$\displaystyle\quad\quad+\int_{\mathcal{A}}\int_{\sigma^{-}(x)}^{\sigma^{+}(x)}\frac{\varepsilon}{2}\left((\overline{\mathbb{H}}^{\varepsilon})^{\prime}(a-tf_{s}(x))\right)^{2}\frac{(1-s)^{2}t^{2}}{r^{2}}|\nabla
h(x,a)|_{(x,a)}^{2}\theta(x,a)da\,d\mathcal{H}^{n}(x).$

Then, as $v_{\varepsilon}=v^{0,s}_{\varepsilon}$ for any $s\in[0,1]$, we have

$\displaystyle
2\sigma(\mathcal{F}_{\varepsilon,\lambda}(v^{t,s}_{\varepsilon})$
$\displaystyle-\mathcal{F}_{\varepsilon,\lambda}(v_{\varepsilon}))=\int_{M}\int_{\sigma^{-}(x)}^{\sigma^{+}(x)}\left[e_{\varepsilon}\left(\overline{\mathbb{H}}^{\varepsilon}(a-tf_{s}(x))\right)-e_{\varepsilon}\left(\overline{\mathbb{H}}^{\varepsilon}(a)\right)\right]\theta(x,a)da\,d\mathcal{H}^{n}(x)$
(37)
$\displaystyle\quad\quad\quad-\int_{M}\int_{\sigma^{-}(x)}^{\sigma^{+}(x)}\sigma\lambda\left[\overline{\mathbb{H}}^{\varepsilon}(a-tf_{s}(x))-\overline{\mathbb{H}}^{\varepsilon}(a)\right]\theta(x,a)da\,d\mathcal{H}^{n}(x)$
(38)
$\displaystyle\quad+\int_{\mathcal{A}}\int_{\sigma^{-}(x)}^{\sigma^{+}(x)}\frac{\varepsilon}{2}\left((\overline{\mathbb{H}}^{\varepsilon})^{\prime}(a-tf_{s}(x))\right)^{2}\frac{(1-s)^{2}t^{2}}{r^{2}}|\nabla
h(x,a)|_{(x,a)}^{2}\theta(x,a)da\,d\mathcal{H}^{n}(x)$ (39)

We now analyse the three terms above separately.

First, consider (5.2), by the Fundamental Theorem of Calculus and Fubini’s
Theorem we see that

$\displaystyle(\ref{eqn: 1d term})$
$\displaystyle=-\int_{M}\int_{\sigma^{-}(x)}^{\sigma^{+}(x)}\sigma\lambda\left[\overline{\mathbb{H}}^{\varepsilon}(a-tf_{s}(x))-\overline{\mathbb{H}}^{\varepsilon}(a)\right]\theta(x,a)da\,d\mathcal{H}^{n}(x)$
$\displaystyle=\int_{0}^{t}\int_{M}f_{s}(x)\int_{\sigma^{-}(x)}^{\sigma^{+}(x)}\sigma\lambda(\overline{\mathbb{H}}^{\varepsilon})^{\prime}(a-rf_{s}(x))\theta(x,a)da\,d\mathcal{H}^{n}(x)\,dr$

Second, consider (37), by the Fundamental Theorem of Calculus, Fubini’s
Theorem and integrating by parts then, by setting
$\theta(x,\sigma^{\pm}(x))=\lim_{a\rightarrow\sigma^{\pm}(x)}\theta(x,a)$ and
recalling (16), we obtain

$\displaystyle(\ref{eqn: energy term})$
$\displaystyle=\int_{M}\int_{\sigma^{-}(x)}^{\sigma^{+}(x)}\left[e_{\varepsilon}\left(\overline{\mathbb{H}}^{\varepsilon}(a-tf_{s}(x))\right)-e_{\varepsilon}\left(\overline{\mathbb{H}}^{\varepsilon}(a)\right)\right]\theta(x,a)da\,d\mathcal{H}^{n}(x)$
$\displaystyle=-\int_{0}^{t}\int_{M}f_{s}(x)\int_{\sigma^{-}(x)}^{\sigma^{+}(x)}e^{\prime}_{\varepsilon}\left(\overline{\mathbb{H}}^{\varepsilon}(a-rf_{s}(x))\right)\theta(x,a)da\,d\mathcal{H}^{n}(x)\,dr$
$\displaystyle=\int_{0}^{t}\int_{M}f_{s}(x)\int_{\sigma^{-}(x)}^{\sigma^{+}(x)}e_{\varepsilon}(\overline{\mathbb{H}}^{\varepsilon}(a-rf_{s}(x)))\partial_{a}\theta(x,a)da\,d\mathcal{H}^{n}(x)\,dr$
$\displaystyle\quad\quad-\int_{0}^{t}\int_{M}f_{s}(x)e_{\varepsilon}\left(\overline{\mathbb{H}}^{\varepsilon}(\sigma^{+}(x)-rf_{s}(x))\right)\theta(x,\sigma^{+}(x))da\,d\mathcal{H}^{n}(x)\,dr$
$\displaystyle\quad\quad\quad+\int_{0}^{t}\int_{M}f_{s}(x)e_{\varepsilon}\left(\overline{\mathbb{H}}^{\varepsilon}(\sigma^{-}(x)-rf_{s}(x))\right)\theta(x,\sigma^{-}(x))da\,d\mathcal{H}^{n}(x)\,dr$
$\displaystyle=\int_{0}^{t}\int_{M}f_{s}(x)\int_{\sigma^{-}(x)}^{\sigma^{+}(x)}e_{\varepsilon}(\overline{\mathbb{H}}^{\varepsilon}(a-rf_{s}(x)))(\lambda-H(x,a))\theta(x,a)da\,d\mathcal{H}^{n}(x)\,dr$
$\displaystyle\quad-\int_{0}^{t}\int_{M}f_{s}(x)\int_{\sigma^{-}(x)}^{\sigma^{+}(x)}\lambda
e_{\varepsilon}(\overline{\mathbb{H}}^{\varepsilon}(a-rf_{s}(x)))\theta(x,a)da\,d\mathcal{H}^{n}(x)\,dr$
$\displaystyle\quad\quad-\int_{0}^{t}\int_{M}f_{s}(x)e_{\varepsilon}\left(\overline{\mathbb{H}}^{\varepsilon}(\sigma^{+}(x)-rf_{s}(x))\right)\theta(x,\sigma^{+}(x))da\,d\mathcal{H}^{n}(x)\,dr$
$\displaystyle\quad\quad\quad+\int_{0}^{t}\int_{M}f_{s}(x)e_{\varepsilon}\left(\overline{\mathbb{H}}^{\varepsilon}(\sigma^{-}(x)-rf_{s}(x))\right)\theta(x,\sigma^{-}(x))da\,d\mathcal{H}^{n}(x)\,dr.$

Note that in the last equality above we added and subtracted the term

$\int_{0}^{t}\int_{M}f_{s}(x)\int_{\sigma^{-}(x)}^{\sigma^{+}(x)}\lambda
e_{\varepsilon}(\overline{\mathbb{H}}^{\varepsilon}(a-rf_{s}(x)))\theta(x,a)da\,d\mathcal{H}^{n}(x)\,dr$

in order to introduce the quantity $\lambda-H(x,a)$ to the calculation; thus
we consider the term

$\int_{0}^{t}\int_{M}f_{s}(x)\int_{\sigma^{-}(x)}^{\sigma^{+}(x)}e_{\varepsilon}(\overline{\mathbb{H}}^{\varepsilon}(a-rf_{s}(x)))(\lambda-H(x,a))\theta(x,a)da\,d\mathcal{H}^{n}(x)\,dr,$
(40)

which we now control by the assumption of positive Ricci curvature.

By the continuity of the $\sigma^{\pm}$ on $M$, Subsection 3.2 and (29), we
may fix a compact $L\subset(M\setminus B_{4})$, $l>0$ sufficiently small so
that $|\sigma^{\pm}(x)|>l$ for $x\in L$,

$\mathcal{H}^{n}(\\{(x,l)\in V_{M}\>|\>x\in
L\\})>\frac{\mathcal{H}^{n}(M)}{2},$ (41)

and so that for some $t_{0}\leq\min\\{t_{1},2d(N)\\}$ we ensure
$\theta(x,a)\geq\theta(x,l)$ for all $a\in[-t_{0},t_{0}]$. Note that as $L\cap
B_{4}=\emptyset$ we have that $f(x)=1$ for all $x\in L$.

By (18) (relying on the positive Ricci curvature assumption) and the above
paragraph we thus have, for $a\in[-t_{0},t_{0}]$ and $x\in L$ that

$\begin{cases}(\lambda-H(x,a))\theta(x,a)\leq-m\,a\,\theta(x,l)\text{ if
}a\in[0,t_{0}]\\\ (\lambda-H(x,a))\theta(x,a)\geq-m\,a\,\theta(x,l)\text{ if
}a\in[-t_{0},0].\end{cases}$

With the above inequalities and (41) we compute for $t\in[-t_{0},t_{0}]$ it
holds that, using (25) and the fact that
$e_{\varepsilon}(\overline{\mathbb{H}}^{\varepsilon}(a-r))\neq 0$ only for
$a\in(r-2\varepsilon\Lambda_{\varepsilon},r+2\varepsilon\Lambda_{\varepsilon})$,
we have

$\displaystyle(\ref{eqn: underlying geometry term})$
$\displaystyle\leq-\mathcal{E}_{\varepsilon}(\overline{\mathbb{H}}^{\varepsilon})\left[\frac{m\sigma}{2}\mathcal{H}^{n}(M)t^{2}-2\varepsilon\Lambda_{\varepsilon}m\sigma\mathcal{H}^{n}(M)\right]$
$\displaystyle\leq-(1-\beta\varepsilon^{2})\left[\frac{m\sigma}{2}\mathcal{H}^{n}(M)t^{2}-2\varepsilon\Lambda_{\varepsilon}m\sigma\mathcal{H}^{n}(M)\right].$

Third, we focus on (39). Using (20), (25), (31) (32) and noting that
$s\in[0,1]$, we see that

$(\ref{eqn: gradient term})\leq 2\sigma
t^{2}\left(\frac{\mathcal{H}^{n}(\mathcal{A})C^{2}_{h}e^{\frac{\lambda^{2}}{2m}}}{r^{2}}\right)\mathcal{E}_{\varepsilon}(\overline{\mathbb{H}}^{\varepsilon})\leq\frac{m\sigma}{4}\mathcal{H}^{n}(M)t^{2}(1-\beta\varepsilon^{2}).$

We now simplify the terms as computed above and see that for
$t\in[-t_{0},t_{0}]$ we have

$\displaystyle
2\sigma(\mathcal{F}_{\varepsilon,\lambda}(v^{t,s}_{\varepsilon})-\mathcal{F}_{\varepsilon,\lambda}(v_{\varepsilon}))$
$\displaystyle\leq-(1-\beta\varepsilon^{2})\left[\frac{m\sigma}{4}\mathcal{H}^{n}(M)t^{2}-2\varepsilon\Lambda_{\varepsilon}m\sigma\mathcal{H}^{n}(M)\right]$
(42)
$\displaystyle\quad+\int_{0}^{t}\int_{M}f_{s}(x)\int_{\sigma^{-}(x)}^{\sigma^{+}(x)}\sigma\lambda(\overline{\mathbb{H}}^{\varepsilon})^{\prime}(a-rf_{s}(x))\theta(x,a)da\,d\mathcal{H}^{n}(x)\,dr$
(43)
$\displaystyle\quad-\int_{0}^{t}\int_{M}f_{s}(x)\int_{\sigma^{-}(x)}^{\sigma^{+}(x)}\lambda
e_{\varepsilon}(\overline{\mathbb{H}}^{\varepsilon}(a-rf_{s}(x)))\theta(x,a)da\,d\mathcal{H}^{n}(x)\,dr$
(44)
$\displaystyle\quad\quad-\int_{0}^{t}\int_{M}f_{s}(x)e_{\varepsilon}\left(\overline{\mathbb{H}}^{\varepsilon}(\sigma^{+}(x)-rf_{s}(x))\right)\theta(x,\sigma^{+}(x))da\,d\mathcal{H}^{n}(x)\,dr$
(45)
$\displaystyle\quad\quad+\int_{0}^{t}\int_{M}f_{s}(x)e_{\varepsilon}\left(\overline{\mathbb{H}}^{\varepsilon}(\sigma^{-}(x)-rf_{s}(x))\right)\theta(x,\sigma^{-}(x))da\,d\mathcal{H}^{n}(x)\,dr.$
(46)

The objective now is to show that the sum of the terms (43) to (46) are errors
uniformly small in $\varepsilon$.

First we treat (43) and (44) together. We define

$m_{\varepsilon}(rf_{s}(x),x)=\max_{a\in[rf_{s}(x)-2\varepsilon\Lambda_{\varepsilon},rf_{s}(x)+2\varepsilon\Lambda_{\varepsilon}]}\theta(x,a)-\min_{a\in[rf_{s}(x)-2\varepsilon\Lambda_{\varepsilon},rf_{s}(x)+2\varepsilon\Lambda_{\varepsilon}]}\theta(x,a)$

so that by (20), $0\leq
m_{\varepsilon}(rf_{s}(x),x)\leq\max_{a\in\mathbb{R}}\theta(x,a)\leq
e^{\frac{\lambda^{2}}{2m}}$ and $m_{\varepsilon}(rf_{s}(x),x)\rightarrow 0$ as
$\varepsilon\rightarrow 0$. Hence, by the Dominated Convergence Theorem we see
that

$M_{\varepsilon}(t)=2\sigma\lambda\int_{M}\int_{0}^{t}m_{\varepsilon}(rf_{s}(x),x)d\mathcal{H}^{n}(x)\,dr\rightarrow
0\text{ as }\varepsilon\rightarrow 0.$

Noting that for $0<\tilde{\varepsilon}<\varepsilon$ we have that $0\leq
m_{\tilde{\varepsilon}}(rf_{s}(x),x)\leq m_{\varepsilon}(rf_{s}(x),x)$ we may
apply Dini’s Theorem (as the $M_{\varepsilon}(t)$ are increasing/decreasing in
$\varepsilon$ for positive/negative values of $t\in\mathbb{R}$ respectively)
to the functions $M_{\varepsilon}(t)$ to conclude that

$M_{\varepsilon}\rightarrow 0\text{ uniformly on compact subsets of
}\mathbb{R}$ (47)

Using the above we compute that for (43) and (44), by
$|f_{s}|,|\overline{\mathbb{H}}^{\varepsilon}|\leq 1$, (25) and Fubini’s
Theorem, we have, for $t\in[-t_{0},t_{0}]$, that

$\displaystyle\left|\int_{0}^{t}\int_{M}f_{s}(x)\int_{\sigma^{-}(x)}^{\sigma^{+}(x)}\sigma\lambda(\overline{\mathbb{H}}^{\varepsilon})^{\prime}(a-rf_{s}(x))\theta(x,a)-\lambda
e_{\varepsilon}(\overline{\mathbb{H}}^{\varepsilon}(a-rf_{s}(x)))\theta(x,a)da\,d\mathcal{H}^{n}(x)\,dr\right|$
$\displaystyle\leq\int_{0}^{t}\int_{M}\left|\int_{\sigma^{-}(x)}^{\sigma^{+}(x)}\sigma\lambda(\overline{\mathbb{H}}^{\varepsilon})^{\prime}(a-rf_{s}(x))\theta(x,a)-\lambda
e_{\varepsilon}(\overline{\mathbb{H}}^{\varepsilon}(a-rf_{s}(x)))\theta(x,a)da\right|d\mathcal{H}^{n}(x)\,dr$
$\displaystyle\leq
2\sigma\lambda\int_{0}^{t}\int_{M}m_{\varepsilon}(rf_{s}(x),x)d\mathcal{H}^{n}(x)\,dr+2\sigma\lambda\mathcal{H}^{n}(M)e^{\frac{\lambda^{2}}{2m}}t_{0}\beta\varepsilon^{2}$
$\displaystyle\leq\max_{t\in[-2d(N),2d(N)]}M_{\varepsilon}(t)+2\sigma\lambda\mathcal{H}^{n}(M)e^{\frac{\lambda^{2}}{2m}}t_{0}\beta\varepsilon^{2}.$

So

$|(\ref{eqn: theta error term 1})+(\ref{eqn: theta error term
2})|\leq\max_{t\in[-2d(N),2d(N)]}M_{\varepsilon}(t)+2\sigma\lambda\mathcal{H}^{n}(M)e^{\frac{\lambda^{2}}{2m}}t_{0}\beta\varepsilon^{2}.$
(48)

Finally we work on (45) and (46). If $t\geq 0$ then as $f_{s},\theta$ and
$e_{\varepsilon}$ are non-negative we have $(\ref{eqn: sigma + term})\leq 0$;
similarly, if $t\leq 0$ then we have $(\ref{eqn: sigma - term})\leq 0$. As
$|f_{s}|\leq 1$ by definition, $\theta\leq e^{\frac{\lambda^{2}}{2m}}$ by (20)
and
$e_{\varepsilon}(\overline{\mathbb{H}}^{\varepsilon}(\sigma^{\pm}(x)-rf_{s}(x))\neq
0$ only when
$\sigma^{\pm}(x)\in(rf_{s}(x)-2\varepsilon\Lambda_{\varepsilon},rf_{s}(x)+2\varepsilon\Lambda_{\varepsilon})$
by definition of $\overline{\mathbb{H}}^{\varepsilon}$, we conclude that

$(\ref{eqn: sigma +
term})\leq(2\sigma+\beta\varepsilon^{2})t_{0}e^{\frac{\lambda^{2}}{2m}}\mathcal{H}^{n}(\\{x\in
M\>|\>\sigma^{+}(x)\leq 2\varepsilon\Lambda_{\varepsilon}\\}$ (49)

and

$(\ref{eqn: sigma -
term})\leq(2\sigma+\beta\varepsilon^{2})t_{0}e^{\frac{\lambda^{2}}{2m}}\mathcal{H}^{n}(\\{x\in
M\>|\>\sigma^{-}(x)\geq-2\varepsilon\Lambda_{\varepsilon}\\}.$ (50)

As $\mathcal{H}^{n}(\\{x\in M\>|\>\sigma^{\pm}(x)=0\\})=0$ by the Dominated
Convergence Theorem we conclude that

$\mathcal{H}^{n}(\\{x\in M\>|\>\sigma^{+}(x)\leq
2\varepsilon\Lambda_{\varepsilon}\\}\rightarrow 0\text{ as
}\varepsilon\rightarrow 0$ (51)

and

$\mathcal{H}^{n}(\\{x\in
M\>|\>\sigma^{-}(x)\geq-2\varepsilon\Lambda_{\varepsilon}\\}\text{ as
}\varepsilon\rightarrow 0.$ (52)

Hence we conclude that for $t\in[-t_{0},t_{0}]$

$\mathcal{F}_{\varepsilon,\lambda}(v^{t,s}_{\varepsilon})\leq\mathcal{F}_{\varepsilon,\lambda}(v_{\varepsilon})-\frac{m\sigma}{4}\mathcal{H}^{n}(M)t^{2}+E(\varepsilon)$

where by (42), (48), (49) and (50) we have noting that as $t_{0}\leq 2d(N)$ we
have

$\displaystyle 2\sigma E(\varepsilon)$
$\displaystyle=\frac{m\sigma}{4}\mathcal{H}^{n}(M)(2d(N))^{2}\beta\varepsilon^{2}+(1-\beta\varepsilon^{2})\left[2\varepsilon\Lambda_{\varepsilon}m\sigma\mathcal{H}^{n}(M)\right]$
$\displaystyle\quad\quad+\max_{t\in[-2d(N),2d(N)]}M_{\varepsilon}(t)+2\sigma\lambda\mathcal{H}^{n}(M)e^{\frac{\lambda^{2}}{2m}}(2d(N))\beta\varepsilon^{2}$
$\displaystyle\quad\quad\quad+(2\sigma+\beta\varepsilon^{2})(2d(N))e^{\frac{\lambda^{2}}{2m}}\mathcal{H}^{n}(\\{x\in
M\>|\>\sigma^{+}(x)\leq 2\varepsilon\Lambda_{\varepsilon}\\}$
$\displaystyle\quad\quad\quad\quad+(2\sigma+\beta\varepsilon^{2})(2d(N))e^{\frac{\lambda^{2}}{2m}}\mathcal{H}^{n}(\\{x\in
M\>|\>\sigma^{-}(x)\geq-2\varepsilon\Lambda_{\varepsilon}\\}$

and so $E(\varepsilon)\rightarrow 0$ as $\varepsilon\rightarrow 0$ by virtue
of (47), (51) and (52). Setting once and for all

$\eta=\frac{m\sigma}{8}\mathcal{H}^{n}(M)t_{0}^{2}$

we conclude, taking $\varepsilon>0$ sufficiently small so that
$E(\varepsilon)<\eta$, the energy bounds as desired. ∎

We now conclude this subsection by showing that the paths
$t\in[-t_{1},t_{1}]\rightarrow v^{t,s}_{\varepsilon}$ and
$s\in[-1,1]\rightarrow v^{t,s}_{\varepsilon}$ are continuous, for fixed
$s\in[0,1]$ and $t\in[-t_{1},t_{1}]$ respectively, in $W^{1,2}(N)$. Recall
that the functions $F$, $\Pi_{V_{M}}$ and $f_{s}$ are continuous. For each
$t\in[-t_{1},t_{1}]$ consider that for $s,\tilde{s}\in[0,1]$ we have, by the
Dominated Convergence Theorem (using
$|\overline{\mathbb{H}}^{\varepsilon}|\leq 1$) and (35), that

$\displaystyle||v^{t,s}_{\varepsilon}-v^{t,\tilde{s}}_{\varepsilon}||^{2}_{L^{2}(N)}$
$\displaystyle=\int_{V_{M}}|\hat{v}^{t,s}_{\varepsilon}-\hat{v}^{t,\tilde{s}}_{\varepsilon}|^{2}d\mathcal{H}^{n+1}_{F^{*}g}$
$\displaystyle=\int_{V_{M}}|\overline{\mathbb{H}}^{\varepsilon}(a-tf_{s}(F(\Pi_{V_{M}}(x,a))))-\overline{\mathbb{H}}^{\varepsilon}(a-tf_{\tilde{s}}(F(\Pi_{V_{M}}(x,a))))|^{2}d\mathcal{H}^{n+1}_{F^{*}g}(x,a)$
$\displaystyle\rightarrow 0\text{ as }\tilde{s}\rightarrow s.$

Also, we see that by the Dominated Convergence Theorem, (35) and (36) we have

$\displaystyle||\nabla v^{t,s}_{\varepsilon}-\nabla
v^{t,\tilde{s}}_{\varepsilon}||^{2}_{L^{2}(N)}$
$\displaystyle=\int_{V_{M}}|\nabla\hat{v}^{t,s}_{\varepsilon}-\nabla\hat{v}^{t,\tilde{s}}_{\varepsilon}|^{2}d\mathcal{H}^{n+1}_{F^{*}g}$
$\displaystyle\leq\int_{V_{M}}\Big{(}(\overline{\mathbb{H}}^{{\varepsilon}})^{\prime}\big{(}a-tf_{s}(F(\Pi_{V_{M}}(x,a)))\big{)}$
$\displaystyle\quad\quad\quad\quad\quad\quad-(\overline{\mathbb{H}}^{{\varepsilon}})^{\prime}\big{(}a-tf_{\tilde{s}}(F(\Pi_{V_{M}}(x,a)))\big{)}\Big{)}^{2}d\mathcal{H}^{n+1}_{F^{*}g}(x,a)$
$\displaystyle\quad+\int_{V_{M}}t^{2}\big{|}\nabla\big{(}f_{\tilde{s}}(F(\Pi_{V_{M}}(x,a)))\big{)}(\overline{\mathbb{H}}^{{\varepsilon}})^{\prime}\big{(}a-tf_{\tilde{s}}(F(\Pi_{V_{M}}(x,a)))\big{)}$
$\displaystyle\quad\quad\quad\quad\quad-\nabla\big{(}f_{s}(F(\Pi_{V_{M}}(x,a)))\big{)}(\overline{\mathbb{H}}^{{\varepsilon}})^{\prime}\big{(}a-tf_{s}(F(\Pi_{V_{M}}(x,a)))\big{)}\big{|}^{2}d\mathcal{H}^{n+1}_{F^{*}g}(x,a)$
$\displaystyle\rightarrow 0\text{ as }\tilde{s}\rightarrow s,$

(where we also use (31) to ensure we can apply the Dominated Convergence
Theorem). Hence, for fixed $t\in[-t_{1},t_{1}]$, the path

$s\in[0,1]\rightarrow v^{t,s}_{\varepsilon}\text{ is continuous in
}W^{1,2}(N).$ (53)

Analogous arguments to those above show that, for fixed $s\in[0,1]$, the path

$t\in[-t_{1},t_{1}]\rightarrow v^{t,s}_{\varepsilon}\text{ is continuous in
}W^{1,2}(N).$ (54)

###  5.2 Sliding the one-dimensional profile

We now define, for $t\in\mathbb{R}$, the functions $v^{t}_{\varepsilon}\in
W^{1,2}(N)$ by setting

$v^{t}_{\varepsilon}(x)=\overline{\mathbb{H}}^{\varepsilon}(d^{\pm}_{\overline{M}}(x)-t),$

and note that, by (33), for $t\in[-t_{1},t_{1}]$ we have
$v^{t}_{\varepsilon}=v^{t,1}_{\varepsilon}$. By the continuity of translations
on $L^{p}$ for $1\leq p<\infty$ we have that $v^{t}_{\varepsilon}\in
W^{1,2}(N)$ for all $t\in\mathbb{R}$, the path $t\rightarrow
v^{t}_{\varepsilon}$ is continuous in $W^{1,2}(N)$ and that
$v^{0}_{\varepsilon}=v_{\varepsilon}$. By choosing $\varepsilon>0$
sufficiently small so that $2\varepsilon\Lambda_{\varepsilon}<d(N)$ we ensure
that both $v^{t}_{\varepsilon}=-1$ everywhere on $N$ for all $t\geq 2d(N)$ and
$v^{t}_{\varepsilon}=+1$ everywhere on $N$ for all $t\leq-2d(N)$.

We now show that, using the assumption of positive Ricci curvature, the path
provided by the functions $v^{t}_{\varepsilon}$, along with the energy
reducing paths from $-1$ and $+1$ provided by negative gradient flow of the
energy to $a_{\varepsilon}$ and $b_{\varepsilon}$ respectively, provides a
“recovery path” for the value $\mathcal{F}_{\lambda}(E)$; this path connects
$a_{\varepsilon}$ to $b_{\varepsilon}$, passing through $v_{\varepsilon}$,
with the maximum value of the energy along this path approximately
$\mathcal{F}_{\lambda}(E)$.

######  Lemma 5.

For all $t\in[-2d(N),2d(N)]$ we have that

$\mathcal{F}_{\varepsilon,\lambda}(v^{t}_{\varepsilon})\leq\mathcal{F}_{\varepsilon,\lambda}(v_{\varepsilon})+E(\varepsilon)$

where $E(\varepsilon)\rightarrow 0$ as $\varepsilon\rightarrow 0$ is as in
Lemma 4. Furthermore, we have that

$\mathcal{F}_{\varepsilon,\lambda}(v^{\pm\tilde{t}}_{\varepsilon})\leq\mathcal{F}_{\varepsilon,\lambda}(v^{\pm
t}_{\varepsilon})+E(\varepsilon),$

whenever $\tilde{t}\geq t\geq 0$; thus, in particular for $t>t_{0}$ we have

$\mathcal{F}_{\varepsilon,\lambda}(v^{\pm
t}_{\varepsilon})\leq\mathcal{F}_{\varepsilon,\lambda}(v_{\varepsilon})-\eta+E(\varepsilon).$

###### Proof.

We compute in an identical manner to the proof of Lemma 4, writing

$\displaystyle
2\sigma\mathcal{F}_{\varepsilon,\lambda}(v^{t}_{\varepsilon})=\int_{M}\int_{\sigma^{-}(x)}^{\sigma^{+}(x)}\left[e_{\varepsilon}\left(\overline{\mathbb{H}}^{\varepsilon}(a-t)\right)-\sigma\lambda\overline{\mathbb{H}}^{\varepsilon}(a-t)\right]\theta(x,a)da\,d\mathcal{H}^{n}(x).$

Assuming that either $\tilde{t}\geq t>0$ or $\tilde{t}\leq t<0$, as before we
compute

$\displaystyle
2\sigma(\mathcal{F}_{\varepsilon,\lambda}(v^{\tilde{t}}_{\varepsilon})-\mathcal{F}_{\varepsilon,\lambda}(v^{t}_{\varepsilon}))$
$\displaystyle=\int_{M}\int_{\sigma^{-}(x)}^{\sigma^{+}(x)}\left[e_{\varepsilon}\left(\overline{\mathbb{H}}^{\varepsilon}(a-\tilde{t})\right)-e_{\varepsilon}\left(\overline{\mathbb{H}}^{\varepsilon}(a-t)\right)\right]\theta(x,a)da\,d\mathcal{H}^{n}(x)$
$\displaystyle\quad-\int_{M}\int_{\sigma^{-}(x)}^{\sigma^{+}(x)}\sigma\lambda\left[\overline{\mathbb{H}}^{\varepsilon}(a-\tilde{t})-\overline{\mathbb{H}}^{\varepsilon}(a-t)\right]\theta(x,a)da\,d\mathcal{H}^{n}(x)$

which yields, noting that $\lambda-H(x,a)\leq 0$ for all $a\in\mathbb{R}$ by
the assumption of positive Ricci curvature, the expression

$\displaystyle
2\sigma(\mathcal{F}_{\varepsilon,\lambda}(v^{\tilde{t}}_{\varepsilon})-\mathcal{F}_{\varepsilon,\lambda}(v^{t}_{\varepsilon}))$
$\displaystyle\leq\int_{t}^{\tilde{t}}\int_{M}\int_{\sigma^{-}(x)}^{\sigma^{+}(x)}\sigma\lambda(\overline{\mathbb{H}}^{\varepsilon})^{\prime}(a-r)\theta(x,a)da\,d\mathcal{H}^{n}(x)\,dr$
$\displaystyle\quad-\int_{t}^{\tilde{t}}\int_{M}\int_{\sigma^{-}(x)}^{\sigma^{+}(x)}\lambda
e_{\varepsilon}(\overline{\mathbb{H}}^{\varepsilon}(a-r))\theta(x,a)da\,d\mathcal{H}^{n}(x)\,dr$
$\displaystyle\quad\quad-\int_{t}^{\tilde{t}}\int_{M}e_{\varepsilon}\left(\overline{\mathbb{H}}^{\varepsilon}(\sigma^{+}(x)-r)\right)\theta(x,\sigma^{+}(x))da\,d\mathcal{H}^{n}(x)\,dr$
$\displaystyle\quad\quad\quad+\int_{t}^{\tilde{t}}\int_{M}e_{\varepsilon}\left(\overline{\mathbb{H}}^{\varepsilon}(\sigma^{-}(x)-r)\right)\theta(x,\sigma^{-}(x))da\,d\mathcal{H}^{n}(x)\,dr.$

Near identical computations to those in the proof of Lemma 4 (for the error
terms (43), (44) giving the bound (48), as well the bounds on (49), (50)
giving (51), (52) respectively) for the above four terms give that

$\mathcal{F}_{\varepsilon,\lambda}(v^{\tilde{t}}_{\varepsilon})\leq\mathcal{F}_{\varepsilon,\lambda}(v^{t}_{\varepsilon})+E(\varepsilon),$

where the expression for $E(\varepsilon)$ is identical to that of Lemma 4
(where here are exploiting the fact that $t\in[-2d(N),2d(N)]$).

Recalling $v^{0}_{\varepsilon}=v_{\varepsilon}$, $v^{\pm
t_{0}}_{\varepsilon}=v^{\pm t_{0},1}_{\varepsilon}$ and the bounds from Lemma
4 then completes the proof. ∎

###  5.3 Paths to local energy minimisers

Recall, $v_{\varepsilon}=\overline{\mathbb{H}}^{\varepsilon}\circ
d^{\pm}_{\overline{M}}$ and that, by Lemma 4 and (33), for a fixed $\eta>0$
and some $r_{0}>0$, as chosen in Subsection 5.1.1, there exists a $t_{0}>0$
and functions $v^{t,0}_{\varepsilon}\in W^{1,2}(N)$ for $t\in[-t_{0},t_{0}]$,
with the following properties:

  * •

$v^{t,0}_{\varepsilon}=v_{\varepsilon}$ in $B_{r_{0}}(p)$.

  * •

$\mathcal{F}_{\varepsilon,\lambda}(v^{t,0}_{\varepsilon})\leq\mathcal{F}_{\varepsilon,\lambda}(v_{\varepsilon})+E(\varepsilon)$,
where $E(\varepsilon)\rightarrow 0$ as $\varepsilon\rightarrow 0$.

  * •

$\mathcal{F}_{\varepsilon,\lambda}(v^{\pm
t_{0},0}_{\varepsilon})\leq\mathcal{F}_{\varepsilon,\lambda}(v_{\varepsilon})-\eta$.

We now produce the local functions described in Step 3 of Subsection 1.3.
These functions are constant outside of $B_{r_{0}}(p)$, providing a continuous
path in $W^{1,2}(N)$ from $v_{\varepsilon}$ to $g_{\varepsilon}$ such that the
maximum energy along this path is bounded from above by
$\mathcal{F}_{\varepsilon,\lambda}(v^{t,0}_{\varepsilon})+\frac{\eta}{2}$.

######  Proposition 3.

For $\eta>0$ as above there exists, for some $R\in(0,r_{0})$ and
$\varepsilon>0$ sufficiently small, functions $g^{t,s}_{\varepsilon}\in
W^{1,2}(N)$, for each $t\in[-t_{0},t_{0}]$ and $s\in[-2,2]$, such that the
following properties hold:

  * •

For each $t\in[-t_{0},t_{0}]$ we have
$g^{t,-2}_{\varepsilon}=v^{t,0}_{\varepsilon}$ on $N$ and
$g^{t,s}_{\varepsilon}=v^{t,0}_{\varepsilon}$ on $N\setminus B_{R}(p)$ for all
$s\in[-2,2]$.

  * •

For each $t\in[-t_{0},t_{0}]$ we have $g^{t,2}_{\varepsilon}=g_{\varepsilon}$
in $B_{R}(p)$ where
$g_{\varepsilon}\in\mathcal{A}_{\varepsilon,\frac{R}{2}}(p)$ arises from Lemma
2 for $\rho=\frac{R}{2}$ and $q=p$.

  * •

For each $s\in[-2,2]$, $t\in[-t_{0},t_{0}]\rightarrow g^{t,s}_{\varepsilon}$
is a continuous path in $W^{1,2}(N)$.

  * •

For each $t\in[-t_{0},t_{0}]$, $s\in[-2,2]\rightarrow g^{t,s}_{\varepsilon}$
is a continuous path in $W^{1,2}(N)$.

  * •

For each $t\in[-t_{0},t_{0}]$ and $s\in[-2,2]$ we have

$\mathcal{F}_{\varepsilon,\lambda}(g^{t,s}_{\varepsilon})\leq\mathcal{F}_{\varepsilon,\lambda}(v^{t,0}_{\varepsilon})+\frac{\eta}{2}.$

Furthermore, if the functions $v_{\varepsilon}$ are such that
$\mathcal{F}_{\varepsilon,\lambda}(v_{\varepsilon})\geq\mathcal{F}_{\varepsilon,\lambda}(g_{\varepsilon})+\tau$
for some $\tau>0$, we have for each $t\in[-t_{0},t_{0}]$ that

$\mathcal{F}_{\varepsilon,\lambda}(g^{t,2}_{\varepsilon})\leq\mathcal{F}_{\varepsilon,\lambda}(v^{t,0}_{\varepsilon})-\tau.$

######  Remark 14.

The paths exhibited in Proposition 3 are “local” in the sense that they remain
constant outside of $B_{R}(p)$, and additionally do not require the assumption
of positive Ricci curvature of $N$. Specifically, in any ambient manifold
(without any curvature assumption), given an $\eta>0$ (not necessarily as
fixed by Lemma 4) and $f\in W^{1,2}(N)$ such that $|f|\leq 1$ and $f$ equal to
$v_{\varepsilon}$ in a ball bi-Lipschitz diffeomorphic to the Euclidean ball
of the same radius, a path as in Proposition 3 may be constructed (replacing
$v^{t,0}_{\varepsilon}$ by $f$ in the conclusions) for a sufficiently small
$R>0$.

###### Proof.

Given some $\eta>0$ we choose $R>0$ sufficiently small to ensure that

$2^{n+2}R^{n}\omega_{n}+C_{p}R^{n}+2\lambda\mathrm{Vol}(B_{\frac{R}{2}}(p))<\frac{\eta}{4},$

where $\omega_{n}$ is the volume of the $n$-dimensional unit ball. We then
potentially re-choose a smaller $0<R\leq\min\\{r_{0},r_{p}\\}$, such that the
closed ball $\overline{B}_{R}(p)$ is $2$-bi-Lipschitz diffeomorphic, via some
smooth chart $\psi$ on $N$, to the Euclidean ball,
$\overline{B}^{\mathbb{R}^{n+1}}_{R}(0)$ of radius $R$, with $\psi(p)=0$.
Consider a sweep-out of $\overline{B}^{\mathbb{R}^{n+1}}_{R}$ by planes
$\Pi_{l}$ defined for $l\in[-1,1]$ by

$\Pi_{l}=\left\\{y\in\overline{B}^{\mathbb{R}^{n+1}}_{R}\>\Big{|}\>y=(y_{1},\dots,y_{n},lR)\right\\},$

which for each $l\in[-1,1]$ is the intersection of
$\overline{B}^{\mathbb{R}^{n+1}}_{R}$ with the plane $\\{y_{n+1}=lR\\}$. We
consider the images, $P_{l}=\psi(\Pi_{l})$ in $N$. The $P_{l}$ then sweep-out
the closure of $B_{R}(p)$, $\overline{B}_{R}(p)$ in the sense that
$\overline{B}_{R}(p)=\cup_{l\in[-1,1]}P_{l}$ and that for $l\neq s$ we have
$P_{l}\cap P_{s}=\emptyset$. Each $\Pi_{l}$ divides
$\overline{B}^{\mathbb{R}^{n+1}}_{R}$ into two connected regions
$\\{y\in\overline{B}^{\mathbb{R}^{n+1}}_{R}\>|\>y_{n+1}>lR\\}$ and
$\\{y\in\overline{B}^{\mathbb{R}^{n+1}}_{R}\>|\>y_{n+1}<lR\\}$; we denote the
images of these sets under the diffeomorphism $\psi$ as
$E_{l}=\psi(\\{y\in\overline{B}^{\mathbb{R}^{n+1}}_{R}\>|\>y_{n+1}>lR\\})$ and
$F_{l}=\psi(\\{y\in\overline{B}^{\mathbb{R}^{n+1}}_{R}\>|\>y_{n+1}<lR\\})$ in
$\overline{B}_{R}(p)$ respectively. Note then that
$\overline{B}_{R}(p)=E_{l}\cup F_{l}\cup P_{l}$, where the unions are all
mutually disjoint.

We now define for $l\in[-1,1]$ functions $p^{l}_{\varepsilon}\in
W^{1,2}(B_{R}(p))$ given by

$p^{l}_{\varepsilon}=\overline{\mathbb{H}}^{\varepsilon}(d^{\pm}_{P_{l}}),$

where here, for $d_{P_{l}}$ the usual distance function on $N$ from the set
$P_{l}$, we define the Lipschitz signed distance to $P_{l}$ by

$d^{\pm}_{P_{l}}=\begin{cases}+d_{P_{l}},\text{ if }x\in E_{l}\\\ 0,\text{ if
}x\in P_{l}\\\ -d_{P_{l}},\text{ if }x\in F_{l}.\end{cases}$

As the sets $P_{l}$ vary continuously in the Haussdorff distance as
$l\in[-1,1]$ varies, the functions $p^{l}_{\varepsilon}$ vary continuously in
$W^{1,2}(B_{R}(p))$ with respect to $l\in[-1,1]$.

Apply Lemma 2 for the choice of $B_{\frac{R}{2}}(p)$ for each
$0<\varepsilon<\varepsilon_{2}$ as in the minimisation section. For
$l\in[-1,1]$ we define on $B_{R}(p)$ the functions

$\check{h}^{l}_{\varepsilon}(x)=\max\Big{\\{}\min\\{v_{\varepsilon}(x),p^{l}_{\varepsilon}(x)\\},\min\\{g_{\varepsilon}(x),v_{\varepsilon}(x)\\}\Big{\\}},$

and

$\hat{h}^{l}_{\varepsilon}(x)=\max\Big{\\{}\min\\{g_{\varepsilon}(x),p^{l}_{\varepsilon}(x)\\},\min\\{g_{\varepsilon}(x),v_{\varepsilon}(x)\\}\Big{\\}}.$

Note then that for $2\varepsilon\Lambda_{\varepsilon}<\frac{R}{2}$ we have
$\check{h}^{-1}_{\varepsilon}=v_{\varepsilon}$,
$\check{h}^{1}_{\varepsilon}=\hat{h}^{1}_{\varepsilon}=\min\\{g_{\varepsilon},v_{\varepsilon}\\}$,
$\hat{h}^{-1}_{\varepsilon}=g_{\varepsilon}$, and both
$|\check{h}^{l}_{\varepsilon}|\leq 1$, $|\hat{h}^{l}_{\varepsilon}|\leq 1$.

For functions $f,g\in W^{1,2}(B_{R}(p))$ we may write
$\max\\{f,g\\}=\max\\{f-g,0\\}+g=\frac{1}{2}(|f-g|+(f-g))+g$ and
$\min\\{f,g\\}=\min\\{f-g,0\\}+g=\frac{1}{2}(|f-g|-(f-g))+g$; thus
$\max\\{f,g\\},\min\\{f,g\\}\in W^{1,2}(N)$. We also note that if
$t\rightarrow f_{t}$ is a continuous path in $W^{1,2}(B_{R}(p))$, then the
paths $t\rightarrow\max\\{f_{t},0\\}$ and $t\rightarrow\min\\{f_{t},0\\}$ are
continuous in $W^{1,2}(B_{R}(p))$. To see this we write
$\max\\{f_{t},0\\}-\max\\{f_{s},0\\}=\frac{1}{2}((|f_{t}|-|f_{s}|)+(f_{t}-f_{s}))$
which by reverse triangle inequality may be controlled by the
$W^{1,2}(B_{R}(p))$ norm of $f_{t}-f_{s}$, showing continuity of the path in
$W^{1,2}(B_{R}(p))$. The proof for the continuity of the path
$t\rightarrow\min\\{f_{t},0\\}$ is identical. With the above in mind, we have
that $\check{h}_{\varepsilon}^{l},\hat{h}^{l}_{\varepsilon}\in
W^{1,2}(B_{R}(p))$ for each $l\in[-1,1]$ and the paths
$l\in[-1,1]\rightarrow\check{h}^{l}_{\varepsilon}$ and
$l\in[-1,1]\rightarrow\hat{h}^{l}_{\varepsilon}$ are continuous in
$W^{1,2}(B_{R}(p))$.

We have that $v_{\varepsilon}=g_{\varepsilon}$ on
$B_{R}(p)\setminus\overline{B_{\frac{R}{2}}(p)}$ (as
$v_{\varepsilon}=g_{\varepsilon}$ on $N\setminus B_{\frac{R}{2}}(p)$) and
hence $\hat{h}^{l}_{\varepsilon}=\check{h}^{l}_{\varepsilon}=v_{\varepsilon}$
on $B_{R}(p)\setminus\overline{B_{\frac{R}{2}}(p)}$ also. Thus we may extend
to well defined functions
$\check{g}^{t,l}_{\varepsilon},\hat{g}^{t,l}_{\varepsilon}\in W^{1,2}(N)$ for
$t\in[-t_{0},t_{0}]$ and $l\in[-1,1]$ by defining

$\check{g}^{t,l}_{\varepsilon}(x)=\begin{cases}v^{t,0}_{\varepsilon}(x),\text{
if }x\in N\setminus B_{R}(p)\\\ \check{h}^{l}_{\varepsilon}(x),\text{ if }x\in
B_{R}(p),\end{cases}$

and

$\hat{g}^{t,l}_{\varepsilon}(x)=\begin{cases}v^{t,0}_{\varepsilon}(x),\text{
if }x\in N\setminus B_{R}(p)\\\ \hat{h}^{l}_{\varepsilon}(x),\text{ if }x\in
B_{R}(p),\end{cases}$

This is well defined as we have that $v^{t,0}_{\varepsilon}=v_{\varepsilon}$
on the set $B_{R}(p)$, hence in defining the functions above we only edit the
functions $v_{\varepsilon}^{t,0}$ in $B_{R}(p)$ and keep them in $W^{1,2}(N)$.
By the above arguments, as the paths
$l\in[-1,1]\rightarrow\check{h}^{l}_{\varepsilon}$ and
$l\in[-1,1]\rightarrow\hat{h}^{l}_{\varepsilon}$ are continuous in
$W^{1,2}(B_{R}(p))$ and so we have that the paths
$t\in[-t_{0},t_{0}]\rightarrow\check{g}^{t,l}_{\varepsilon}$,
$l\in[-1,1]\rightarrow\check{g}^{t,l}_{\varepsilon}$,
$t\in[-t_{0},t_{0}]\rightarrow\hat{g}^{t,l}_{\varepsilon}$ and
$l\in[-1,1]\rightarrow\hat{g}^{t,l}_{\varepsilon}$ are continuous in
$W^{1,2}(N)$.

Note that then as
$\check{h}^{1}_{\varepsilon}=\hat{h}^{1}_{\varepsilon}=\min\\{g_{\varepsilon},v_{\varepsilon}\\}$
we have for each $t\in[-t_{0},t_{0}]$ that
$\check{g}^{t,1}_{\varepsilon}=\hat{g}^{t,-1}_{\varepsilon}=\min\\{g_{\varepsilon},v_{\varepsilon}\\}$
in $B_{R}(p)$. We thus define, for $s\in[-2,2]$, the functions

$g^{t,s}_{\varepsilon}=\begin{cases}\check{g}^{t,s+1},\text{ if }s\in[-2,0]\\\
\hat{g}^{t,s-1}_{\varepsilon},\text{ if }s\in[0,2],\end{cases}$

so that
$g^{t,0}_{\varepsilon}=\check{g}^{t,1}_{\varepsilon}=\hat{g}^{t,-1}_{\varepsilon}=\min\\{g_{\varepsilon},v_{\varepsilon}\\}$,
$g^{t,-2}_{\varepsilon}=\check{g}^{0,-1}_{\varepsilon}=v^{t,0}_{\varepsilon}$
and $g^{0,2}_{\varepsilon}=\hat{g}^{0,1}_{\varepsilon}=g_{\varepsilon}$ in
$B_{R}(p)$. By the continuity of the paths mentioned above we have that the
paths, $s\in[-2,2]\rightarrow g^{t,s}_{\varepsilon}$ and
$t\in[-t_{0},t_{0}]\rightarrow g^{t,s}_{\varepsilon}$, are continuous in
$W^{1,2}(N)$.

We now show that we may bound the energy of all the functions
$g^{t,s}_{\varepsilon}$ above by
$\mathcal{F}_{\varepsilon,\lambda}(v^{t,0}_{\varepsilon})$ plus errors
depending only on the geometry of the ball $B_{R}(p)$. As
$\\{g_{\varepsilon}\neq v_{\varepsilon}\\}\subset B_{\frac{R}{2}}(p)$ we note
that $g^{t,s}_{\varepsilon}=v^{t,0}_{\varepsilon}$ on $N\setminus
B_{\frac{R}{2}}(p)$. We then compute that, as $\check{h}^{s+1}_{\varepsilon}$
is always equal to one of $g_{\varepsilon},v_{\varepsilon}$ or
$p_{\varepsilon}^{s+1}$ in $B_{\frac{R}{2}}(p)$, for $s\in[-2,0]$ we have

$\displaystyle\mathcal{F}_{\varepsilon,\lambda}(g^{t,s}_{\varepsilon})$
$\displaystyle=\frac{1}{2\sigma}\int_{N}e_{\varepsilon}(g^{t,s}_{\varepsilon})-\frac{\lambda}{2}\int_{N}g^{t,s}_{\varepsilon}=\frac{1}{2\sigma}\int_{N\setminus
B_{\frac{R}{2}}(p)}e_{\varepsilon}(v^{t,0}_{\varepsilon})+\frac{1}{2\sigma}\int_{B_{\frac{R}{2}}(p)}e_{\varepsilon}(\check{h}^{s+1}_{\varepsilon})$
$\displaystyle-\frac{\lambda}{2}\int_{N\setminus
B_{\frac{R}{2}}(p)}v^{t,0}_{\varepsilon}-\frac{\lambda}{2}\int_{B_{\frac{R}{2}}(p)}\check{h}^{s+1}_{\varepsilon}$
$\displaystyle=\frac{1}{2\sigma}\int_{N\setminus
B_{\frac{R}{2}}(p)}e_{\varepsilon}(v^{t,0}_{\varepsilon})+\frac{1}{2\sigma}\int_{B_{\frac{R}{2}}(p)\cap\\{\check{h}^{s+1}_{\varepsilon}=v_{\varepsilon}\\}}e_{\varepsilon}(v_{\varepsilon})+\frac{1}{2\sigma}\int_{B_{\frac{R}{2}}(p)\cap\\{\check{h}^{s+1}_{\varepsilon}=g_{\varepsilon}\\}}e_{\varepsilon}(g_{\varepsilon})$
$\displaystyle+\frac{1}{2\sigma}\int_{B_{\frac{R}{2}}(p)\cap\\{\check{h}^{s+1}_{\varepsilon}=p^{s+1}_{\varepsilon}\\}}e_{\varepsilon}(p^{s+1}_{\varepsilon})-\frac{\lambda}{2}\int_{N\setminus
B_{\frac{R}{2}}(p)}v^{t,0}_{\varepsilon}-\frac{\lambda}{2}\int_{B_{\frac{R}{2}}(p)}\check{h}^{s+1}_{\varepsilon}$
$\displaystyle\leq\frac{1}{2\sigma}\int_{N\setminus
B_{\frac{R}{2}}(p)}e_{\varepsilon}(v^{t,0}_{\varepsilon})+\frac{1}{2\sigma}\int_{B_{\frac{R}{2}}(p)}e_{\varepsilon}(v_{\varepsilon})+\frac{1}{2\sigma}\int_{B_{\frac{R}{2}}(p)}e_{\varepsilon}(g_{\varepsilon})+\frac{1}{2\sigma}\int_{B_{\frac{R}{2}}(p)}e_{\varepsilon}(p^{s+1}_{\varepsilon})$
$\displaystyle-\frac{\lambda}{2}\int_{N\setminus
B_{\frac{R}{2}}(p)}v^{t,0}_{\varepsilon}-\frac{\lambda}{2}\int_{B_{\frac{R}{2}}(p)}\check{h}^{s+1}_{\varepsilon}$
$\displaystyle=\mathcal{E}_{\varepsilon}(v^{t,0}_{\varepsilon})+\frac{1}{2\sigma}\int_{B_{\frac{R}{2}}(p)}e_{\varepsilon}(g_{\varepsilon})+\frac{1}{2\sigma}\int_{B_{\frac{R}{2}}(p)}e_{\varepsilon}(p^{s+1}_{\varepsilon})-\frac{\lambda}{2}\int_{N\setminus
B_{\frac{R}{2}}(p)}v^{t,0}_{\varepsilon}-\frac{\lambda}{2}\int_{B_{\frac{R}{2}}(p)}\check{h}^{s+1}_{\varepsilon}$
$\displaystyle=\mathcal{F}_{\varepsilon,\lambda}(v^{t,0}_{\varepsilon})+\frac{1}{2\sigma}\int_{B_{\frac{R}{2}}(p)}e_{\varepsilon}(g_{\varepsilon})+\frac{1}{2\sigma}\int_{B_{\frac{R}{2}}(p)}e_{\varepsilon}(p^{s+1}_{\varepsilon})+\frac{\lambda}{2}\int_{B_{\frac{R}{2}}(p)}(v^{t,0}_{\varepsilon}-\check{h}^{s+1}_{\varepsilon}),$

where in the final line we have added and subtracted the term
$\frac{\lambda}{2}\int_{B_{\frac{R}{2}}(p)}v^{t,0}_{\varepsilon}$. Similarly
for $s\in[0,2]$ we have

$\mathcal{F}_{\varepsilon,\lambda}(g^{t,s}_{\varepsilon})\leq\mathcal{F}_{\varepsilon,\lambda}(v^{t,0}_{\varepsilon})+\frac{1}{2\sigma}\int_{B_{\frac{R}{2}}(p)}e_{\varepsilon}(g_{\varepsilon})+\frac{1}{2\sigma}\int_{B_{\frac{R}{2}}(p)}e_{\varepsilon}(p^{s-1}_{\varepsilon})+\frac{\lambda}{2}\int_{B_{\frac{R}{2}}(p)}(v^{t,0}_{\varepsilon}-\hat{h}^{s-1}_{\varepsilon}).$

Thus as for all $l\in[-1,1]$ we have $|\check{h}^{l}_{\varepsilon}|\leq 1$,
$|\hat{h}^{l}_{\varepsilon}|\leq 1$ and $|v^{t,0}_{\varepsilon}|\leq 1$ we see
that for each $t\in[-t_{0},t_{0}]$ and $s\in[-2,2]$

$\mathcal{F}_{\varepsilon,\lambda}(g^{t,s}_{\varepsilon})\leq\mathcal{F}_{\varepsilon,\lambda}(v^{t,0}_{\varepsilon})+\frac{1}{2\sigma}\int_{B_{\frac{R}{2}}(p)}e_{\varepsilon}(g_{\varepsilon})+\sup_{l\in(-1,1)}\left(\frac{1}{2\sigma}\int_{B_{\frac{R}{2}}(p)}e_{\varepsilon}(p^{l}_{\varepsilon})\right)+\lambda\mathrm{Vol}(B_{\frac{R}{2}}(p)).$
(55)

Similarly, we note that as $v_{\varepsilon}=g_{\varepsilon}$ on $N\setminus
B_{\frac{R}{2}}(p)$, $|g_{\varepsilon}|,|v_{\varepsilon}|\leq 1$ on $N$ and
$\mathcal{F}_{\varepsilon,\lambda}(g_{\varepsilon})\leq\mathcal{F}_{\varepsilon,\lambda}(v_{\varepsilon})$
by construction of $g_{\varepsilon}$ we have that

$\frac{1}{2\sigma}\int_{B_{\frac{R}{2}}(p)}e_{\varepsilon}(g_{\varepsilon})\leq\frac{1}{2\sigma}\int_{B_{\frac{R}{2}}(p)}e_{\varepsilon}(v_{\varepsilon})+\lambda\mathrm{Vol}(B_{\frac{R}{2}}(p)).$

and so by similar calculations to Subsection 3.3 and (25) we see that

$\displaystyle\frac{1}{2\sigma}\int_{B_{\frac{R}{2}}(p)}e_{\varepsilon}(v_{\varepsilon})$
$\displaystyle\leq(1+\beta\varepsilon^{2}){\mathrm{ess\,sup}}_{a\in[-2\varepsilon\Lambda_{\varepsilon},2\varepsilon\Lambda_{\varepsilon}]}\mathcal{H}^{n}(\\{d^{\pm}_{\overline{M}}=a\\}\cap
B_{R}(p))$ $\displaystyle\rightarrow\mathcal{H}^{n}(M\cap B_{R}(p))\text{ as
}\varepsilon\rightarrow 0.$

By (28) we conclude that, for sufficiently small $\varepsilon>0$, we have

$\frac{1}{2\sigma}\int_{B_{\frac{R}{2}}(p)}e_{\varepsilon}(g_{\varepsilon})\leq
C_{p}R^{n}+\lambda\mathrm{Vol}(B_{\frac{R}{2}}(p))+\frac{\eta}{8}.$ (56)

We now compute upper $\mathcal{E}_{\varepsilon}$ bounds on the term in (55)
involving the functions $p^{l}_{\varepsilon}$ in terms of the geometry of $N$.
Applying the co-area formula, slicing with $d^{\pm}_{P_{l}}$, and using (25)
we see that for $l\in(-1,1)$ we have

$\displaystyle\frac{1}{2\sigma}\int_{B_{\frac{R}{2}}(p)}e_{\varepsilon}(p^{l}_{\varepsilon})$
$\displaystyle=\frac{1}{2\sigma}\int_{B_{\frac{R}{2}}(p)}e_{\varepsilon}(\overline{\mathbb{H}}^{\varepsilon}(d^{\pm}_{P_{l}}))=\frac{1}{2\sigma}\int_{\mathbb{R}}\int_{\\{d^{\pm}_{P_{l}}=a\\}\cap
B_{\frac{R}{2}}(p)}e_{\varepsilon}(\overline{\mathbb{H}}^{\varepsilon}(a))d\mathcal{H}^{n}\,da$
$\displaystyle=\frac{1}{2\sigma}\int_{\mathbb{R}}\mathcal{H}^{n}(\\{d^{\pm}_{P_{l}}=a\\}\cap
B_{\frac{R}{2}}(p))e_{\varepsilon}(\overline{\mathbb{H}}^{\varepsilon}(a))\,da$
$\displaystyle\leq(1+\beta\varepsilon^{2}){\mathrm{ess\,sup}}_{a\in[-2\varepsilon\Lambda_{\varepsilon},2\varepsilon\Lambda_{\varepsilon}]}\mathcal{H}^{n}(\\{d^{\pm}_{P_{l}}=a\\}\cap
B_{\frac{R}{2}}(p)).$

We now bound
${\mathrm{ess\,sup}}_{a\in[-2\varepsilon\Lambda_{\varepsilon},2\varepsilon\Lambda_{\varepsilon}]}\mathcal{H}^{n}(\\{d^{\pm}_{P_{l}}=a\\}\cap
B_{\frac{R}{2}}(p))$ from above independently of $\varepsilon$. Recall that
$P_{l}=\psi(\Pi_{l})$ where $\psi$ is a smooth 2-bi-Lipschitz map from
$\overline{B}_{R}^{\mathbb{R}^{n+1}}(0)$ to $\overline{B}_{R}(p)$ and
$\Pi_{l}=\left\\{y\in\overline{B}^{\mathbb{R}^{n+1}}_{R}\>\Big{|}\>y=(y_{1},\dots,y_{n},lR)\right\\}$.
For $l\in(-1,1)$ we have that $\Pi_{l}$ is a smooth embedded $n$-dimensional
sub-manifold of $\overline{B}_{R}^{\mathbb{R}^{n+1}}$ and hence its image,
$P_{l}$, in $N$ is a smooth embedded $n$-dimensional sub-manifold of
$\overline{B}_{R}(p)\subset N$.

We define the tubular hypersurface at distance $s$ from $P_{l}$ to be the set

$P_{l}(a)=\\{x\in N\>|\>\text{ there exists a geodesic of length }a\text{
meeting }P_{l}\text{ orthogonally}\\}.$

Here we choose $\varepsilon>0$ possibly smaller to ensure that
$2\varepsilon\Lambda_{\varepsilon}<\frac{R}{2}$, so that
$\\{|d^{\pm}_{P_{l}}|=a\\}\cap B_{\frac{R}{2}}(p)\subset P_{l}(a)$ for all
$l\in(-1,1)$ such that $B_{\frac{R}{2}}(p)\cap P_{l}\neq\emptyset$. We then
apply [Gra04, Lemma 3.12/8.2] to compute that, for each $l\in(-1,1)$ we have

$\mathcal{H}^{n}(P_{l}(s))\leq\int_{P_{l}}\int_{S^{0}}\theta_{u}(q,s)du\,d\mathcal{H}^{n}(q).$

In the above we are adopting the notation that $S^{0}$ is the $0$ dimensional
unit sphere, and for $u\in S^{0}$ we define $\theta_{u}(q,a)$ to be the
Jacobian of the exponential map $\exp_{q}$ at the point $\exp_{q}(au)$. Note
then that for each $q\in P_{l}$ we have that $\theta_{u}(q,0)=1$ and
$\theta_{u}(q,a)\rightarrow 1$ as $a\rightarrow 0$. For all $\varepsilon>0$
sufficiently small we ensure that $\theta_{u}(q,a)\leq 2$ for any
$q\in\overline{B}_{R}(p)$; this may be seen by noticing that as the
exponential map is smooth, its Jacobian varies continuously in each of its
variables and hence its maximum, for a fixed $a\in\mathbb{R}$ on
$\overline{B}_{R}(p)\times\mathbb{R}$, is achieved and converges to $1$ as
$a\rightarrow 0$. Thus, we have that

$\mathcal{H}^{n}(P_{l}(a))\leq 4\mathcal{H}^{n}(P_{l}).$

Note that as $\psi$ is $2$-bi-Lipschitz we may apply [EG15, Theorem 2.8] to
see that

$\mathcal{H}^{n}(P_{l})\leq 2^{n}\mathcal{H}^{n}(\Pi_{l})\leq
2^{n}R^{n}\omega_{n},$

where $\omega_{n}$ is the volume of the $n$-dimensional unit ball. Hence, for
all $l\in[-1,1]$ and $\varepsilon>0$ sufficiently small we have that

$\mathcal{H}^{n}(P_{l}(s))\leq 2^{n+2}R^{n}\omega_{n}.$

We thus conclude that as $\\{|d^{\pm}_{P_{l}}|=a\\}\cap
B_{\frac{R}{2}}(p)\subset P_{l}(a)$ for all $l\in(-1,1)$ such that
$B_{\frac{R}{2}}(p)\cap P_{l}\neq\emptyset$ we have

${\mathrm{ess\,sup}}_{a\in[-2\varepsilon\Lambda_{\varepsilon},2\varepsilon\Lambda_{\varepsilon}]}\mathcal{H}^{n}(\\{d^{\pm}_{P_{l}}=a\\}\cap
B_{\frac{R}{2}}(p))\leq 2^{n+2}R^{n}\omega_{n},$

and hence

$\frac{1}{2\sigma}\int_{{B_{\frac{R}{2}}}(p)}e_{\varepsilon}(p^{l}_{\varepsilon})\leq(1+\beta\varepsilon^{2})2^{n+2}R^{n}\omega_{n}.$
(57)

Choosing $\varepsilon>0$ again possibly smaller we ensure that

$2^{n+2}R^{n}\omega_{n}\beta\varepsilon^{2}<\frac{\eta}{8}.$

Recall that, given some $\eta>0$, we chose $R>0$ sufficiently small to ensure
that

$2^{n+2}R^{n}\omega_{n}+C_{p}R^{n}+2\lambda\mathrm{Vol}(B_{\frac{R}{2}}(p))<\frac{\eta}{4}.$

Combining the above two bounds with (55), (56) and (57) we conclude that for
each $t\in[-t_{0},t_{0}]$ and $s\in[-2,2]$ we have

$\mathcal{F}_{\varepsilon,\lambda}(g^{t,s}_{\varepsilon})\leq\mathcal{F}_{\varepsilon,\lambda}(v^{t,0}_{\varepsilon})+\frac{\eta}{2},$

for $\varepsilon>0$ sufficiently small.

As $\\{g_{\varepsilon}\neq v_{\varepsilon}\\}\subset B_{\frac{R}{2}}(p)$ we
have $g^{t,s}_{\varepsilon}=v^{t,0}_{\varepsilon}$ on $N\setminus
B_{\frac{R}{2}}(p)$, so using the fact that in $B_{R}(p)$ we have
$v^{t,0}_{\varepsilon}=v_{\varepsilon}$ and
$g^{t,2}_{\varepsilon}=g_{\varepsilon}$ we thus note that

$\mathcal{F}_{\varepsilon,\lambda}(g^{t,2}_{\varepsilon})=\mathcal{F}_{\varepsilon,\lambda}(v^{t,0}_{\varepsilon})+(\mathcal{F}_{\varepsilon,\lambda}(g_{\varepsilon})-\mathcal{F}_{\varepsilon,\lambda}(v_{\varepsilon})).$

Using this, if the functions $v_{\varepsilon}$ are such that
$\mathcal{F}_{\varepsilon,\lambda}(v_{\varepsilon})\geq\mathcal{F}_{\varepsilon,\lambda}(g_{\varepsilon})+\tau$
for some $\tau>0$, we therefore conclude that

$\mathcal{F}_{\varepsilon,\lambda}(g^{t,2}_{\varepsilon})\leq\mathcal{F}_{\varepsilon,\lambda}(v^{t,0}_{\varepsilon})-\tau,$

as desired. ∎

##  6 Proof of Theorems 1, 2 & 3

We recall our setup. Let $(N,g)$ be a smooth compact Riemannian manifold of
dimension $3$ or higher with positive Ricci curvature. We consider $M\subset
N$ a closed embedded hypersurface of constant mean curvature
$\lambda\in\mathbb{R}$, smooth away from a closed singular set of Hausdorff
dimension at most $n-7$, as produced by the one-parameter Allen–Cahn min-max
in [BW20a], with constant prescribing function $\lambda$.

###  6.1 Proof of Theorem 3

###### Proof of Theorem 3.

For each isolated singularity $p\in\mathrm{Sing}(M)$ we may choose, positive
$r_{0}$, $r$ and $R$, as in Subsection 5.1.1 and Proposition 3 in order to
define the various paths constructed in Subsections 5.1.2, 5.2 and 5.3.
Assuming that for
$g_{\varepsilon}\in\mathcal{A}_{\varepsilon,\frac{R}{2}}(p)$, defined as in
Lemma 2 (setting $\rho=\frac{R}{2}$ and $q=p$), there exists a $\tau>0$ such
that
$\mathcal{F}_{\varepsilon,\lambda}(v_{\varepsilon})\geq\mathcal{F}_{\varepsilon,\lambda}(g_{\varepsilon})+\tau$,
for all $\varepsilon>0$ sufficiently small, then using the constructions in
Section 5 as mentioned above we define the following nine paths:

  * •

First, a path from $+1$ to the constant $b_{\varepsilon}$,

$s\in[1,b_{\varepsilon}]\rightarrow s,$

through constant functions which, by the construction of $b_{\varepsilon}$
through negative gradient flow of $\mathcal{F}_{\varepsilon,\lambda}$ in
Subsection 1.2, has $\mathcal{F}_{\varepsilon,\lambda}$ energy along the path
$\leq\mathcal{F}_{\varepsilon,\lambda}(1)$.

  * •

Second, a path from $+1$ to
$v^{-t_{0}}_{\varepsilon}=v^{-t_{0},1}_{\varepsilon}$,

$t\in[-2d(N),-t_{0}]\rightarrow v^{t}_{\varepsilon},$

which by Lemma 5 has $\mathcal{F}_{\varepsilon,\lambda}$ energy along the path
$\leq\mathcal{F}_{\varepsilon,\lambda}(v^{-t_{0},1})\leq\mathcal{F}_{\lambda}(E)-\eta+E(\varepsilon)$.
This path varies continuously by the reasoning in Subsection 5.2.

  * •

Third, a path from $v^{-t_{0},1}_{\varepsilon}$ to
$v^{-t_{0},0}_{\varepsilon}=g^{-t_{0},-2}_{\varepsilon}$,

$s\in[0,1]\rightarrow v^{-t_{0},s}_{\varepsilon},$

which by Lemma 4 has $\mathcal{F}_{\varepsilon,\lambda}$ energy along the path
$\leq\mathcal{F}_{\varepsilon,\lambda}(v_{\varepsilon})-\eta$. This path
varies continuously by (53).

  * •

Fourth, a path from $g^{-t_{0},-2}_{\varepsilon}$ to
$g^{-t_{0},2}_{\varepsilon}$,

$s\in[-2,2]\rightarrow g^{-t_{0},s}_{\varepsilon},$

which by Proposition 3 and Lemma 4 has $\mathcal{F}_{\varepsilon,\lambda}$
energy along the path
$\leq\mathcal{F}_{\varepsilon,\lambda}(v_{\varepsilon})-\frac{\eta}{2}$. This
path varies continuously by Proposition 3.

  * •

Fifth, a path from $g^{-t_{0},2}_{\varepsilon}$ to
$g^{t_{0},2}_{\varepsilon}$,

$t\in[-t_{0},t_{0}]\rightarrow g^{t,2}_{\varepsilon},$

which by Proposition 3 and Lemma 4 has $\mathcal{F}_{\varepsilon,\lambda}$
energy along the path
$\leq\mathcal{F}_{\varepsilon,\lambda}(v_{\varepsilon})-\tau+E(\varepsilon)$.
This path varies continuously by (54) and Proposition 3.

  * •

Sixth, a path from $g^{t_{0},-2}_{\varepsilon}=v^{t_{0},0}_{\varepsilon}$ to
$g^{t_{0},2}_{\varepsilon}$,

$s\in[-2,2]\rightarrow g^{t_{0},s}_{\varepsilon},$

which by by Proposition 3 and Lemma 4 has $\mathcal{F}_{\varepsilon,\lambda}$
energy along the path
$\leq\mathcal{F}_{\varepsilon,\lambda}(v_{\varepsilon})-\frac{\eta}{2}$. This
path varies continuously by Proposition 3.

  * •

Seventh, a path from $v^{t_{0},0}_{\varepsilon}$ to
$v^{t_{0},1}_{\varepsilon}=v^{t_{0}}_{\varepsilon}$,

$s\in[0,1]\rightarrow v^{t_{0},s}_{\varepsilon}$

which by Lemma 4 has $\mathcal{F}_{\varepsilon,\lambda}$ energy along the path
$\leq\mathcal{F}_{\varepsilon,\lambda}(v_{\varepsilon})-\eta$. This path
varies continuously by (53).

  * •

Eighth, a path from $v^{t_{0}}_{\varepsilon}$ to $-1$,

$t\in[t_{0},2d(N)]\rightarrow v^{t}_{\varepsilon}$

which by Lemma 5 has $\mathcal{F}_{\varepsilon,\lambda}$ energy along the path
$\leq\mathcal{F}_{\varepsilon,\lambda}(v_{\varepsilon})-\eta+E(\varepsilon)$.
This path varies continuously by the reasoning in Subsection 5.2.

  * •

Ninth, a path from $-1$ to the constant $a_{\varepsilon}$,

$s\in[-1,a_{\varepsilon}]\rightarrow s,$

through constant functions which, by the construction of $a_{\varepsilon}$
through negative gradient flow of $\mathcal{F}_{\varepsilon,\lambda}$ in
Subsection 1.2, has $\mathcal{F}_{\varepsilon,\lambda}$ energy along the path
$\leq\mathcal{F}_{\varepsilon,\lambda}(-1)$.

Consider the above paths in the following order: first (reversed), second,
third (reversed), fourth, fifth, sixth (reversed), seventh, eighth and ninth;
this is the path depicted in Figure 2 in Subsection 1.3. In the order just
given, the endpoint of each partial path matches the starting point of the
next, therefore their composition in the same order provides a continuous path
in $W^{1,2}(N)$, for all $\varepsilon>0$ sufficiently small, from the constant
$a_{\varepsilon}$ to the constant $b_{\varepsilon}$ with

$\text{$\mathcal{F}_{\varepsilon,\lambda}$ energy along the
path}\leq\mathcal{F}_{\varepsilon,\lambda}(v_{\varepsilon})-\min\left\\{\frac{\eta}{2},\tau\right\\}+E(\varepsilon),$

by Subsection 1.2, Lemma 4, Lemma 5 and Proposition 3. By (2) and the fact
that $E(\varepsilon)\rightarrow 0$ by Lemma 4, by choosing $\varepsilon>0$
sufficiently small we ensure that we have

$\text{$\mathcal{F}_{\varepsilon,\lambda}$ energy along the
path}\leq\mathcal{F}_{\lambda}(E)-\min\left\\{\frac{\eta}{4},\frac{\tau}{2}\right\\}.$

Note that as (1) holds, by (2) and the path provided in Lemma 5, we ensure
$\mathcal{F}_{\varepsilon,\lambda}(u_{\varepsilon_{j}})\rightarrow\mathcal{F}_{\lambda}(E)$
as $\varepsilon_{j}\rightarrow 0$. Thus, as the above path is admissible in
the min-max construction of [BW20a], we contradict the assumption that
$\mathcal{F}_{\varepsilon,\lambda}(v_{\varepsilon})\geq\mathcal{F}_{\varepsilon,\lambda}(g_{\varepsilon})+\tau$
for some $\tau>0$. We therefore conclude that for any such $M$ as produced by
the Allen–Cahn min-max procedure in Ricci positive curvature must be such that

$\mathcal{F}_{\varepsilon,\lambda}(v_{\varepsilon})\leq\mathcal{F}_{\varepsilon,\lambda}(g_{\varepsilon})+\tau_{\varepsilon}\text{
for some sequence }\tau_{\varepsilon}\rightarrow 0\text{ as
}\varepsilon\rightarrow 0.$

With the above in mind and applying Lemma 3, by setting $r_{1}=\frac{R}{4}$
and $r_{2}=\frac{R}{2}$, we have that $E$ satisfies

$\mathcal{F}_{\lambda}(E)=\inf_{G\in\mathcal{C}(N)}\\{\mathcal{F}_{\lambda}(G)\>|\>G\setminus
B_{\frac{R}{4}}(p)=E\setminus B_{\frac{R}{4}}(p)\\}.$

In particular, by Remark 13, we note that every tangent cone at an isolated
singularity of $M$ is thus area-minimising. Applying the above reasoning for
each isolated singularity of $M$ then concludes the proof of Theorem 3. ∎

###  6.2 Proof of Theorems 1 & 2

###### Proof of Theorem 2.

To prove Theorem 2 we exploit the results of Section 2, using the fact that
the proof of Theorem 3 ensures $M$ is one-sided minimising in the sense of
Definition 1 by Remark 7. Because the set of isolated singular points of $M$
with regular tangent cones is discrete, but not necessarily closed when $n\geq
8$, it suffices to index the isolated singularities with regular tangent cones
and make a small change to the metric around each point so that the sum of the
resulting perturbations is arbitrarily small.

With the above in hand, we now make an arbitrarily small change to the metric
at each isolated singular point, $p\in\mathrm{Sing}(M)$, with regular tangent
cone. By applying Proposition 1 in a ball, $B_{\rho}(p)$, for some $\rho>0$
sufficiently small, there exists both:

  * •

A metric, $\tilde{g}$, arbitrarily close to $g$ in the $C^{k,\alpha}$ norm for
each $k\geq 1$ and $\alpha\in(0,1)$, agreeing with $g$ on $N\setminus
B_{\rho}(p)$.

  * •

A closed embedded hypersurface, $\widetilde{M}$, of constant mean curvature
$\lambda$, which is smooth in $B_{\rho}(p)$, and agrees with $M$ on
$N\setminus B_{\rho}(p)$.

In this manner we are able to locally smooth $M$ up to an arbitrarily small
perturbation of the metric $g$. Thus we have shown that for each $k\geq 1$ and
$\alpha\in(0,1)$ there exists a dense set of metrics,
$\mathcal{G}_{k}\subset\text{Met}^{k,\alpha}_{\mathrm{Ric}_{g}>0}$, such that
for each $h\in\mathcal{G}_{k}$, $(N,h)$ admits a closed embedded hypersurface
of constant mean curvature $\lambda$, smooth away from a closed singular set
of Hausdorff dimension at most $n-7$, containing no singularities with regular
tangent cones. By [Whi15, Theorem 2.10] there thus exists a dense set
$\mathcal{G}$ of the smooth metrics with Ricci positive curvature such that
for each $h\in\mathcal{G}$, $(N,h)$ admits a closed embedded hypersurface of
constant mean curvature $\lambda$, smooth away from a closed singular set of
Hausdorff dimension at most $n-7$, containing no singularities with regular
tangent cones, concluding the proof of Theorem 2. ∎

###### Proof of Theorem 1.

In dimension $8$ all singularities are isolated with regular tangent cone.
Hence, for each $g\in\mathcal{G}$, as produced in the proof of Theorem 2,
there exists a smooth hypersurface of constant mean curvature. The fact that
$\mathcal{G}$ as above is then open in dimension $8$ follows from the results
of [Whi91, Section 7]. ∎

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CB: Department of Mathematics, University College London, 25 Gordon St,
London, WC1H 0AY

Email address<EMAIL_ADDRESS>
KMS: Department of Mathematics, University College London, 25 Gordon St,
London, WC1H 0AY

Email address<EMAIL_ADDRESS>


††thanks: corresponding author

# An N-Body Solution to the Problem of Fock Exchange

Matt Challacombe<EMAIL_ADDRESS>http://www.freeon.org Nicolas
Bock  Theoretical Division, Los Alamos National Laboratory

###### Abstract

We report an N-Body approach to computing the Fock exchange matrix with and
without permutational symmetry. The method achieves an $O(N{\rm lg}N)$
computational complexity through an embedded metric-query, allowing
hierarchical application of direct SCF criteria. The advantages of
permutational symmetry are found to be 4-fold for small systems, but
decreasing with increasing system size and/or more permissive neglect
criteria. This work sets the stage for: (1) the introduction of range queries
in multi-level multipole schemes for rank reduction, and (2) recursive task
parallelism.

††preprint: LA-UR-14-20354

##  I introduction

For physical problems that find compact representation supported by fast
transforms, such as the fast wavelet transform and the fast Fourier transform,
recursion and reduced complexity are intrinsic. For problems that do not find
compact representation, there may be fast N-Body solutions. N-Body solvers
combine recursive subdivision, elements of database theory that include
efficient metric- and range-queries, as well as multi-level approaches to rank
reduction and culling. In addition to a reduced computational complexity,
these database elements enable the exploitation of temporal and spatial
locality in high performance implementations, making the N-Body programming
model one of the most successful in large scale scientific simulation Warren
and Salmon (1992); Taiji et al. (1994); Fukushige and Makino (1996); Warren et
al. (1997); Kawai et al. (1999); Makino et al. (2003); Hamada et al. (2009);
Hamada and Nitadori (2010); Ishiyama et al. (2012). So far, it has been
possible to develop reduced complexity, N-Body solutions for many aspects of
conventional quantum chemical self consistent field theory Szabo and Ostlund
(1982), including the Hartree problem White et al. (1996); Challacombe et al.
(1996a); Challacombe and Schwegler (1997); Challacombe et al. (1996b),
exchange correlation cubature Challacombe (2000a), inverse factorization and
spectral projection based on the SpAMM algorithm Challacombe and Bock (2010);
Bock and Challacombe (2013).

Beyond the recursive multi-wavelet work of Yanai et al. Yanai et al. (2004),
N-Body solutions to Fock exchange in the conventional, Gaussian Type Atomic
Orbital (GTAO) representation remains an open and important problem. It’s
important because the Fock exchange is a key ingredient in hybrid theories
Becke (1993a, b) that yield qualitatively better results for many challenging
problems (relative to “pure” DFT), from metal oxides Perdew and Levy (1983);
Sham and Schlüter (1985); Xiao et al. (2011a); Bredow and Gerson (2000); Meyer
and Marx (2003); Muscat et al. (2001); Wang et al. (2011); Wilson (2009), to
chemical reactions Brittain et al. (2009); Zhao and Truhlar (2011) battery
materials Ramzan et al. (2011), photovoltaic semiconductors Xiao et al.
(2011b), and in biochemistry Ames and Larsen (2009); Rao et al. (2008);
Marianski et al. (2012).

Reduced complexity approaches to GTAO Fock exchange, such as ONX and its
variants Schwegler and Challacombe (1996); Schwegler et al. (1997, 1998);
Ochsenfeld et al. (1998); Schwegler and Challacombe (1999a, b); Schwegler
(1998); Schwegler and Challacombe (2000), remain predicated on density matrix
truncation (sparsification), preordered skip-out lists and/or matrix range
estimates, as well as list-of-lists matrix structures such as BCSR Challacombe
(2000b). These preprocessing steps and their associated data structures
greatly complicate aspects of the problem involving range-queries and metric-
queries necessary to implement multi-level methods for rank reduction and
culling Schwegler and Challacombe (1999b); Maurer et al. (2012), and also
force a choice between “integral driven” and “index driven” schemes for domain
decomposition Takashima et al. (2002); Weber and Challacombe (2006). In this
contribution, we outline an N-body reformulation of Fock exchange matrix
construction that enables task decomposition in the recursive space of tensor
products, and which also provides an embedded framework for metric- and range-
queries.

##  II RECURSIVE QUANTUM CHEMISTRY

We begin our development with recursive bisection of the indicial space of
GTAO basis functions:

$\left|\boldsymbol{\mu}\right)^{k}=\left[\begin{array}[]{c}\left|\boldsymbol{\mu}_{0}\right)^{k+1}\\\\[2.84544pt]
\hline\cr\left|\boldsymbol{\mu}_{1}\right)^{k+1}\end{array}\right]=\left[\begin{array}[]{c}\phi_{\mu^{L}_{0}}\\\
\phi_{\mu^{L+1}_{0}}\\\ \vdots\\\ \phi_{\mu^{R}_{0}}\\\
\hline\cr\phi_{\mu^{L}_{1}}\\\ \phi_{\mu^{L+1}_{1}}\\\ \vdots\\\
\phi_{\mu^{R}_{1}}\end{array}\right]\;,$ (1)

where $\left|\boldsymbol{\mu}\right)^{k}$ denotes a vector (block) of
functions $\phi$ at depth $k$ with span $\mu\in\left[\mu^{L},\mu^{R}\right]$.
This bisection can be accomplished in a variety of ways. Here, we consider
bisection with ragged edges, rather than by powers of two, so that the
underlying atomic-orbital shell structure (ie. $p,sp,d,f\dots$) is preserved,
greatly simplifying the associated computation of two-electron integrals.

The fundamental telescoping quantities are matrix quadtrees, discussed in
Refs. [Challacombe and Bock, 2010; Bock and Challacombe, 2013] and references
there in, and shell pairs, quadtrees that obtain recursively from shell-shell
outerproducts:

$\displaystyle\left|\boldsymbol{\mu\nu}\right)^{k}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}\left|\boldsymbol{\mu}_{0}\boldsymbol{\nu}_{0}\right)^{k+1}&\left|\boldsymbol{\mu}_{0}\boldsymbol{\nu}_{1}\right)^{k+1}\\\\[4.26773pt]
\left|\boldsymbol{\mu}_{1}\boldsymbol{\nu}_{0}\right)^{k+1}&\left|\boldsymbol{\mu}_{1}\boldsymbol{\nu}_{1}\right)^{k+1}\end{array}\right)$
(4) $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}\left|\boldsymbol{\mu}_{0}\right)^{k+1}\otimes\left|\boldsymbol{\nu}_{0}\right)^{k+1}&\left|\boldsymbol{\mu}_{0}\right)^{k+1}\otimes\left|\boldsymbol{\nu}_{1}\right)^{k+1}\\\\[4.26773pt]
\left|\boldsymbol{\mu}_{1}\right)^{k+1}\otimes\left|\boldsymbol{\nu}_{0}\right)^{k+1}&\left|\boldsymbol{\mu}_{1}\right)^{k+1}\otimes\left|\boldsymbol{\nu}_{1}\right)^{k+1}\end{array}\right)$
(7)

In the large system limit, the complexity with respect to number of GTAO basis
functions $N$ becomes ${\cal O}(N)$ due to non-overlapping functions,

$\left|\boldsymbol{\mu\nu}\right)^{k}=\begin{cases}0&{\tt
if}\;\left|\boldsymbol{\mu}\right)^{k}\cap\left|\boldsymbol{\nu}\right)^{k}=\emptyset\\\
\left|\boldsymbol{\mu}\right)^{k}\otimes\left|\boldsymbol{\nu}\right)^{k}&\tt
else\end{cases}$ (8)

where non-intersection obtains if all overlap integrals between shell-blocks
are sufficiently small,

$\left|\boldsymbol{\mu}\right)^{k}\cap\left|\boldsymbol{\nu}\right)^{k}=\emptyset\quad{\tt
if}\quad\left(\phi_{\mu},\phi_{\nu}\right)<\tau_{\tt ovlp}\\\
\forall\;\mu\in\left[\mu^{L},\mu^{R}\right],\;\nu\in\left[\nu^{L},\nu^{R}\right]\;,$
(9)

as determined by an overlap threshold $\tau_{\tt ovlp}$.

##  III EXCHANGE AS HEXTREE TRAVERSAL

At the top level ($k=0$), the Fock exchange matrix can be written simply with
the less than orthodox bra-ket notation:

${\bf
K}^{0}=-\frac{1}{2}\left(\boldsymbol{\mu}_{0}\boldsymbol{\nu}_{0}\right|^{0}{\bf
P}^{0}\left|\boldsymbol{\lambda}_{0}\boldsymbol{\sigma}_{0}\right)^{0}\;,$
(10)

which is useful shorthand for the tensor contraction

$K_{\mu\sigma}=-\frac{1}{2}\sum_{\nu,\lambda}P_{\nu,\lambda}\left(\mu\nu\right|\left.\lambda\sigma\right)\;,$
(11)

where $\left(\mu\nu\right|\left.\lambda\sigma\right)$ is the standard two-
electron integral over GTAO basis functions Szabo and Ostlund (1982).

In the case of naive recursion, where permutational symmetry of the two-
electron integrals is unexploited, shell-pair quadtrees maintain their
relationship with sub-indices on recursion, permitting the simplified
notation:
$\left|\boldsymbol{00}\right)^{k}\equiv\left|\boldsymbol{\mu_{0}\nu_{0}}\right)^{k}$,
$\left|\boldsymbol{01}\right)^{k}\equiv\left|\boldsymbol{\mu_{0}\nu_{1}}\right)^{k}$
and so on. Then, at all levels, sub-blocks of the Fock exchange matrix are

$\displaystyle{\bf K}^{k}_{\boldsymbol{00}}$ $\displaystyle\leftarrow$
$\displaystyle{\bf
K}^{k}_{\boldsymbol{00}}-\left[\left(\boldsymbol{00}\right|^{k}{\bf
P}^{k}_{\boldsymbol{00}}\left|\boldsymbol{00}\right)^{k}+\left(\boldsymbol{00}\right|^{k}{\bf
P}^{k}_{\boldsymbol{01}}\left|\boldsymbol{10}\right)^{k}\right.$ (12a)
$\displaystyle\left.+\left(\boldsymbol{10}\right|^{k}{\bf
P}^{k}_{\boldsymbol{10}}\left|\boldsymbol{00}\right)^{k}+\left(\boldsymbol{01}\right|^{k}{\bf
P}^{k}_{\boldsymbol{11}}\left|\boldsymbol{10}\right)^{k}\right]/2$
$\displaystyle{\bf K}^{k}_{\boldsymbol{01}}$ $\displaystyle\leftarrow$
$\displaystyle{\bf
K}^{k}_{\boldsymbol{01}}-\left[\left(\boldsymbol{00}\right|^{k}{\bf
P}^{k}_{\boldsymbol{00}}\left|\boldsymbol{01}\right)^{k}+\left(\boldsymbol{01}\right|^{k}{\bf
P}^{k}_{\boldsymbol{10}}\left|\boldsymbol{01}\right)^{k}\right.$ (12b)
$\displaystyle\left.+\left(\boldsymbol{00}\right|^{k}{\bf
P}^{k}_{\boldsymbol{01}}\left|\boldsymbol{11}\right)^{k}+\left(\boldsymbol{01}\right|^{k}{\bf
P}^{k}_{\boldsymbol{11}}\left|\boldsymbol{11}\right)^{k}\right]/2$
$\displaystyle{\bf K}^{k}_{\boldsymbol{10}}$ $\displaystyle\leftarrow$
$\displaystyle{\bf
K}^{k}_{\boldsymbol{10}}-\left[\left(\boldsymbol{10}\right|^{k}{\bf
P}^{k}_{\boldsymbol{00}}\left|\boldsymbol{00}\right)^{k}+\left(\boldsymbol{11}\right|^{k}{\bf
P}^{k}_{\boldsymbol{10}}\left|\boldsymbol{00}\right)^{k}\right.$ (12c)
$\displaystyle\left.+\left(\boldsymbol{10}\right|^{k}{\bf
P}^{k}_{\boldsymbol{01}}\left|\boldsymbol{10}\right)^{k}+\left(\boldsymbol{11}\right|^{k}{\bf
P}^{k}_{\boldsymbol{11}}\left|\boldsymbol{10}\right)^{k}\right]/2$
$\displaystyle{\bf K}^{k}_{\boldsymbol{11}}$ $\displaystyle\leftarrow$
$\displaystyle{\bf
K}^{k}_{\boldsymbol{11}}-\left[\left(\boldsymbol{10}\right|^{k}{\bf
P}^{k}_{\boldsymbol{00}}\left|\boldsymbol{01}\right)^{k}+\left(\boldsymbol{11}\right|^{k}{\bf
P}^{k}_{\boldsymbol{10}}\left|\boldsymbol{01}\right)^{k}\right.$ (12d)
$\displaystyle\left.+\left(\boldsymbol{10}\right|^{k}{\bf
P}^{k}_{\boldsymbol{01}}\left|\boldsymbol{11}\right)^{k}+\left(\boldsymbol{11}\right|^{k}{\bf
P}^{k}_{\boldsymbol{11}}\left|\boldsymbol{11}\right)^{k}\right]/2$

equivalent to hextree traversal in the recursive task space of exchange tensor
contraction. An advantage of this construction is that, together with
quadtrees that are informed at each level by the Frobenius norm
$\|\cdot\|_{{}_{F}}$, the blocked Almlöf-Alhrichs criteria Almlöf et al.
(1982); Häser and Ahlrichs (1989)

$\displaystyle\|\left({\boldsymbol{\mu}}{\boldsymbol{\nu}}\right|\left.{\boldsymbol{\mu}}{\boldsymbol{\nu}}\right)^{k}\|_{{}_{F}}\,\cdot\,\|{\bf
P}^{k}_{{\boldsymbol{\nu}}{\boldsymbol{\lambda}}}\|_{{}_{F}}\,\cdot\,\|\left({\boldsymbol{\lambda}}{\boldsymbol{\sigma}}\right|\left.{\boldsymbol{\lambda}}{\boldsymbol{\sigma}}\right)^{k}\|_{{}_{F}}\leq{\tau}_{2e}\;,$
(13)

can be carried out naturally via embedded metric-query, enabling recursion
termination when the bound is satisfied. Because $\|\cdot\|_{{}_{F}}$ is sub-
multiplicative, this procedure is rigorously equivalent to the standard direct
SCF method, with the integral threshold ${\tau}_{2e}$ retaining its
conventional meaning.

The efficiency of this query in culling negligible integral contributions is
dependent on the numerical structure of the the underlying data. A simple
solution to this problem involves ordering shells with a locality preserving
space filling curve, effectively clustering elements of like magnitude as
shown in Figs. 1 and 2 of Ref. [Bock and Challacombe, 2013]. Note that methods
relying on random permutation to make use of Cannon’s algorithm for the
parallel multiplication of sparse matrices Buluç and Gilbert (2008); Buluc and
Gilbert (2011) destroy these locality properties. As with the SpAMM
Challacombe and Bock (2010); Bock and Challacombe (2013) solver, truncation of
the vector space (sparsification) is not a prerequisite for achieving reduced
complexities, but well structured density matrices with decay are.

##  IV RECURSION WITH SYMMETRY

Extending the recursive approach outlined in the previous section to the
exploitation of 4-fold permutational symmetry is more involved than in
conventional schemes Ochsenfeld et al. (1998); Schwegler and Challacombe
(2000). Naively, we wish to employ

$\displaystyle{\bf K}^{k}_{{\boldsymbol{\mu}}{\boldsymbol{\sigma}}}\\!\\!\\!$
$\displaystyle\leftarrow$ $\displaystyle\\!\\!\\!{\bf
K}^{k}_{{\boldsymbol{\mu}}{\boldsymbol{\sigma}}}\\!\\!-\left({\boldsymbol{\mu}}{\boldsymbol{\nu}}\right|^{k}{\bf
P}^{k}_{{\boldsymbol{\nu}}{\boldsymbol{\lambda}}}\left|{\boldsymbol{\lambda}}{\boldsymbol{\sigma}}\right)^{k}/2$
(14a) $\displaystyle{\bf
K}^{k}_{{\boldsymbol{\nu}}{\boldsymbol{\sigma}}}\\!\\!\\!$
$\displaystyle\leftarrow$ $\displaystyle\\!\\!\\!{\bf
K}^{k}_{{\boldsymbol{\nu}}{\boldsymbol{\sigma}}}\\!\\!-\left({\boldsymbol{\nu}}{\boldsymbol{\mu}}\right|^{k}{\bf
P}^{k}_{{\boldsymbol{\mu}}{\boldsymbol{\lambda}}}\left|{\boldsymbol{\lambda}}{\boldsymbol{\sigma}}\right)^{k}/2$
(14b) $\displaystyle{\bf
K}^{k}_{{\boldsymbol{\mu}}{\boldsymbol{\lambda}}}\\!\\!\\!$
$\displaystyle\leftarrow$ $\displaystyle\\!\\!\\!{\bf
K}^{k}_{{\boldsymbol{\mu}}{\boldsymbol{\lambda}}}\\!\\!-\left({\boldsymbol{\mu}}{\boldsymbol{\nu}}\right|^{k}{\bf
P}^{k}_{{\boldsymbol{\nu}}{\boldsymbol{\sigma}}}\left|{\boldsymbol{\sigma}}{\boldsymbol{\lambda}}\right)^{k}/2$
(14c) $\displaystyle{\bf
K}^{k}_{{\boldsymbol{\nu}}{\boldsymbol{\lambda}}}\\!\\!\\!$
$\displaystyle\leftarrow$ $\displaystyle\\!\\!\\!{\bf
K}^{k}_{{\boldsymbol{\nu}}{\boldsymbol{\lambda}}}\\!\\!-\left({\boldsymbol{\nu}}{\boldsymbol{\mu}}\right|^{k}{\bf
P}^{k}_{{\boldsymbol{\mu}}{\boldsymbol{\sigma}}}\left|{\boldsymbol{\sigma}}{\boldsymbol{\lambda}}\right)^{k}/2$
(14d)

which has the potential to yield speedups of up to 4x in the evaluation of
two-electron integrals, but which does not reduce the cost of tensor
contraction. To avoid overcomputing however, the blockwise restriction

$\displaystyle\mu\leq\nu,\;\;\lambda\leq\sigma,\;\;\mu+\nu(\nu-1)/2\leq\lambda+\sigma(\sigma-1)/2$
$\displaystyle\forall\;\mu\in\left[\mu^{L},\mu^{R}\right],\quad\nu\in\left[\nu^{L},\nu^{R}\right],$
(15) $\displaystyle\lambda\in\left[\lambda^{L},\lambda^{R}\right],\quad{\rm
and}\quad\sigma\in\left[\sigma^{L},\sigma^{R}\right].$

must be observed at each level of recursion. Satisfying these inequalities
without carrying along an explosion of auxiliary source and sink sub-matrices
requires that we untether the strict 1-to-1 relationship between target
symmetries and matrix sub-indices, putting the transpose operation into play
for shell-pairs and matrices. These complications are resolved by introducing
an intermediate level of recursion, where target symmetries are determined and
links to density and exchange matrix sub-blocks are set, up to 4 of 16
possible links each. Including Eq. (9), there are 8 conditions that exploit
the full 4-fold symmetry of Fock exchange under recursion (the remaining 7 are
given in the Appendix), which involve non-standard, “across the bar”
(transpose) permutations as well as additional factors of two, (e.g. due to
restricting the density matrix above the diagonal. In addition to cases of
4-fold symmetry, there are many other intermediate and terminal cases where
there are fewer than 4 sub-matrices involved in recursion, for example due to
cases were blockwise symmetry operations do not yield a full 4-fold
compliment, and also in cases where the density matrix is sparse and sub-
blocks are not available. These later cases that fall outside of Eqs. (14) &
(16-22) are referred to as “sparse” in the following. Also, the blockwise
culling of negligible integral contributions is carried out recursively, as in
Eq. (13), but instead using the maximum density matrix norms as they occur at
each level.

Figure 1: Onset of linear scaling for the symmetry enhanced N-Body Fock
exchange solver using two screening regimes: $({\tau}_{\rm 2e},{\tau}_{\rm
ovlp})=$ (1d-8,1d-11) and (1d-10,1d-13). The inset is the ratio of CPU times
for naive recursion relative to symmetry enhanced recursion. Figure 2:
Percentage breakdown by contraction blocks, Eqs. (14) & (16-22), including
also “sparse” blocks, in the symmetry enhanced N-Body Fock exchange solver in
the tighter (1d-10,1d-13) regime. Inset shows blocks with fewer occurrences,
corresponding to Eqs. (17-22).

##  V IMPLEMENTATION AND RESULTS

Recursive construction of the Fock exchange matrix was implemented with and
without permutational symmetry in a development version of FreeON Bock et al.
(2014), using the Head-Gordon Pople algorithm for two-electron integrals Head-
Gordon and Pople (1988). Non-sparse contractions, involving full 4-fold matrix
compliments, are carried out in a single code block, different from
conventional contraction schemes in that there is no auxiliary “batch”
dimension available for vector optimization. In the current implementation,
recursion extends to blocks with dimension 10x10 or smaller. All calculations
were carried out on an Intel Xeon CPU E5-2687W @ 3.10GHz using v. 14.0.1 of
the Intel Fortran and C compilers.

Results are shown in Fig. 1 and 2 for the standard sequence of water droplets
corresponding to STP conditions, (H2O)n with $n=10,30,50,70,90,110,130$, and
density matrices tightly converged using an atom blocked threshold
${\tau}_{\rm mtrx}=$1d-6 Challacombe (2000b) at the B3LYP/6-31G** level of
theory. All systems were ordered using the Hilbert-curve to maximize locality
as shown in Figs. 1 and 2 of Ref. [Bock and Challacombe, 2013]. In the largest
droplet, (H2O)130, ragged bisection with $\sim$10x10 blocks yields a depth
$k=10$, comprising $\sim$6d+5 nodes in the shell-pair quadtree.

In Fig. 1, an early approach to linear scaling is shown for two values of the
integral screening parameter ${\tau}_{\rm 2e}$, together with the relative
performance of the naive vs symmetry enhanced methods as inset. In the case of
permutational symmetry, it is found that the cost of integral evaluation and
contraction tend towards 1:1. Figure 2 shows the % occurrence by symmetry
block, Eqs. (14) & (16-22), for the symmetry enhanced method. The dominant
occurrences correspond to Eq. (14) and (16) at about 60% and 20% respectively,
followed by the sparse case at 10%. Because this recursive task space
corresponds to a hextree, there are ample opportunities for Certainly, the
task parallel features of OpenMP 3.0 Duran et al. (2009) are ideally suited to
this formulation, and middleware for distributed memory task parallelism, such
as charm++ Kale and Krishnan (1993), are becoming increasing powerful and
simple to use. Also, the cost of integral contraction with and without
symmetry is about 2:3, reflecting loop overheads and higher levels of
optimization that favor the combined 4-fold contraction blocks.

##  VI Summary

We’ve presented a novel, $N$-Body formulation for the naive and symmetry
enhanced construction of the Fock exchange matrix, which achieves a reduced,
linear scaling complexity for density matrices with decay. The method does not
employ any of the matrix truncation (sparsification), integral skipout lists,
matrix range estimation or list-of-lists data structures employed by
conventional methods. Also, the method may enable transcendence of
conventional “index driven” vs “integral” driven paradigms for parallel Fock
matrix construction Takashima et al. (2002); Weber and Challacombe (2006),
through decomposition in the recursive task space of the tensor contraction.
Even though this task space is sparse and irregular due to culling, because it
is higher dimensional, corresponding to hextree traversal, it offers ample
opportunity for parallelism through well developed middleware such as OpenMP
3.0 Duran et al. (2009) in shared memory environments and charm++ Kale and
Krishnan (1993) in distributed environments.

With this contribution, all minimally essential components of reduced
complexity electronic structure theory at the self consistent field level of
theory have been reformulated as N-Body solvers. It remains to be seen how
tightly these solvers can be integrated using a common infrastructure and
programming model. Finally, it also remains to be seen if the ability to embed
range- and metric-queries into this structure can be exploited to achieve true
multi-level rank reduction for computation of the Fock exchange matrix.

###### Acknowledgements.

This work was supported by the U. S. Department of Energy under Contract No.
DE-AC52-06NA25396 and LDRD-ER grant 20110230ER. The authors acknowledge the
stimulating environment of the Ten-Bar Café and expert administrations of the
barista.

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*

##  Appendix A Appendix

Additional symmetry blocks:

$\displaystyle{\bf K}^{k}_{{\boldsymbol{\mu}}{\boldsymbol{\sigma}}}\\!\\!\\!$
$\displaystyle\leftarrow$ $\displaystyle\\!\\!\\!{\bf
K}^{k}_{{\boldsymbol{\mu}}{\boldsymbol{\sigma}}}\\!\\!-\left({\boldsymbol{\mu}}{\boldsymbol{\nu}}\right|^{k}{\bf
P}^{k}_{{\boldsymbol{\nu}}{\boldsymbol{\lambda}}}\left|{\boldsymbol{\lambda}}{\boldsymbol{\sigma}}\right)^{k}/2$
(16a) $\displaystyle{\bf
K}^{k}_{{\boldsymbol{\nu}}{\boldsymbol{\sigma}}}\\!\\!\\!$
$\displaystyle\leftarrow$ $\displaystyle\\!\\!\\!{\bf
K}^{k}_{{\boldsymbol{\nu}}{\boldsymbol{\sigma}}}\\!\\!-\left({\boldsymbol{\nu}}{\boldsymbol{\mu}}\right|^{k}{\bf
P}^{k}_{{\boldsymbol{\mu}}{\boldsymbol{\lambda}}}\left|{\boldsymbol{\lambda}}{\boldsymbol{\sigma}}\right)^{k}/2$
(16b) $\displaystyle{\bf
K}^{k}_{{\boldsymbol{\lambda}}{\boldsymbol{\mu}}}\\!\\!\\!$
$\displaystyle\leftarrow$ $\displaystyle\\!\\!\\!{\bf
K}^{k}_{{\boldsymbol{\lambda}}{\boldsymbol{\mu}}}\\!\\!-\left({\boldsymbol{\lambda}}{\boldsymbol{\sigma}}\right|^{k}{\bf
P}^{k}_{{\boldsymbol{\sigma}}{\boldsymbol{\nu}}}\left|{\boldsymbol{\nu}}{\boldsymbol{\mu}}\right)^{k}/2$
(16c) $\displaystyle{\bf
K}^{k}_{{\boldsymbol{\nu}}{\boldsymbol{\lambda}}}\\!\\!\\!$
$\displaystyle\leftarrow$ $\displaystyle\\!\\!\\!{\bf
K}^{k}_{{\boldsymbol{\nu}}{\boldsymbol{\lambda}}}\\!\\!-\left({\boldsymbol{\nu}}{\boldsymbol{\mu}}\right|^{k}{\bf
P}^{k}_{{\boldsymbol{\mu}}{\boldsymbol{\sigma}}}\left|{\boldsymbol{\sigma}}{\boldsymbol{\lambda}}\right)^{k}/2$
(16d) $\displaystyle{\bf
K}^{k}_{{\boldsymbol{\mu}}{\boldsymbol{\sigma}}}\\!\\!\\!$
$\displaystyle\leftarrow$ $\displaystyle\\!\\!\\!{\bf
K}^{k}_{{\boldsymbol{\mu}}{\boldsymbol{\sigma}}}\\!\\!-\left({\boldsymbol{\mu}}{\boldsymbol{\nu}}\right|^{k}{\bf
P}^{k}_{{\boldsymbol{\nu}}{\boldsymbol{\lambda}}}\left|{\boldsymbol{\lambda}}{\boldsymbol{\sigma}}\right)^{k}/2$
(17a) $\displaystyle{\bf
K}^{k}_{{\boldsymbol{\nu}}{\boldsymbol{\lambda}}}\\!\\!\\!$
$\displaystyle\leftarrow$ $\displaystyle\\!\\!\\!{\bf
K}^{k}_{{\boldsymbol{\nu}}{\boldsymbol{\lambda}}}\\!\\!-\left({\boldsymbol{\nu}}{\boldsymbol{\mu}}\right|^{k}{\bf
P}^{k}_{{\boldsymbol{\mu}}{\boldsymbol{\sigma}}}\left|{\boldsymbol{\sigma}}{\boldsymbol{\lambda}}\right)^{k}/2$
(17b) $\displaystyle{\bf
K}^{k}_{{\boldsymbol{\mu}}{\boldsymbol{\lambda}}}\\!\\!\\!$
$\displaystyle\leftarrow$ $\displaystyle\\!\\!\\!{\bf
K}^{k}_{{\boldsymbol{\mu}}{\boldsymbol{\lambda}}}\\!\\!-\left({\boldsymbol{\mu}}{\boldsymbol{\nu}}\right|^{k}{\bf
P}^{k}_{{\boldsymbol{\nu}}{\boldsymbol{\sigma}}}\left|{\boldsymbol{\sigma}}{\boldsymbol{\lambda}}\right)^{k}/2$
(17c) $\displaystyle{\bf
K}^{k}_{{\boldsymbol{\lambda}}{\boldsymbol{\nu}}}\\!\\!\\!$
$\displaystyle\leftarrow$ $\displaystyle\\!\\!\\!{\bf
K}^{k}_{{\boldsymbol{\lambda}}{\boldsymbol{\nu}}}\\!\\!-\left({\boldsymbol{\lambda}}{\boldsymbol{\sigma}}\right|^{k}{\bf
P}^{k}_{{\boldsymbol{\sigma}}{\boldsymbol{\mu}}}\left|{\boldsymbol{\mu}}{\boldsymbol{\nu}}\right)^{k}/2$
(17d) $\displaystyle{\bf
K}^{k}_{{\boldsymbol{\mu}}{\boldsymbol{\sigma}}}\\!\\!\\!$
$\displaystyle\leftarrow$ $\displaystyle\\!\\!\\!{\bf
K}^{k}_{{\boldsymbol{\mu}}{\boldsymbol{\sigma}}}\\!\\!-\left({\boldsymbol{\mu}}{\boldsymbol{\nu}}\right|^{k}{\bf
P}^{k}_{{\boldsymbol{\nu}}{\boldsymbol{\lambda}}}\left|{\boldsymbol{\lambda}}{\boldsymbol{\sigma}}\right)^{k}/2$
(18a) $\displaystyle{\bf
K}^{k}_{{\boldsymbol{\nu}}{\boldsymbol{\sigma}}}\\!\\!\\!$
$\displaystyle\leftarrow$ $\displaystyle\\!\\!\\!{\bf
K}^{k}_{{\boldsymbol{\nu}}{\boldsymbol{\sigma}}}\\!\\!-\left({\boldsymbol{\nu}}{\boldsymbol{\mu}}\right|^{k}{\bf
P}^{k}_{{\boldsymbol{\mu}}{\boldsymbol{\lambda}}}\left|{\boldsymbol{\lambda}}{\boldsymbol{\sigma}}\right)^{k}/2$
(18b) $\displaystyle{\bf
K}^{k}_{{\boldsymbol{\mu}}{\boldsymbol{\lambda}}}\\!\\!\\!$
$\displaystyle\leftarrow$ $\displaystyle\\!\\!\\!{\bf
K}^{k}_{{\boldsymbol{\mu}}{\boldsymbol{\lambda}}}\\!\\!-\left({\boldsymbol{\mu}}{\boldsymbol{\nu}}\right|^{k}{\bf
P}^{k}_{{\boldsymbol{\nu}}{\boldsymbol{\sigma}}}\left|{\boldsymbol{\sigma}}{\boldsymbol{\lambda}}\right)^{k}/2$
(18c) $\displaystyle{\bf
K}^{k}_{{\boldsymbol{\lambda}}{\boldsymbol{\nu}}}\\!\\!\\!$
$\displaystyle\leftarrow$ $\displaystyle\\!\\!\\!{\bf
K}^{k}_{{\boldsymbol{\lambda}}{\boldsymbol{\nu}}}\\!\\!-\left({\boldsymbol{\lambda}}{\boldsymbol{\sigma}}\right|^{k}{\bf
P}^{k}_{{\boldsymbol{\sigma}}{\boldsymbol{\mu}}}\left|{\boldsymbol{\mu}}{\boldsymbol{\nu}}\right)^{k}/2$
(18d) $\displaystyle{\bf
K}^{k}_{{\boldsymbol{\mu}}{\boldsymbol{\sigma}}}\\!\\!\\!$
$\displaystyle\leftarrow$ $\displaystyle\\!\\!\\!{\bf
K}^{k}_{{\boldsymbol{\mu}}{\boldsymbol{\sigma}}}\\!\\!-\left({\boldsymbol{\mu}}{\boldsymbol{\nu}}\right|^{k}{\bf
P}^{k}_{{\boldsymbol{\nu}}{\boldsymbol{\lambda}}}\left|{\boldsymbol{\lambda}}{\boldsymbol{\sigma}}\right)^{k}/2$
(19a) $\displaystyle{\bf
K}^{k}_{{\boldsymbol{\nu}}{\boldsymbol{\sigma}}}\\!\\!\\!$
$\displaystyle\leftarrow$ $\displaystyle\\!\\!\\!{\bf
K}^{k}_{{\boldsymbol{\nu}}{\boldsymbol{\sigma}}}\\!\\!-\left({\boldsymbol{\nu}}{\boldsymbol{\mu}}\right|^{k}{\bf
P}^{k}_{{\boldsymbol{\mu}}{\boldsymbol{\lambda}}}\left|{\boldsymbol{\lambda}}{\boldsymbol{\sigma}}\right)^{k}/2$
(19b) $\displaystyle{\bf
K}^{k}_{{\boldsymbol{\lambda}}{\boldsymbol{\mu}}}\\!\\!\\!$
$\displaystyle\leftarrow$ $\displaystyle\\!\\!\\!{\bf
K}^{k}_{{\boldsymbol{\lambda}}{\boldsymbol{\mu}}}\\!\\!-\left({\boldsymbol{\lambda}}{\boldsymbol{\sigma}}\right|^{k}{\bf
P}^{k}_{{\boldsymbol{\sigma}}{\boldsymbol{\nu}}}\left|{\boldsymbol{\nu}}{\boldsymbol{\mu}}\right)^{k}/2$
(19c) $\displaystyle{\bf
K}^{k}_{{\boldsymbol{\lambda}}{\boldsymbol{\nu}}}\\!\\!\\!$
$\displaystyle\leftarrow$ $\displaystyle\\!\\!\\!{\bf
K}^{k}_{{\boldsymbol{\lambda}}{\boldsymbol{\nu}}}\\!\\!-\left({\boldsymbol{\lambda}}{\boldsymbol{\sigma}}\right|^{k}{\bf
P}^{k}_{{\boldsymbol{\sigma}}{\boldsymbol{\mu}}}\left|{\boldsymbol{\mu}}{\boldsymbol{\nu}}\right)^{k}/2$
(19d) $\displaystyle{\bf
K}^{k}_{{\boldsymbol{\mu}}{\boldsymbol{\sigma}}}\\!\\!\\!$
$\displaystyle\leftarrow$ $\displaystyle\\!\\!\\!{\bf
K}^{k}_{{\boldsymbol{\mu}}{\boldsymbol{\sigma}}}\\!\\!-\left({\boldsymbol{\mu}}{\boldsymbol{\nu}}\right|^{k}{\bf
P}^{k}_{{\boldsymbol{\nu}}{\boldsymbol{\nu}}}\left|{\boldsymbol{\nu}}{\boldsymbol{\sigma}}\right)^{k}/2$
(20a) $\displaystyle{\bf
K}^{k}_{{\boldsymbol{\nu}}{\boldsymbol{\sigma}}}\\!\\!\\!$
$\displaystyle\leftarrow$ $\displaystyle\\!\\!\\!{\bf
K}^{k}_{{\boldsymbol{\nu}}{\boldsymbol{\sigma}}}\\!\\!-\left({\boldsymbol{\nu}}{\boldsymbol{\mu}}\right|^{k}{\bf
P}^{k}_{{\boldsymbol{\mu}}{\boldsymbol{\nu}}}\left|{\boldsymbol{\nu}}{\boldsymbol{\sigma}}\right)^{k}/2$
(20b) $\displaystyle{\bf
K}^{k}_{{\boldsymbol{\mu}}{\boldsymbol{\nu}}}\\!\\!\\!$
$\displaystyle\leftarrow$ $\displaystyle\\!\\!\\!{\bf
K}^{k}_{{\boldsymbol{\mu}}{\boldsymbol{\nu}}}\\!\\!-\left({\boldsymbol{\mu}}{\boldsymbol{\nu}}\right|^{k}{\bf
P}^{k}_{{\boldsymbol{\nu}}{\boldsymbol{\sigma}}}\left|{\boldsymbol{\sigma}}{\boldsymbol{\nu}}\right)^{k}/2$
(20c) $\displaystyle{\bf
K}^{k}_{{\boldsymbol{\nu}}{\boldsymbol{\nu}}}\\!\\!\\!$
$\displaystyle\leftarrow$ $\displaystyle\\!\\!\\!{\bf
K}^{k}_{{\boldsymbol{\nu}}{\boldsymbol{\nu}}}\\!\\!-\left[\left({\boldsymbol{\nu}}{\boldsymbol{\sigma}}\right|^{k}{\bf
P}^{k}_{{\boldsymbol{\sigma}}{\boldsymbol{\mu}}}\left|{\boldsymbol{\mu}}{\boldsymbol{\nu}}\right)^{k}\right.$
$\displaystyle\left.\qquad+\left({\boldsymbol{\nu}}{\boldsymbol{\mu}}\right|^{k}{\bf
P}^{k}_{{\boldsymbol{\mu}}{\boldsymbol{\sigma}}}\left|{\boldsymbol{\sigma}}{\boldsymbol{\nu}}\right)^{k}\right]/2$
$\displaystyle{\bf K}^{k}_{{\boldsymbol{\mu}}{\boldsymbol{\nu}}}\\!\\!\\!$
$\displaystyle\leftarrow$ $\displaystyle\\!\\!\\!{\bf
K}^{k}_{{\boldsymbol{\mu}}{\boldsymbol{\nu}}}\\!\\!-\left({\boldsymbol{\mu}}{\boldsymbol{\nu}}\right|^{k}{\bf
P}^{k}_{{\boldsymbol{\nu}}{\boldsymbol{\nu}}}\left|{\boldsymbol{\nu}}{\boldsymbol{\nu}}\right)^{k}/2$
(21a) $\displaystyle{\bf
K}^{k}_{{\boldsymbol{\nu}}{\boldsymbol{\nu}}}\\!\\!\\!$
$\displaystyle\leftarrow$ $\displaystyle\\!\\!\\!{\bf
K}^{k}_{{\boldsymbol{\nu}}{\boldsymbol{\nu}}}\\!\\!-\left[\left({\boldsymbol{\nu}}{\boldsymbol{\nu}}\right|^{k}{\bf
P}^{k}_{{\boldsymbol{\nu}}{\boldsymbol{\mu}}}\left|{\boldsymbol{\mu}}{\boldsymbol{\nu}}\right)^{k}\right.$
$\displaystyle\left.\qquad+\left({\boldsymbol{\nu}}{\boldsymbol{\mu}}\right|^{k}{\bf
P}^{k}_{{\boldsymbol{\mu}}{\boldsymbol{\nu}}}\left|{\boldsymbol{\nu}}{\boldsymbol{\nu}}\right)^{k}\right]/2$
$\displaystyle{\bf K}^{k}_{{\boldsymbol{\mu}}{\boldsymbol{\lambda}}}\\!\\!\\!$
$\displaystyle\leftarrow$ $\displaystyle\\!\\!\\!{\bf
K}^{k}_{{\boldsymbol{\mu}}{\boldsymbol{\nu}}}\\!\\!-\left({\boldsymbol{\mu}}{\boldsymbol{\nu}}\right|^{k}{\bf
P}^{k}_{{\boldsymbol{\nu}}{\boldsymbol{\nu}}}\left|{\boldsymbol{\nu}}{\boldsymbol{\lambda}}\right)^{k}/2$
(21c) $\displaystyle{\bf
K}^{k}_{{\boldsymbol{\lambda}}{\boldsymbol{\nu}}}\\!\\!\\!$
$\displaystyle\leftarrow$ $\displaystyle\\!\\!\\!{\bf
K}^{k}_{{\boldsymbol{\lambda}}{\boldsymbol{\nu}}}\\!\\!-\left({\boldsymbol{\lambda}}{\boldsymbol{\nu}}\right|^{k}{\bf
P}^{k}_{{\boldsymbol{\nu}}{\boldsymbol{\mu}}}\left|{\boldsymbol{\mu}}{\boldsymbol{\nu}}\right)^{k}/2$
(21d) $\displaystyle{\bf
K}^{k}_{{\boldsymbol{\mu}}{\boldsymbol{\sigma}}}\\!\\!\\!$
$\displaystyle\leftarrow$ $\displaystyle\\!\\!\\!{\bf
K}^{k}_{{\boldsymbol{\mu}}{\boldsymbol{\sigma}}}\\!\\!-\left({\boldsymbol{\mu}}{\boldsymbol{\nu}}\right|^{k}{\bf
P}^{k}_{{\boldsymbol{\nu}}{\boldsymbol{\mu}}}\left|{\boldsymbol{\mu}}{\boldsymbol{\sigma}}\right)^{k}/2$
(22a) $\displaystyle{\bf
K}^{k}_{{\boldsymbol{\nu}}{\boldsymbol{\sigma}}}\\!\\!\\!$
$\displaystyle\leftarrow$ $\displaystyle\\!\\!\\!{\bf
K}^{k}_{{\boldsymbol{\nu}}{\boldsymbol{\sigma}}}\\!\\!-\left({\boldsymbol{\nu}}{\boldsymbol{\mu}}\right|^{k}{\bf
P}^{k}_{{\boldsymbol{\mu}}{\boldsymbol{\mu}}}\left|{\boldsymbol{\mu}}{\boldsymbol{\sigma}}\right)^{k}/2$
(22b) $\displaystyle{\bf
K}^{k}_{{\boldsymbol{\mu}}{\boldsymbol{\mu}}}\\!\\!\\!$
$\displaystyle\leftarrow$ $\displaystyle\\!\\!\\!{\bf
K}^{k}_{{\boldsymbol{\mu}}{\boldsymbol{\mu}}}\\!\\!-\left[\left({\boldsymbol{\mu}}{\boldsymbol{\nu}}\right|^{k}{\bf
P}^{k}_{{\boldsymbol{\nu}}{\boldsymbol{\sigma}}}\left|{\boldsymbol{\sigma}}{\boldsymbol{\mu}}\right)^{k}\right.$
(22d)
$\displaystyle\left.\qquad+\left({\boldsymbol{\mu}}{\boldsymbol{\sigma}}\right|^{k}{\bf
P}^{k}_{{\boldsymbol{\sigma}}{\boldsymbol{\nu}}}\left|{\boldsymbol{\nu}}{\boldsymbol{\mu}}\right)^{k}\right]/2$
$\displaystyle{\bf K}^{k}_{{\boldsymbol{\nu}}{\boldsymbol{\mu}}}\\!\\!\\!$
$\displaystyle\leftarrow$ $\displaystyle\\!\\!\\!{\bf
K}^{k}_{{\boldsymbol{\nu}}{\boldsymbol{\mu}}}\\!\\!-\left({\boldsymbol{\nu}}{\boldsymbol{\mu}}\right|^{k}{\bf
P}^{k}_{{\boldsymbol{\mu}}{\boldsymbol{\sigma}}}\left|{\boldsymbol{\sigma}}{\boldsymbol{\mu}}\right)^{k}/2$
(22e)