denote an closed orientable surface of genus $g$
with $p$ marked or puncture points. $\partial Y$ denotes the boundary of a
compact surface, $Y$. $|X|$ denotes the cardinality of the set $X$.

### Acknowledgements

We thank the authors of [2] for the use of Fig. 1. This work has its genesis
in the second author’s collaboration with Tarik Aougab and Mark Nieland and he
thanks them for numerous discussion on the topic of this note.

##  2\. Surgeries on coherent filling pairs on $S_{1}$

Our strategy for constructing a minimal coherent intersecting filling pair for
a genus $g\geq 3$ oriented closed surface is to start with a coherent
intersecting filling pair of curves, $(\alpha,\beta)$, on the torus, $S_{1}$,
that intersect $g$-times. (Using the ordered $2$-tuple, $\langle
m,l\rangle(=\langle{\rm
meridian,longitude}\rangle)\in\mathbb{Z}\times\mathbb{Z}$ that specifies curve
isotopy classes on $S_{1}$, the reader might think of $\alpha\subset S_{1}$ as
being the $\langle 0,1\rangle$ curve and $\beta\subset S_{1}$ being the
$\langle g,1\rangle$ curve.) Our construction requires that we consider two
cases, when $g$ is odd and when $g$ is even. For the odd case, through a
simply surgery operation on the graph, $\alpha\cup\beta\subset S_{1}$, we will
add in $g-1$ new vertices. Initially, $S_{1}\setminus(\alpha\cup\beta)$ has
$g$ disc components. Our simple surgery on $\beta$ will trade disc components
for genus—each surgery decreases the number of disc components by one while
increasing the genus of the resulting surface by one. The genus of the final
surface will be $2g-1$ and there will be exactly one disc component in the
complement of the resulting filling pair, $(\alpha,\beta^{\prime})$, for
$S_{2g-1}$. For the case when $g$ is even, will need one additional simply
surgery to trade disc-for-genus all the way up to $2g-1$ genus.

###  2.1. Two simple surgeries on filling pairs

$\partial_{1}$ [tl] at 8 328 $p$ [tl] at 109 334 $\partial_{2}$ [t] at 15 360
$\partial_{4}$ [tl] at 225 328 $\partial_{3}$ [t] at 235 360

Figure 2. The first illustration of the sequence shows the extended core of
the band-to-be-added. Its endpoints are on $\alpha$. The second illustration
shows the added band. The third and fourth illustration in the sequence show
how the “shear” the intersection point in $\alpha\cap\beta$ and adjoin, or
“splice”, to the endpoints of the extended core of the band. The salient
feature is the $\partial_{1}$ and $\partial_{3}$ are band connected.

Let $(\alpha,\beta)$ be a filling pair for $S_{g\geq 1}$ and consider a closed
regular neighborhood, $\mathbf{N}$, of the graph, $\alpha\cup\beta\subset
S_{g}$. We assume that $\alpha$ and $\beta$ are positioned so as to intersect
minimally within the isotopy class of, say, $\beta$. As such, each boundary
component of $\partial\mathbf{N}$ bounds a disc in $S_{g}$. In particular, we
focus on any small neighborhood, $\nu(p)\subset\mathbf{N}$, around
$p\in\alpha\cap\beta$, one of the $4$-valent intersection points—the first
illustration in the sequence of Fig. 2 depicts $\nu(p)$. $\nu(p)$ will have
segment portions of four boundary components, $\partial_{1}$, $\partial_{2}$,
$\partial_{3}$ and $\partial_{4}$ as shown in first illustration in the
sequence of Fig. 2. We remark that some of $\partial_{i}{\rm{}^{\prime}s}$ may
be the same component of $\partial N$. Taking $\alpha$ near $p$ as a west/east
axis and $\beta$ as a north/south axis, the four boundary segment are
positioned so that $\partial_{1}$ is Southwest (SW), $\partial_{2}$ is NW,
$\partial_{3}$ is NE, and $\partial_{4}$ is SE.

The single $1$-handle surgery—We now glue to the sub-surface, $\mathbf{N}$, a
$1$-handle, $B(\cong[0,1]\times[0,1])$, that is attached to $\partial_{1}$
(SW) and $\partial_{3}$ (NE).

Referring to the second illustration in the sequence in Fig. 2, we take an
arc, $\gamma$, to be the extended core of the attached $B$. The salient
feature is that $\gamma$ is attached to the south side (north side) of the
west portion (east portion) of $\alpha\cap\nu$. Then
$\alpha\cup\beta\cup\gamma$ will be a graph in $\mathbf{N}\cup B$ that has
some number of $4$-valent vertices—same number as $|\alpha\cap\beta|$—and two
$3$-valent vertices—the two endpoints of $\gamma$.

The third illustration in the sequence in Fig. 2 show a shearing of $\beta$ at
the point $p$, creating two new $3$-valent vertices. The reader should observe
that we now have four $3$-valent vertices in succession on $\alpha$. The
fourth illustration shows how these four $3$-valence vertices are realigned
and spliced to create two new $4$-valence vertices and a new $\beta^{\prime}$.
The key feature of the final fourth illustration is that the orientation at
the two intersections of $\alpha\cap\beta^{\prime}$ created by this splice is
consistent with the original orientation intersection point,
$\nu\cap(\alpha\cap\beta)$—crossing $\alpha$ south to north.

We observe that if $\partial_{1}\not=\partial_{3}$ then the
$\partial(\mathbf{N}\cup B)$ has one less boundary component and the genus of
$\mathbf{N}\cup B$ is increased by one. Moreover, the curve pair
$(\alpha,\beta^{\prime})$, will be a filling pair in the surface obtains by
capping off each component of $\partial(\mathbf{N}\cup B)$ with a disc, i.e.
$S_{g+1}$. Additionally, $|\alpha\cap\beta|+1=|\alpha\cap\beta^{\prime}|$.

The surgery sequence obviously is generalized by rotation and reflection.

We will refer back to this shear and splice construction numerous times in
this note.

$\partial_{1}$ [tl] at -20 85 $p$ [tl] at 140 89 $\partial_{2}$ [t] at -12 110
$\partial_{4}$ [tl] at 240 85 $\partial_{3}$ [t] at 250 110

Figure 3. Banding three boundary components with one arc

The two $1$-handle surgery—For this surgery we refer the reader to Fig. 3 on
how we will alter the initial $\nu(p)$ neighborhood. Specifically, we glue in
two $1$-handles: a $1$-handle, $B_{NW/SW}$, that is attached to $\partial_{2}$
(NW) and $\partial_{1}$ (SW); and, a $1$-handle, $B_{NW/SE}$ attached to,
again, $\partial_{1}$ (SW) and $\partial_{3}$ (NE). Next, we take a core arc
of each $1$-handle and extend them into $\nu(p)$ so as to create a single arc,
$\gamma$, that is attached to $\alpha$ on the north (south) side of the west
(east) portion in $\nu(p)$. The blue arc in the left illustration of Fig. 2
corresponds to $\gamma$. Note that at this stage $\alpha\cup\beta\cup\gamma$
is a graph in $\mathbf{N}\cup B_{NW/SW}\cup B_{SW/NE}$ having
$|\alpha\cap\beta|+1$ $4$-valent vertices and two $3$-valent vertices.

Finally, we shear $\beta$ at the point $p\in\alpha\cap\beta$ to create two
$3$-valent vertices. As with our first surgery, we will then have four
$3$-valent vertices in succession on $\alpha$. The right illustration of Fig.
2 shows the realignment of these four vertices creating two new $4$-valent
vertices and a new $\beta^{\prime}$ curve by splicing into $\beta$ the
extended core arc. As with our first surgery, the two new vertices of
$\beta^{\prime}$ are intersections with $\alpha$ that are consistent with the
manner of intersection of our original point $p$—crossing $\alpha$ south to
north. Thus, again we have a shear and splice construction, going from $\beta$
to $\beta^{\prime}$.

If we assume that $\partial_{1},\partial_{2},\partial_{3}$ are all distinct
boundary curves of $\mathbf{N}$ then $|\partial(\mathbf{N}\cap B_{NW/SW}\cap
B_{SW/NE})|=|\partial\mathbf{N}|-2$. Thus, the curve pair
$(\alpha,\beta^{\prime})$, will be a filling pair in the surface obtains by
capping off each component of $\partial(\mathbf{N}\cap B_{NW/SW}\cap
B_{SW/NE}))$ with a disc, i.e. $S_{g+2}$. Additionally,
$|\alpha\cap\beta|+2=|\alpha\cap\beta^{\prime}|$

Finally, both the surgery sequences are generalized by rotation and
reflection.

###  2.2. Constructing minimal coherent filling pairs.

With our two surgeries in hand we are now in a position to construct minimal
coherent intersecting filling pairs for genus, $g\geq 3$.

As stated at the beginning of § 2, we start with a filling pair on $S_{1}$
that intersect $g$-times. Again, $\alpha$ is a $\langle 0,1\rangle$ curve and
$\beta$ as a $\langle g,1\rangle$ curve. We give an orientation to $\alpha$
and label the $g$ intersection points,
$\\{p_{1},p_{2},\cdots,p_{g}\\}=\alpha\cap\beta$, such that the cyclic order
of the points on $\alpha$ correspondence to the cyclic order given by indices
of the $\nu(p_{i})$-labels. Next we orient $\beta$ similarly—traversing
$\beta$, $mod(g)$ the $i^{\rm th}$ intersection point is $p_{i}$.

Our construction requires that we consider the cases when $g$ is odd and even
separately. The top illustration in Fig. 4 has $g=3$ and is representative of
the cases having $g$ odd. The bottom illustration of Fig. 4 has $g=4$ and is
representative of the cases having $g$ even.

$\alpha$ [t] at 45 278

$p_{1}$ [tl] at 103 272 $p_{2}$ [t] at 243 273 $p_{3}$ [t] at 372 273

$1$ [tl] at 113 366 $2$ [t] at 249 366 $3$ [t] at 380 366

$3$ [tl] at 113 193 $1$ [t] at 249 193 $2$ [t] at 380 193

$\alpha$ [t] at -2 87

$p_{1}$ [tl] at 70 81 $p_{2}$ [t] at 193 81 $p_{3}$ [tl] at 314 81 $p_{4}$ [t]
at 472 83

$1$ [tl] at 65 172 $2$ [t] at 202 172 $3$ [t] at 332 172 $4$ [t] at 461 172

$4$ [tl] at 64 2 $1$ [t] at 202 2 $2$ [t] at 332 2 $3$ [t] at 463 2

Figure 4. The top illustration has $g=3$ and is representative of the odd
case. The bottom illustration has $g=4$ and is representative of the even
case. The horizontal segments in each has the right/left endpoints identified
and corresponds to the $\alpha$ curve. The labels on the endpoints of the
vertical segments correspond to the identification of their endpoints so as to
form the $\beta$ curve.

Referring to Fig. 4, it is convenient to represent the $\alpha$ curve by a
horizontal line segment which has its left and right endpoints identified.
Then we can represent the $\beta$ curve by $g$ vertical line segment, each one
of which intersects our $\alpha$ representation once at its midpoint.
Assigning labels—$1$ through $g$, left to right—to the top endpoints of our
$g$ vertical segments and labels, $g$ then $1$ through $g-1$, to the bottom
ends of the vertical segments, we realize $\beta$ by a gluing that matches the
top endpoint labels with the bottom endpoint labels.

It is also helpful to assign labels, $p_{1}$ through $p_{g}$, to the points of
intersection of the horizontal $\alpha$ segment with the vertical
segments—$p_{i}$ will be in the vertical segment have $i$ as a top endpoint
label. Next, when we consider a regular neighborhood, $\mathbf{N}$ of
$\alpha\cup\beta\subset S_{1}$, near $p_{i}$ we have the four “compass”
boundary curves, $NE_{i},NW_{i},SW_{i},SE_{i}$, where, due to indexing scheme
for connecting the labels of the vertical segments,
$NE_{i}=NW_{i+1},SE_{i}=SW_{i+1},NW_{i}=SE_{i}$ To help to reader with this
identification in Fig. 4 we have distinguished the components of
$\partial\mathbf{N}$ by a color assignment. The reader should observe that
$|\partial\mathbf{N}|=g$.

The key issue is when a “$1$-handle attaching scheme” to $\partial\mathbf{N}$
results in a surface with one boundary. (In § 5 we will expand and make
precise our notion of “scheme”.) To that end we define an attaching graph or
A-graph, $G$. The vertices of $G$ are the components of $\partial\mathbf{N}$.
And, two vertices share an edge if they share the attaching ends of a
specified $1$-handle. Then $G$ will have $g$ vertices and $g-1$ edges. We then
have the following lemma.

######  Lemma 2.

Given an attaching scheme of $(g-1)$ $1$-handles to $\partial\mathbf{N}$, the
resulting surface will have exactly one boundary component if and only if the
A-graph, $G$, is a connected tree.

The proof of the following lemma will be delayed until § 3.

We now give two schemes—one for $g$ odd and one for $g$ even—attaching
$1$-handles to $\mathbf{N}$, both utilize the $1$-handle surgeries of § 2.1.

Case where $g$ is odd. In a neighborhood of each intersection point, $p_{i},\
2\leq i\leq g$, we perform a single $1$-handle surgery attached to ${SW}_{i}$
to ${NE}_{i}$ for $2\leq i\leq g$. It is readily observed that the graph, $G$,
is a linear tree. (The reader may wish to consult the top of Fig. 4.) Thus, by
Lemma 2 the resulting surface has one boundary component and is of genus
$(2g-1)$.

As previously observed, the resulting filling pair will still have coherent
intersection. $\square$

Case where $g$ is even. In a neighborhood of $p_{i}$ we perform a two
$1$-handel surgery: attaching a $1$-handle between ${NW}_{1}$ and ${SW}_{1}$;
and, ${NW}_{1}$ and ${SE}_{1}$. Then, in a neighborhood of each intersection
point, $p_{i},\ 4\leq i\leq g$, we perform a single $1$-handle surgery
attached to ${SW}_{i}$ to ${NE}_{i}$ for $4\leq i\leq g$. Again, it is readily
observed that the associated graph, $G$, is a linear tree. (The reader may
wish to consult the bottom of Fig. 4.) And, Lemma 2 again gives us that the
resulting surface has one boundary component and is of genus $(2g-1)$.

And again, the resulting filling pair will still have coherent intersection.
$\square$

Based upon the above two surgery schemes we can state the following result.

######  Theorem 3.

For genus $g\geq 3$, we can create minimal coherent filling pairs utilizing
the two $1$-handle surgeries of § 2.1.

By now the reader may have realized that there are other choices one may make
for attaching $1$-handles, shearing vertices and slicing in the extended
handles cores so as to obtain a single boundary curve and a new
$\beta^{\prime}$. In the next sections we investigate other such choices.

######  Remark 4.

For $g=2$, if we attempt to attach a single $1$-handles to
$\partial\mathbf{N}$ using the single $1$-handle surgery we “run out of room”.
That is, the associated graph, $G$, will not be a connected tree since both
points, $p_{1},p_{2}\subset\alpha\cup\beta\subset\mathbf{N}$, are adjacent to
just the two boundary curves of $\partial\mathbf{N}$. Thus, we cannot realize
a filling pair such that $\alpha\cap\beta=2\cdot g-1=2\cdot 2-1=3$. This
“failure to construct” is consistent with the fact the for genus $2$ we need
$|\alpha\cap\beta|=4$.

##  3\. $1$-handle attaching schemes and filling pair

We now supply the proof of our previously used lemma.

###### Proof of Lemma 3..

First, we will assume that the resulting surface has exactly one boundary
component. We will argue that $G$ must be a tree.

We observe that our definition of the graph $G$ is really independent of the
filling pair and only dependent on the surface type of $\mathbf{N}$. That is,
$\mathbf{N}$ is homeomophic to $S_{1,0,g}$, a compact surface of genus one
having no puncture (or marked points) and $g$ boundary components. By
attaching a $1$-handle to any surface with boundary we either increase by one
or decrease by one the number of boundary components of the resulting surface.
To do the former (latter), both ends of the $1$-handle must be attached to the
same (different) boundary component(s).

With the above in mind, we take a $\hat{g}$ to be of minimal value for which
the theorem is not true. Then for $\mathbf{N}\cong S_{1,o,\hat{g}}$, there is
an attaching scheme of $(\hat{g}-1)$ handle on the $\hat{g}$ boundary
components that produce a single boundary curve, but the associated A-graph is
not a tree.

We next take a maximal sub-collection of handles that results in an A-graph,
$G^{\prime}$, having each component of $G^{\prime}$ is a tree. (By assumption
this sub-collection has fewer than $(\hat{g}-1)$ handles.) The cardinality of
our sub-collection of handles is $(\hat{g}-|G^{\prime}|)$. And, there are
$|G^{\prime}|-1$ remaining handles to attach. Moreover, each component
accounts for one boundary component of the resulting sub-surface, i.e. there
are $|G^{\prime}|$ boundary component.

If we now attach a one of the handles not in our maximal sub-collection, it
must result in a component of our A-graph being not a tree. This implies that
both ends of this handle are attached to the same boundary component. But,
this is not possible. We are in the situation where we have a surface with
$|G^{\prime}|(<\hat{g})$ boundary curves and $|G^{\prime}|-1$ handles to be
attach resulting in a single boundary component. By the assumption that
$\hat{g}$ is minimal for realizing a counterexample, we have a contradiction.

For the other direction of our theorem we proceeding inductively. If there are
initially two components of $\partial\mathbf{N}$, attaching a $1$-handle
between them will produce a single boundary component and the associated $G$
is a tree. Now suppose we can attach

Now consider the associated graph, $G$, coming from an attachment scheme of
$(g-1)$ $1$-handles to the $g$ boundary components of $\mathbf{N}$. And,
assume that $G$ is a connected tree. We use $\mathbf{N}^{\prime}$ to denote
the resulting surface and we observe that $\mathbf{N}$ is naturally seen as a
sub-surface in $\mathbf{N}^{\prime}$. Moreover, we can obtain $\mathbf{N}$
from $\mathbf{N}^{\prime}$ by deleting the open sets in $\mathbf{N}^{\prime}$
that correspond to the “interior” of the $1$-handles—homeomorphically
equivalent to $(0,1)\times[0,1]$.

We then have a similar behavior to that described in the first half of our
argument. By deleting the interior of a $1$-handle to any surface with
boundary we either increase or decrease by one the number of boundary
components of the resulting surface. To do the former (latter), both
components of $\\{1{\rm-handle}\\}\cap\partial\mathbf{N}^{\prime}$ must on the
same (different) boundary component(s) of $\mathbf{N}^{\prime}$. Since the
deletion of $(g-1)$ interiors of $1$-handles in $\mathbf{N}^{\prime}$ produces
a surface with $g$ boundary components, by $G$ being connected—every boundary
of $\mathbf{N}$ has at least one $1$-handle attached—we conclude that
$|\partial\mathbf{N}^{\prime}|=1$. ∎

As previously observed, the $1$-handle attaching scheme of Theorem 3 is not
unique in that one can readily construct other $1$-handle attaching schemes
whose associated graph $G$ is a tree. In Fig. 5 we offer such an example.

$\gamma$ [t] at 59 160

$p_{1}$ [tl] at 17 120 $p_{2}$ [t] at 65 120 $p_{3}$ [tl] at 102 120 $p_{4}$
[t] at 150 120 $p_{5}$ [tl] at 187 120 $p_{6}$ [t] at 236 120

Figure 5. The arc, $\gamma$, contains the extended core of five $1$-handles.
There is only one shear and splice which is at $p_{2}$. It splices $\gamma$
into $\beta$ to produce $\beta^{\prime}$. The lower illustration depicts the
associated graph, $G$, which is a tree.

######  _Example_ 5 (An attaching scheme for $S_{11}$.).

Initially, the $\beta$ curve is a $(6,1)$ curve on $S_{1}$ with $\alpha$ again
being the $(0,1)$ curve. As before, we will have
$\\{p_{1},\cdots,p_{6}\\}=\alpha\cap\beta$ which we indicate in the top
illustration of Fig. 5. To reduce the clutter we do not depict the vertical
arcs associated with $\beta$. Finally, we double-label the boundary components
of $\partial\mathbf{N}$ with numeric-colored labels.

We depict an oriented blue arc, $\gamma$, that has endpoints in $\alpha$ to
the left and right of the point, $p_{2}(\subset\alpha\cap\beta)$. We will use
$\gamma$ for a scheme of attaching five $1$-handles to the components of
$\partial\mathbf{N}$. Specifically, $\gamma$ is the union of five extended
$1$-handled cores. These five extended cores have their endpoints in $\alpha$:
one to the right of $p_{1}$; one to the right of $p_{3}$; one to the left of
$p_{6}$; and, the two endpoints that are to the left and right of $p_{2}$.

The last two listed points, the endpoints of $\gamma$ are of particular
interest. They are positioned on $\alpha$ so that we can splice $\gamma$ into
$\beta$ by a shearing of $\beta$ at $p_{2}$ followed by a reconnecting of
endpoints as previously depicted in Fig.’s 2 & 3. The resulting curve will be
our new $\beta^{\prime}$. Returning to the orientation assignment of $\gamma$,
it is consistent with the orientation of $\beta$ that we have been
assigning—edges of $\beta$ are depicted as coming into $\alpha$ from below and
going out of $\alpha$ from above. Thus, the new $\beta^{\prime}$ will have
coherent intersection with $\alpha$.

The bottom illustration of Fig. 5 depicts the associated A-graph, $G$. (The
validity of $G$ we leave it to the reader to check.) Since $G$ is a tree by
our Lemma 2 we conclude that this scheme for attaching $1$-handles yields a
minimal coherent filling pair for $S_{11}$. Moreover, the extended cores of
the $1$-handles inherent an orientation from $\gamma$ that results in giving
each edge of $G$ an orientation, e.g. the core of the $1$-handle goes from the
$1$/red boundary curve to the $6$/black boundary curve giving us a edge in $G$
going from the $1$/red vertex to the $6$/black vertex. $\diamond$

From this example we see that an attaching scheme for a collection of
$1$-handles can be described by specifying a disjoint collection of oriented
arcs, $\\{\gamma_{1},\cdots,\gamma_{n}\\}$, that have their endpoints on our
initial $\alpha$ curve—a $\langle 0,1\rangle$ curve on $S_{1}$—and intersect
$\alpha$ is a coherent manner that is consistent with that of $\beta$—a
$\langle g,1\rangle$ curve on $S_{1}$. Then, $1$-handles are attached to
$\mathbf{N}$ so as to have their extended cores equal $\cup_{1\leq i\leq
n}\gamma_{i}$. Each $\gamma_{i}$ satisfies the following conditions. (We
continue our appeal to the setup: $\alpha$ and $\beta$ curves in $S_{1}$
coherently intersect and $\mathbf{N}$ is a regular neighborhood
$\alpha\cup\beta$. We also visually depict the setup in the same manner:
$\alpha$ a single horizontal arc with left/right endpoints identified; and,
$\beta$ as $g$ vertical segments oriented pointing bottom-to-top.)

  * 1.

For $\partial\gamma_{i}$ there exists an intersection point
$p\in\alpha\cap\beta$ such that on $\alpha$ these two endpoints are to the
immediate left/right of $p$.

  * 2.

$\gamma_{i}$ intersects $\alpha$ in a coherent manner.

  * 3.

$\gamma_{i}$ is attached to $\alpha$ and oriented such that a shear and splice
operation at the point $p$ (previous condition) yields a consistently oriented
curve coherently intersecting $\alpha$.

A collection of $\gamma$ arcs satisfying the above three conditions are said
to be a $1$-handle attaching scheme.

Performing the shear and splice operation for each $\gamma$ arc of a
$1$-handle attaching scheme will yield a curve pair,
$(\alpha,\beta^{\prime})$, for some oriented some closed surface.
Additionally, a corresponding A-graph, $G$, can be constructed.

We have the following theorem whose proof is now self-evident.

######  Theorem 6.

Let $\\{\gamma_{1},\cdots,\gamma_{n}\\}$ be a $1$-handle attaching scheme.
Suppose

$|\cup^{n}_{1}\gamma_{i}\cap\alpha|-n=g-1.$

Then the resulting curve pair is a minimal coherent filling pair for a $S_{g}$
if and only if the A-graph $G$ is a connected tree.

##  4\. Cases with punctures

We now extend our minimal coherent filling pair construction to orientable
finite type surfaces, $S_{g,p}$, where the genus is $g\geq 3$ and there are
$p(>0)$, punctures (or marked) points. A pair of curves,
$\bar{\alpha},\bar{\beta}\subset S_{g,p}$, is filling if, when $\bar{\alpha}$
and $\bar{\beta}$ are positioned to intersect minimally within their isotopy
classes, $S_{g,p}\setminus(\bar{\alpha}\cup\bar{\beta})$ is a collection of
discs and once punctured discs. By an Euler characteristic argument, the
minimal number of intersections needed for a pair of curves to fill is
$2g+p-2$ [7]. If $(\bar{\alpha},\bar{\beta})$ is a minimal filling pair then
$|S_{g,p}\setminus(\bar{\alpha}\cup\bar{\beta})|=p$. Alternatively, if we
consider a regular neighborhood,
$\mathbf{N}(\bar{\alpha}\cup\bar{\beta})\subset S_{g,o,p}$ we would have
$|\partial\mathbf{N}|=p$.

We slightly modify our initial setup of $\alpha,\beta\subset S_{1}$ by
requiring $|\alpha\cap\beta|=g+p-1$. Thus, $\beta$ is now a
$\langle(g+p-1),1\rangle$ curve on $S_{1}$ while $\alpha$ is still a $\langle
0,1\rangle$ curve. This implies that $S_{1}\setminus(\alpha\cup\beta)$ has
$(g+p-1)$ discs components. We again denote a regular neighborhood of
$\alpha\cup\beta$ by $\mathbf{N}(\subset S_{1})$.

We define a collection of $\gamma$ arcs, $\\{\gamma_{1},\cdots,\gamma_{n}\\}$,
as being a $1$-handle attaching scheme in exactly the same manner as that of §
3. As such we can consider the A-graph, $G$, of an attaching scheme. If
$|G|=p$ with each connected sub-graph component being a tree where exactly one
of its vertices corresponds to a $\partial_{i}$ boundary component of
$\mathbf{N}$, then the resulting curve pair, $(\alpha,\beta^{\prime})$, will
be filling. And additionally, if

$|\cup_{1}^{n}\gamma_{i}\cap\alpha|-n=g-1,$

the resulting pair, $(\alpha,\beta^{\prime})$, will be minimal, i.e.
$(g+p-1)+(g-1)=2g+p-2$. Since our third condition for an attaching scheme
requires that the shear and splice operation produce a $\beta^{\prime}$ that
coherently intersects $\alpha$, we have our conditions for constructing
minimal coherent filling pairs for $S_{g,p},g\geq 3,p>0$.

######  Theorem 7 (Also see [7], Theorem 1.3.).

Minimal coherent filling pairs for $S_{g,p},g\geq 3,p>0$, exist for all such
$g$ and $p$.

###### Proof.

Utilizing only the two $1$-handle surgeries of § 2, we take any $1$-handle
attaching scheme of $\gamma$ arcs that yields a minimal coherent filling pair
for $S_{g+p}$. We throw away any $p-1$ edges of the associated tree graph,
$G$, to produce a non-connected graph, $G^{\prime}$, which has $p$ sub-graphs.
We restrict our choices of discarded edges to those that correspond to single
$1$-handle surgery and the $B_{NW/SW}$ handle of the two $1$-handle surgery.
The reader should observe that throwing away the $B_{NW/SW}$ alters the two
$1$-handle surgery to a single $1$-handle surgery.

Thus, by construction there is a sub-collection of our original collection of
$\gamma$ arcs that yield an attaching scheme that producing a surface of
genus, $g$, with $p$ boundary components. Moreover, $G^{\prime}$ is the
associated A-graph and this surface will be a regular neighborhood of the
resulting coherent filling pair four valent graph, $\alpha\cup\beta^{\prime}$.
Now by capping off each of the $p$ boundary components with a once punctured
disc, we obtain $S_{g,p}$. Necessarily, $|\alpha\cap\beta^{\prime}|=2g+p-2$.

One last subtle observation. We can designate one arbitrary vertex from each
component of $G^{\prime}$ as corresponding to a one of the $p$ designated
boundary components, $\partial_{i}$. Placing a puncture in each associated
disc of $S_{1}\setminus(\alpha\cup\beta)$, we can then have this $S_{1,p}$
with its coherent filling pair, $(\alpha,\beta)$, as the initial starting
setup. The previous sub-collection of $\gamma$ arcs will then be an attaching
scheme that yields $(\alpha,\beta^{\prime})$ as a minimal coherent filling
pair in a $S_{g,p}$ surface. ∎

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