# The Avoidance Principle for Noncompact Hypersurfaces Moving by Mean
Curvature Flow

Brian White  Department of Mathematics
Stanford University
Stanford, CA 94305

(Date: 1 February, 2024)

###### Abstract.

Consider a pair of smooth, possibly noncompact, properly immersed
hypersurfaces moving by mean curvature flow, or, more generally, a pair of
weak set flows. We prove that if the ambient space is Euclidean space and if
the distance between the two surfaces is initially nonzero, then the surfaces
remain disjoint at all subsequent times.

We prove the same result when the ambient space is a complete Riemannian
manifold of nonzero injectivity radius, provided the curvature tensor (of the
ambient space) and all its derivatives are bounded.

###### Key words and phrases:

Mean curvature flow, avoidance, level set flow, weak set flow

###### 2020 Mathematics Subject Classification:

Primary 53E10

##  1\. introduction

The classical avoidance principle for mean curvature flow says that two
smooth, properly immersed, initially disjoint hypersurfaces moving by mean
curvature flow in Euclidean space remain disjoint as long as they are smooth,
if at least one of them is compact. That avoidance principle is an easy
consequence of the strong maximum principle. Ilmanen ([ilmanen-elliptic]*§10,
[ilmanen-proc]) generalized the avoidance principle to arbitrary “set-
theoretic subsolutions of mean curvature flow” or (in the terminology of
[white-topology] and [hershkovits-w-avoid]) “weak set flows”. A special case
is that of surfaces moving by the level set flow of [chen-giga-goto] and
[evans-spruck]. The support of a codimension-one, integral Brakke flow is a
weak set flow, so Ilmanen’s avoidance principle also applies to such Brakke
flows. In [hershkovits-w-avoid], Ilmanen’s avoidance principle for weak set
flows in Euclidean space was extended to weak set flows in a complete
Riemannian manifold, provided the Ricci curvature of the ambient manifold is
bounded below.

Those papers left open, even in the case of smooth hypersurfaces in Euclidean
space, the question of whether there is an avoidance principle when neither
surface is compact. The correct hypothesis for such an avoidance principle is
not that the surfaces are initially disjoint. There is, for example, a smooth
curvature flow $t\in[0,T)\mapsto C(t)$ in the plane such that at time $0$ the
curve is the union of the graphs $y=1/(1+x^{2})$ and $y=-1/(1+x^{2})$ and such
that at times $t\in(0,T)$, the curve is a simple closed curve. (See [ilmanen-
indiana]*7.3.) Of course the static flow $t\mapsto
X(t):=\mathbf{R}\times\\{0\\}$ is also a curvature flow. Note that $C(t)$ and
$X(t)$ are disjoint at time $0$ but intersect for all $t\in(0,T)$.

The correct hypothesis is that the distance between the two surfaces is
initially positive. Note that if either surface is compact, this hypothesis is
equivalent to disjointness.

This paper proves that the avoidance principle does indeed remain true for
noncompact weak set flows in Euclidean space, or, more generally, in a
Riemannian manifold, provided the manifold is well-behaved at infinity:

######  Theorem 1 (Avoidance Theorem).

Suppose that $N$ is a complete, connected Riemannian manifold with positive
injectivity radius such that $|\nabla^{k}\operatorname{Riem}|$ is bounded for
each nonnegative integer $k$. Let $\Lambda$ be a lower bound for the Ricci
curvature of $N$. Suppose that $t\in[0,\infty)\mapsto X(t),Y(t)$ are weak set
flows in $N$. Then

$e^{-\Lambda t}d(X(t),Y(t))$

is an increasing function of $t$.

Here $d(P,Q)=\inf\\{d(p,q):p\in P,\,q\in Q\\}$, where $d(p,q)$ is geodesic
distance from $p$ to $q$. (Thus $d(p,q)=|p-q|$ in Euclidean space.)

We now describe Ilmanen’s proof that the avoidance principle holds in
Euclidean space if at least one of the flows is compact. By definition of weak
set flow, a weak set flow cannot bump into a smooth, compact mean curvature
flow, provided the two flows are initially disjoint. It follows easily (in
Euclidean space) that the distance between the two is an increasing function
of time. Suppose that two weak set flows $X(\cdot)$ and $Y(\cdot)$, one
compact, are initially disjoint. Ilmanen proved a $C^{1,1}$ interpolation
theorem, according to which there is a compact $C^{1,1}$ hypersurface $M$
between $X(0)$ and $Y(0)$ such that

(1) $d(X(0),M)=R=d(Y(0),M),$

where $R=\frac{1}{2}d(X(0),Y(0))$. Now let $M$ evolve by mean curvature flow.
Then $M(t)$ will be smooth for $t\in(0,\epsilon]$ for some $\epsilon>0$. It
follows that $d(X(t),M(t))\geq R$ and $d(Y(t),M)\geq R$ for all
$t\in[0,\epsilon]$, and therefore that

$d(X(t),Y(t))\geq d(X(t),M(t))+d(M(t),Y(t))\geq 2R$

for all $t\in[0,\epsilon]$. The result follows rather directly.

The same idea was used in [hershkovits-w-avoid] for general ambient manifolds.
That paper replaced Ilmanen’s $C^{1,1}$ interpolation theorem by an easier
$C^{1}$ interpolation theorem.

If one tries to adapt Ilmanen’s proof to noncompact $X(t)$ and $Y(t)$, one can
at least get started: there is a locally $C^{1,1}$ surface $M$ between $X(0)$
and $Y(0)$ satisfying (1). However, it is not clear that there is uniformly
$C^{1,1}$ interpolating surface, and it seems that uniform $C^{1,1}$ bounds
are needed in order to use it to prove avoidance.

Likewise, to adapt the proof in [hershkovits-w-avoid] to noncompact $X(t)$ and
$Y(t)$, it would seem that uniform $C^{1}$ bounds on the interpolating surface
$M$ are needed. However, this paper shows that uniform $C^{1}$ bounds are not
required.

Let $K$ be the set of points in $N$ at distance $\geq R$ from $X(0)\cup Y(0)$,
where $R=(1/2)d(X(0),Y(0))$. Thus the interpolating surface $M$ of
[hershkovits-w-avoid] is contained in $K$. This paper does not prove uniform
$C^{1}$ bounds on $M$. However, it does prove uniform $C^{1}$ bounds on the
points of $M$ near $\partial K$, and it shows that those bounds suffice for
proving avoidance.

The organization of the paper is as follows. Sections 2 and 3 describe
conventions in the paper and prove some preliminary facts about weak set
flows. Section 4 gives the proof of the avoidance principle. The proof uses
various facts about the distance function, about certain harmonic functions,
and about separating Brakke flows. Those facts are proved in Sections 5, 6,
and 7.

To understand the main ideas of the proof, it may be helpful to focus on the
case when the ambient space is Euclidean. In that case, it is not necessary to
use the exponential map. Where the proof uses the exponential map at a point
$p$, in the Euclidean case one simply translates by $-p$ (and, in some cases,
also rotates and dilates).

The definition and basic properties of weak set flows may be found in
[hershkovits-w-avoid]. Sections 6 and 7 of [ilmanen-elliptic] give a concise
introduction to Brakke flows.

##  2\. Conventions

If $t\in[0,\infty)\mapsto M(t)$ is a integral Brakke flow in the Riemannian
manifold $N$, we let $\mathbf{M}$ denote its spacetime support:

$\mathbf{M}:=\overline{\\{(x,t):t\in[0,\infty),\,x\in\operatorname{spt}M(t)\\}},$

and we let

$\mathbf{M}(t)=\\{x:(x,t)\in\mathbf{M}\\}.$

Thus $t\mapsto\mathbf{M}(t)$ is the weak set flow associated to the Brakke
flow $M(\cdot)$.

If $p$ and $q$ are points in a connected Riemannian manifold $N$ and if $X$
and $Y$ are subsets of $N$, we let $d(p,q)$ be the geodesic distance from $p$
to $q$, and we let

$\displaystyle d(p,X)=d(X,p)=\inf_{x\in X}d(p,x),$ $\displaystyle
d(X,Y)=\inf_{x\in X,y\in Y}d(x,y).$

##  3\. Preliminaries

Throughout the paper, $N$ is a complete, connected Riemannian $(m+1)$-manifold
(without boundary) such that $\operatorname{inj}(N)>0$ (where
$\operatorname{inj}(N)$ is the injectivity radius of $N$), and such that

$\sup_{N}|\nabla^{k}\operatorname{Riem}|<\infty$

for each nonnegative integer $k$.

######  Lemma 2 (Finite Speed Lemma).

For $r>0$, there is an $h=h(r,\Lambda,m)$ with the following property. If
$S\subset N$, if $t\in[0,\infty)\mapsto X(t)$ is a weak set flow, and if

$r<R:=d(X(0),S),$

then

$d(X(t),S)\geq R-ht\qquad\text{for $0\leq t\leq(R-r)/h$}.$

See [hershkovits-w-avoid]*Theorem 5.

The Finite Speed Lemma 2 implies control on how fast a weak set flow can move
away from its initial set:

######  Corollary 3.

For every $\epsilon>0$, there is a $\delta>0$ with the following property. If
$t\in[0,\infty)\mapsto Z(t)$ is a weak set flow in $N$, then

$\cup_{t\in[0,\delta]}Z(t)\subset\\{p:d(p,Z(0))\leq\epsilon\\}.$

The following corollary gives a bound on the rate at which two weak set flows
can approach each other.

######  Corollary 4.

If $t\in[t_{0},\infty)\mapsto X(t),\,Y(t)$ are weak set flows with

$d(X(t_{0}),Y(t_{0}))>r>0,$

then

$d(X(t_{0}+t),Y(t_{0}+t))\geq d(X(t_{0}),Y(t_{0}))-2ht$

for $t\leq(R-r)/h$, where $h$ is as in Lemma 2.

###### Proof.

It suffices to prove it for $t_{0}=0$. Let

$R=\frac{1}{2}d(X(t_{0}),Y(t_{0})),$

and let $\tilde{X}=\\{p:d(p,X(0))\geq R\\}$ and $\tilde{Y}=\\{p:d(p,Y(0))\geq
R\\}$. Then

$\displaystyle d(X(t),\tilde{X})\geq R-ht,$ $\displaystyle
d(Y(t),\tilde{Y})\geq R-ht$

for $t\in[0,(R-r)/h]$. Thus

$d(X(t),Y(t))\geq d(X(t),\tilde{X})+d(\tilde{Y},Y(t))\geq 2R-2ht.$

∎

######  Proposition 5.

Suppose that $\lambda\in\mathbf{R}$. The following are equivalent.

  1. (1)

If $t\in[t_{0},\infty)\mapsto X(t),Y(t)$ are weak set flows in $N$, then there
is an $\epsilon>0$ such that

$e^{-\lambda(t-t_{0})}d(X(t),Y(t))\geq d(X(t_{0}),Y(t_{0}))$

for all $t\in[t_{0},t_{0}+\epsilon]$.

  2. (2)

If $t\in[t_{0},\infty)\mapsto X(t),Y(t)$ are weak set flows that in $N$, then

$e^{-\lambda(t-t_{0})}d(X(t),Y(t))\geq d(X(t_{0}),Y(t_{0}))$

for all $t\in[t_{0},\infty)$.

  3. (3)

If $t\in[0,\infty)\mapsto X(t),Y(t)$ are weak set flows, then

$t\in[0,\infty)\mapsto e^{-\lambda t}d(X(t),Y(t))$

is an increasing function of $t$.

Furthermore, if

$t\in[0,\infty)\mapsto e^{-\lambda t}d(X(t),Y(t))$

is an increasing function of $t$ for each $\lambda<\Lambda$, then it is also
an increasing function of $t$ for $\lambda=\Lambda$.

###### Proof.

Suppose that (1) holds. In (2), let $\mathcal{T}$ be the set of
$T\in[t_{0},\infty)$ such that the inequality in (2) holds for all
$t\in[t_{0},T]$. Then $\mathcal{T}$ is closed (by Corollary 4), and it is
nonempty since $t_{0}\in\mathcal{T}$. By (1), $\mathcal{T}$ is relatively open
in $[t_{0},\infty)$. Thus $\mathcal{T}=[t_{0},\infty)$. Hence (1) implies (2).

Trivially, (2) implies (3) and (3) implies (1). The “furthermore” assertion is
also trivially true. ∎

The proof of the Avoidance Theorem uses the following version of the maximum
principle for weak set flows:

######  Proposition 6.

Suppose that $\lambda\in\mathbf{R}$, that $W$ is a smooth (not necessarily
complete) Riemannian manifold, and that $X(\cdot)$ and $Y(\cdot)$ are weak set
flows in $W$. Suppose that $T>0$ and that

$\displaystyle e^{-\lambda t}d(X(t),Y(t))$ $\displaystyle>\eta\quad\text{for
$0\leq t<T$},$ $\displaystyle e^{-\lambda T}d(X(T),Y(T))$
$\displaystyle=\eta.$

Suppose also that there is a geodesic $\Gamma$ of length $d(X(T),Y(T))$ from a
point $x\in X(T)$ to a point $y\in Y(T)$, and that $(y,T)$ is a regular point
of the flow $Y(\cdot)$. Then there are points in $W$ (indeed, points on
$\Gamma$) where the Ricci curvature is $\leq\lambda$.

See [hershkovits-w-avoid]*Lemma 10. In the statement of [hershkovits-w-
avoid]*Lemma 10, the flow $K(\cdot)$ in the lemma is assumed to be smooth
everywhere. But the proof only requires $K(\cdot)$ to be smooth in a spacetime
neighborhood of the point $(p,b)$ in that lemma. (Indeed, by replacing $W$ by
a suitable open subset $W^{\prime}$, $[0,T]$ by a suitable subinteral
$[t_{0},T]$, and $X(\cdot)$ and $Y(\cdot)$ by $X(\cdot)\cap W^{\prime}$ and
$Y(\cdot)\cap W^{\prime}$, one can reduce the general case to the case when
$Y(\cdot)$ is smooth everywhere.)

######  Corollary 7.

Suppose $W$ is a smooth, complete Riemannian manifold with Ricci curvature
bounded below by $\Lambda$. If $t\mapsto X(t)$ is a weak set flow in $W$, if
$\lambda<\Lambda$, and if $c\geq 0$, then

$t\mapsto\\{p:d(p,X(t))\leq ce^{\lambda t}\\},$

is also a weak set flow.

Of course, since a limit of weak set flows is a weak set flow, Corollary 7
also holds for $\lambda=\Lambda$.

##  4\. The Avoidance Theorem

######  Theorem 8 (Avoidance Theorem).

Suppose that $N$ is a complete, connected Riemannian $(m+1)$-manifold (without
boundary) such that $\operatorname{inj}(N)>0$ and such that

$\sup_{N}|\nabla^{k}\operatorname{Riem}|<\infty$

for each nonnegative integer $k$. Let $\Lambda$ be a lower bound for the Ricci
curvature of $N$. Suppose that $t\in[0,\infty)\mapsto X(t),\,Y(t)$ are weak
set flows in $N$. Then

$e^{-\Lambda t}d(X(t),Y(t))$

is an increasing function of $t\in[0,\infty)$.

###### Proof.

By Proposition 5, it suffices to prove that if $\lambda<\Lambda$, and if

$R=\frac{1}{2}d(X(0),Y(0)),$

then there is an $\epsilon>0$ such that

(2) $\text{ $e^{-\lambda t}d(X(t),Y(t))\geq 2R$ for $t\in[0,\epsilon]$}.$

We may assume that $R>0$, as (2) is trivially true for $R=0$. We may also
assume that $\Lambda\leq 0$, as otherwise $N$ is compact, and the avoidance
principle is already known in that case. See [hershkovits-w-avoid].

######  Case 1.

$d(X(0),Y(0)\leq r:=\frac{1}{2}\operatorname{inj}(N)$.

Let $X=X(0)$ and $Y=Y(0)$. Let $K$ be the set of points $p$ such that
$d(p,X\cup Y)\geq R$. Let $S$ be the set of points $p$ such that
$d(p,X)=d(p,Y)=R$. Thus $S\subset\partial K$.

We will use the following interpolation theorem.

######  Theorem 9 (Interpolation Theorem).

There is a closed region $\Omega$ in $N$ with the following properties:

  1. (1)

$\Omega$ is a $C^{1}$ manifold-with-boundary.

  2. (2)

$\Omega$ contains $\\{p:d(p,X)\leq R\\}$ and is disjoint from
$\\{p:d(p,Y)<R\\}$.

Furthermore, $\Sigma:=\partial\Omega$ is uniformly $C^{1}$ near $\partial K$
in the following sense. For every $\epsilon>0$, there is a $\delta>0$ such
that if $p,q\in\Sigma$, if $d(p,\partial K)<\delta$, and if $d(p,q)<\delta$,
then

$d(\nu(p),\nu(q))\leq\epsilon.$

Here, $\nu$ is the unit normal to $\Sigma$ that points out from $\Omega$. The
meaning of $d(\nu(p),\nu(q))$ is given by:

######  Definition 10.

If $\mathbf{u}$ and $\mathbf{v}$ are tangent vectors to $N$, we let
$d(\mathbf{u},\mathbf{v})$ be the supremum of
$|\mathbf{u}\cdot\mathbf{w}-\mathbf{v}\cdot\mathbf{w}|$ among $C^{1}$
vectorfields $\mathbf{w}$ on $N$ such that $|\mathbf{w}|\leq 1$ and
$|\nabla\mathbf{w}|\leq 1$ at all points.

The Interpolation Theorem is proved in §6.

Let $t\in[0,\infty)\mapsto M(t)$ be a separating Brakke flow such that $M(0)$
is the Radon measure associated to $\Sigma$. See §7 for the definition of
“separating flow” and for a proof that the flow $M(\cdot)$ exists.

If $(p,t)$ is a regular point of $\mathbf{M}$, we let $a(\mathbf{M},p,t)$ be
the norm of the second fundamental form of $\mathbf{M}(t)$ at $p$. If $(p,t)$
is a singular point of $\mathbf{M}$, we let $a(\mathbf{M},p,t)=\infty$. We let

$A(\mathbf{M},p,t)=\sup\\{a(\mathbf{M},p^{\prime},t^{\prime}):d(p^{\prime},p)\leq|t|^{1/2},\,t/2\leq
t^{\prime}\leq 2t\\}.$

######  Claim 1.

There is an $\epsilon>0$ with the following property. If $0<T\leq\epsilon$, if
$p\in\mathbf{M}(T)$, and if $d(p,K^{c})\leq T^{1/2}$, then

$T^{1/2}A(\mathbf{M},p,T)<1,$

###### Proof of Claim 1.

Suppose not. Then there exist $t_{i}\to 0$ and $p_{i}\in\mathbf{M}(t_{i})$
such that

$d(p_{i},K^{c})\leq t_{i}^{1/2}$

and such that

$t_{i}^{1/2}A(\mathbf{M},p_{i},t_{i})\geq 1.$

Let $L_{i}:\mathbf{R}^{m+1}\to\operatorname{Tan}(N,p_{i})$ be a linear
isometry, and let

$\displaystyle\phi_{i}:B_{i}=B^{m+1}(0,r/\sqrt{t_{i}})\to N,$
$\displaystyle\phi_{i}(v)=\exp_{p_{i}}(L_{i}(t_{i}^{1/2}v)),$

where $r=\operatorname{inj}(N)/2$. We give $B_{i}$ the Riemmannian metric
$g_{i}$ such that a curve of $g_{i}$-length $s$ in $B_{i}$ is mapped (by
$\phi_{i}$) to a curve of length $t_{i}^{1/2}s$ in $N$.

Let $\Sigma_{i}=\phi_{i}^{-1}(\Sigma)$.

Let $M_{i}$ be the Brakke flow in $B_{i}$ obtained by applying $\phi_{i}^{-1}$
(parabolically) to the Brakke flow $M$.

After passing to a subsequence, $\Sigma_{i}$ converges to a closed subset
$\Sigma^{\prime}$ of $\mathbf{R}^{m+1}$.

We claim that $\Sigma^{\prime}$ is nonempty. For if it were empty, then (after
passing to a further subsequence) the flows $\mathbf{M}_{i}$ would converge to
a weak set flow $t\mapsto Z(t)$ in $\mathbf{R}^{m+1}$ with $0\in Z(1)$ and
with $Z(0)=\emptyset$, which is impossible (by Lemma 2, for example).

Thus $\Sigma^{\prime}$ is nonempty. Thus if $\tilde{p}_{i}$ is a point in
$\Sigma$ closest to $p_{i}$, then

$d(\tilde{p}_{i},p_{i})\leq kt_{i}^{1/2}$

for some $k<\infty$ independent of $i$. Hence

$d(\tilde{p}_{i},K^{c})\leq d(\tilde{p}_{i},p_{i})+d(p_{i},K^{c})\to 0.$

Consequently, if $\tilde{q}_{i}\in\Sigma$ and
$d(\tilde{q}_{i},\tilde{p}_{i})\to 0$, then

(3) $d(\nu(\tilde{q}_{i}),\nu(\tilde{p}_{i}))\to 0$

by Theorem 9.

It follows that $\Sigma^{\prime}$ is a multiplicity-$1$ plane. (To see that
there is only one plane, counting multiplicity, note that $\Sigma^{\prime}$ is
a union of oriented planes having the same normal $\nu$, by (3). In
particular, all the planes in $\Sigma^{\prime}$ are parallel. If there were
more than one plane, counting multiplicity, then some of the planes would have
normal $\nu$ and others would have normal $-\nu$, since $\Sigma$ bounds a
region $\Omega$.)

After passing to a further subsequence, the $M_{i}$ converge to an integral
Brakke flow $M^{\prime}$, where $M^{\prime}(0)$ is given by the plane
$\Sigma^{\prime}$ with multiplicity $1$. Since the flow is separating, it
follows (see Lemma 25) that $M^{\prime}(t)$ is the plane $\Sigma^{\prime}$
with multiplicity $1$ for all $t\geq 0$.

By local regularity [white-local], the convergence
$\mathbf{M}_{i}\to\mathbf{M}^{\prime}$ is smooth on compact subsets of
$\mathbf{R}^{m+1}\times(0,\infty)$. Thus

$A(\mathbf{M}_{i},0,1)\to 0,$

But, by hypothesis, $A(\mathbf{M}_{i},0,1)\geq 1$ for all $i$. The
contradiction proves Claim 1. ∎

######  Claim 2.

There is a $\delta>0$ such that

$e^{-\lambda t}d(X(t),\mathbf{M}(t))\geq R$

and

$e^{-\lambda t}d(Y(t),\mathbf{M}(t))\geq R$

for $0\leq t\leq\delta$.

###### Proof.

It suffices to prove it for $X(\cdot)$. Recall that there exist $C<\infty$ and
$\eta<0$ such that

(4) $d(X(t),K)\geq R-Ct$ for $0\leq t\leq\eta$.

(See Lemma 2.) Choose $\delta>0$ small enough that

$\displaystyle\delta$ $\displaystyle\leq\epsilon,$ $\displaystyle\delta$
$\displaystyle\leq\eta,$ $\displaystyle C\delta^{1/2}$
$\displaystyle\leq\frac{1}{2}$

where $\epsilon$ is as in Claim 1 and $\eta$ and $C$ are as in (4). Thus if
$0\leq t\leq\delta$, then

(5) $Ct\leq C\delta^{1/2}t^{1/2}\leq\frac{1}{2}t^{1/2}.$

Let

$\displaystyle f:[0,\delta]\to\mathbf{R},$ $\displaystyle f(T)=\inf_{0\leq
t\leq T}e^{-\lambda t}d(X(t),\mathbf{M}(t)).$

Suppose the claim is false. Then $f(T)<R$ for some $T\in(0,\delta]$. By
Corollary 4, $f$ is continuous. Thus, by replacing $T$ by a smaller $T>0$, we
can assume that

$0<f(T)<R.$

We can also assume that $f(t)>f(T)$ for $t<T$; otherwise replace $T$ by the
smallest $t$ such that $f(t)=f(T)$. Thus

$\eta:=f(T)<f(t)\qquad\text{for $t<T$}.$

In particular, $e^{-\lambda T}d(X(T),\mathbf{M}(T))=\eta$, so

(6) $d(X(T),\mathbf{M}(T))=e^{\lambda T}\eta<\eta<R\leq\frac{1}{2}r$

since we are assuming that $\lambda<0$ and (in Case 1) that
$R\leq\frac{1}{2}r$.

Choose $p_{i}\in\mathbf{M}(T)$ and $q_{i}\in X(T)$ so that

$d(p_{i},q_{i})\to d(\mathbf{M}(T),X(T))=e^{\lambda T}\eta.$

By passing to a subsequence, we can assume that $d(K^{c},q_{i})$ converges to
a limit in $[0,\infty]$. We wish to show that

(7) $\lim_{i}d(q_{i},K^{c})\leq\frac{1}{2}T^{1/2}.$

We may assume that $q_{i}$ is in the interior of $K$ for all sufficiently
large $i$, as otherwise the inequality (7) is trivially true. For such $i$,

$\displaystyle d(p_{i},q_{i})$ $\displaystyle\geq d(p_{i},\partial
K)+d(\partial K,q_{i})$ $\displaystyle\geq R-CT+d(K^{c},q_{i}).$

Thus

$e^{\lambda T}\eta\geq R-CT+\lim_{i}d(K^{c},q_{i}).$

Now $e^{\lambda T}\leq 1$ (since we are assuming that $\Lambda\leq 0$) and
$\eta<R$, so

$\lim_{i}d(q_{i},K^{c})\leq CT<\frac{1}{2}T^{1/2}$

by (5). This completes the proof that the inequality (7) holds.

By Claim 1 and (7),

(8) $|T|^{1/2}A(\mathbf{M}_{i},q_{i},T)<1$

for all sufficiently large $i$.

Now let $L_{i}:\mathbf{R}^{m+1}\to\operatorname{Tan}(N,p_{i})$ be a a linear
isometry, and let

$\displaystyle F_{i}:B=B^{m+1}(0,r)\to N,$ $\displaystyle
F_{i}=\exp_{p_{i}}\circ L_{i}.$

Let $g_{i}$ be the pull-back by $F_{i}$ of the metric on $N$, so that $F_{i}$
maps $(B,g_{i})$ isometrically onto $B(p_{i},r)$.

For large $i$, $d(p_{i},q_{i})<r$ (by (6)), so there is a $\tilde{q}_{i}\in B$
with $F_{i}(\tilde{q}_{i})=q_{i}$. Let

$\displaystyle\tilde{X}_{i}(t)$ $\displaystyle=F_{i}^{-1}(X(t)),$
$\displaystyle\tilde{\mathbf{M}}_{i}(t)$
$\displaystyle=F_{i}^{-1}(\mathbf{M}(t)).$

Thus $\tilde{X}_{i}(\cdot)$ and $\tilde{\mathbf{M}}_{i}(\cdot)$ are weak set
flows in $B$ with respect to $g_{i}$.

After passing to a subsequence, the $g_{i}$ converge to a smooth Riemannian
metric $\tilde{g}$, $\tilde{X}_{i}(\cdot)$ and $\tilde{\mathbf{M}}_{i}(\cdot)$
converge to weak set flows $\tilde{X}(\cdot)$ and $\tilde{\mathbf{M}}(\cdot)$
in $(B,\tilde{g})$, and $\tilde{q}_{i}$ converges to point $\tilde{q}$ with

(9) $d(0,\tilde{q})=|\tilde{q}|=d(X(T),\mathbf{M}(T))=e^{\lambda T}\eta.$

For $0\leq t<T$,

(10) $e^{-\lambda t}d(\tilde{X}(t),\tilde{\mathbf{M}}(t))\geq e^{-\lambda
t}d(X(t),\mathbf{M}(t))>\eta.$

By (8), the spacetime point $(0,T)$ is a regular point of the flow
$\tilde{\mathbf{M}}$. Thus by (9), (10), and Proposition 6, $(B,\tilde{g})$
has points of Ricci curvature $\leq\lambda$, contrary to our choice of
$\lambda<\Lambda$. This completes the proof of Claim 2. ∎

######  Claim 3.

There is an $\epsilon>0$ with the following property:

$\displaystyle\cup_{t\in[0,\epsilon]}X(t)\subset\\{p:d(p,X)<R/8\\},$
$\displaystyle\cup_{t\in[0,\epsilon]}\mathbf{M}(t)\subset\\{p:d(p,\mathbf{M}(0))<R/8\\},$
$\displaystyle\cup_{t\in[0,\epsilon]}Y(t)\subset\\{p:d(p,Y)<R/8\\}.$

Claim 3 follows immediately from Corollary 3.

######  Claim 4.

If $0<T\leq\epsilon$, where $\epsilon$ is as in Claim 3, then

$d(X(T),Y(T))\geq d(X(T),\mathbf{M}(T))+d(\mathbf{M}(T),Y(T)).$

###### Proof of Claim 4.

Let $p\in X(T)$ and $q\in Y(T)$. Let $\alpha:[0,T]\to N$ be a geodesic such
that

$\displaystyle\alpha(0)\in X,$ $\displaystyle\alpha(T)=p,$
$\displaystyle\operatorname{Length}(\alpha)=d(p,X).$

Let $\beta:[T,2T]\to N$ be a shortest geodesic from $p$ to $q$. Let
$\gamma:[2T,3T]\to N$ be a geodesic such that

$\displaystyle\gamma(2T)=q,$ $\displaystyle\gamma(3T)\in Y,$
$\displaystyle\operatorname{Length}(\gamma)=d(q,Y).$

Consider the path

$\mu:s\in[0,3T]\mapsto\begin{cases}(\alpha(s),s)&\text{if $s\in[0,T]$},\\\
(\beta(s),T)&\text{if $s\in[T,2T]$},\\\ (\gamma(s),3T-s)&\text{if
$s\in[2T,3T]$}.\end{cases}$

Thus $\mu$ is a path in $N\times[0,\infty)$ from $(p,0)$ to $(q,0)$. Since
$M(\cdot)$ is separating, there must be an $s\in[0,3T]$ such that

$\mu(s)\in\mathbf{M}.$

By Claim 3, $s$ cannot be in $[0,T]$ or in $[2T,3T]$. Thus $s\in(T,2T)$, so

$\beta(s)\in\mathbf{M}(T).$

Therefore,

$\displaystyle d(X(T),Y(T))$ $\displaystyle\geq d(p,q)$
$\displaystyle=d(p,\beta(s))+d(\beta(s),q)$ $\displaystyle\geq
d(p,\mathbf{M}(T))+d(\mathbf{M}(T),q).$

Taking the infimum over $p\in X(T)$ and $q\in Y(T)$ gives

$d(X(T),Y(T))\geq d(X(T),\mathbf{M}(T))+d(\mathbf{M}(T),Y(T)).$

This completes the proof of Claim 4. ∎

Now let $\tilde{\epsilon}$ be the minimum of the $\delta$ in Claim 2 and the
$\epsilon$ in Claim 4. Then for $0\leq t\leq\tilde{\epsilon}$,

$\displaystyle d(X(t),Y(t))$
$\displaystyle\geq\d{(}X(t),\mathbf{M}(t))+d(\mathbf{M}(t),Y(t))$
$\displaystyle\geq 2Re^{\lambda t}$

This completes the proof of 2 in Case 1 of the Avoidance Theorem.

######  Case 2.

$d(X(0),Y(0))>r=\frac{1}{2}\operatorname{inj}(N)$.

Let

$c=d(X(0),Y(0))-r,$

and let

$\tilde{X}(t)=\\{p:d(p,X(t))\leq ce^{\lambda t}\\}$

for $t\geq 0$. Then $\tilde{X}(\cdot)$ is a weak set flow (by Corollary 7),
and

$d(\tilde{X}(0),Y(0))=r.$

Thus by Case 1,

(11) $e^{-t}d(\tilde{X}(t),Y(t))\geq d(\tilde{X}(0),Y(0))$

for $t\in[0,\epsilon]$, for some $\epsilon>0$.

If $d(\tilde{X}(t),Y(t))>0$, then

$d(X(t),Y(t))=d(\tilde{X}(t),Y(t))+ce^{\lambda t}.$

Thus (11) can be written as

$e^{-\lambda t}d(X(t),Y(t))+c\geq d(X(0),Y(0))+c$

or

$e^{-\lambda t}d(X(t),Y(t))\geq d(X(0),Y(0))$

for $t\in[0,\epsilon]$. This completes the proof of (2) in Case 2. ∎

##  5\. Properties of the Distance Function

This section proves properties of the distance function that are used in the
proof of the Interpolation Theorem 6 in Section 6.

Throughout this section, $X$ and $Y$ are closed sets in $N$ such that
$d(X,Y)>0$. Let

$\displaystyle R$ $\displaystyle=\frac{1}{2}d(X,Y),$ $\displaystyle K$
$\displaystyle=\\{p:d(p,X\cup Y)\geq R\\},$ $\displaystyle S$
$\displaystyle=\\{p:d(p,X)=d(p,Y)=R\\}.$

We abbreviate $d(p,X)$ by $x(p)$ and $d(p,Y)$ by $y(p)$.

First, we prove a useful fact about geodesics in $N$.

######  Proposition 11.

For every $\epsilon>0$ and $\sigma>0$, there is a $\delta>0$ with the
following property. If $\alpha:[-a,0]\to N$ and $\beta:[0,b]\to N$ are unit-
speed geodesics with $a,b\geq\sigma$, and if

(12) $\begin{gathered}d(\alpha(0),\beta(0))\leq\delta,\\\
d(\alpha(-a),\beta(b))\geq a+b-\delta,\end{gathered}$

then $d(\alpha^{\prime}(0),\beta^{\prime}(0))<\epsilon$.

(See Definition 10 for the meaning of
$d(\alpha^{\prime}(0),\beta^{\prime}(0))$.)

###### Proof.

If the theorem holds for one $\sigma$, then it holds for any large $\sigma$.
Thus we may assume that $\sigma<r:=\frac{1}{2}\operatorname{inj}(N)$.

Suppose that $\alpha$ and $\beta$ are unit-speed geodesics that satisfy (12).
Then

$\displaystyle a+b-\delta$ $\displaystyle\leq d(\alpha(-a),\beta_{i}(b))$
$\displaystyle\leq
d(\alpha_{i}(-a),\alpha(-\sigma))+d(\alpha(-\sigma),\beta(\sigma))+d(\beta(\sigma),\beta(b))$
$\displaystyle\leq a-\sigma+d(\alpha(-\sigma),\beta(\sigma))+b-\sigma,$

So

$2\sigma-\delta\leq d(\alpha(-\sigma),\beta(\sigma)).$

Thus if we replace $a$ and $b$ by $\sigma$ and $\alpha$ and $\beta$ by their
restrictions to $[-\sigma,0]$ and $[0,\sigma]$, the conditions (12) still
hold. Consequently, it suffices to prove the proposition when $a=b=\sigma$.

Thus it suffices to show that if $\alpha_{i}:[-\sigma,0]\to N$ and
$\beta_{i}:[0,\sigma]\to N$ are unit-speed geodesics, and if

$\displaystyle d(\alpha_{i}(0),\beta_{i}(0))\to 0$
$\displaystyle\liminf_{i}d(\alpha_{i}(-\sigma),\beta_{i}(\sigma))\geq
2\sigma,$

then

$d(\alpha_{i}^{\prime}(0),\beta_{i}^{\prime}(0))\to 0.$

Let $p_{i}=\alpha_{i}(0)$, let
$L_{i}:\mathbf{R}^{m+1}\to\operatorname{Tan}(N,p_{i})$ be a linear isometry,
and let

$\displaystyle F_{i}:\mathbf{R}^{m+1}\to N,$ $\displaystyle
F_{i}(v)=\exp_{p_{i}}(L_{i}v).$

Let $g_{i}$ be the metric on $B=B^{m+1}(0,r)$ obtained by pulling-back the
metric on $N$ by $F_{i}$. Define $\tilde{\alpha}_{i}:[-\sigma,0]\to B$ and
$\tilde{\beta}_{i}:[0,\sigma]\to B$ by

$\displaystyle\tilde{\alpha}_{i}(s)=F_{i}^{-1}\alpha_{i}(s),$
$\displaystyle\tilde{\beta}_{i}(s)=F_{i}^{-1}\beta_{i}(s)$

Then $\tilde{\alpha}_{i}$ and $\tilde{\beta}_{i}$ are unit-speed geodesics
with respect to $g_{i}$.

By passing to a subsequence, we can assume that $g_{i}$ converges smoothly to
a metric $g$, and that $\tilde{\alpha}_{i}$ and $\tilde{\beta}_{i}$ converge
smoothly to unit-speed $g$-geodesics $\alpha:[-\sigma,0]\to B$ and
$\beta:[0,\sigma]\to B$. Note that $\alpha(0)=\beta(0)$ and that

$d_{g}(\alpha(-\sigma),\beta(\sigma))\geq 2\sigma.$

Consider the map

$\gamma:s\in[-\sigma,\sigma]\mapsto\begin{cases}\alpha(s)&\text{if
$s\in[-\sigma,0]$},\\\ \beta(s)&\text{if $s\in[0,\sigma]$}.\end{cases}$

Then

$2\sigma\leq
d_{g}(\gamma(-\sigma),\gamma(\sigma))\leq\operatorname{Length}(\gamma)=2\sigma.$

Thus $\gamma$ is a unit-speed geodesic, so it is smooth, and thus

$\alpha^{\prime}(0)=\beta^{\prime}(0).$

∎

For the following definition, recall that $x(p)=d(p,X)$ and $y(p)=d(p,Y)$.

######  Definition 12.

For $p\in N\setminus(X\cup Y)$, let $\mathcal{X}(p)$ be the set of unit
vectors $\mathbf{u}\in\operatorname{Tan}(N,p)$ such that

$\exp_{p}(-x(p)\mathbf{u})\in X,$

and let $\mathcal{Y}(p)$ be the set of unit vectors
$\mathbf{v}\in\operatorname{Tan}(N,p)$ such that

$\exp_{p}(y(p)\mathbf{v})\in Y.$

Note that for each $x\in(X\cup Y)$, $\mathcal{X}(p)$ and $\mathcal{Y}(p)$ are
nonempty. Note also that if $\alpha:[-a,0]\to X$ is a unit-speed geodesic with
with $\alpha(0)=p$ and with length $a=d(X,p)=x(p)$, then
$\alpha^{\prime}(0)\in\mathcal{X}(p)$. Conversely, if
$\mathbf{u}\in\mathcal{X}(p)$, then

$s\in[-x(p),0]\mapsto\exp_{p}(s\mathbf{u})$

is such a geodesic. Of course, the analogous statements hold for
$\mathbf{v}\in\mathcal{Y}(p)$.

######  Remark 13.

Suppose that $\mathbf{u}\in\mathcal{X}(p)$ and let
$\alpha(s)=\exp_{p}(s\mathbf{u})$ for $-x(p)\leq s\leq 0$. Note that
$x(\cdot)\leq x(p)$ on ball $B(\alpha(s),-s)$. Likewise, if
$\mathbf{v}\in\mathcal{Y}(p)$ and if $\beta(s)=\exp_{p}(s\mathbf{v})$ for
$0\leq s\leq y(p)$, then $y(\cdot)\leq s$ on the ball $B(\beta(s),s)$.

######  Theorem 14.

For every $\epsilon>0$, there is an $\eta>0$ with the following property.
Suppose that

$\displaystyle\mathbf{u}\in\mathcal{X}(p),$
$\displaystyle\mathbf{v}\in\mathcal{Y}(q),$
$\displaystyle|x(p)-R|\leq\frac{1}{2}\eta,$
$\displaystyle|y(q)<R|\leq\frac{1}{2}\eta,$ $\displaystyle
d(p,q)\leq\frac{1}{2}\eta.$

Then $d(\mathbf{u},\mathbf{v})<\epsilon.$

###### Proof.

Let $\sigma=R/2$, and let $\delta$ be as in Proposition 11. Let $\eta$ be the
smaller of $\delta$ and $R/2$.

Let $a=x(p)$, $b=y(q)$, and let

$\displaystyle\alpha(s)$
$\displaystyle=\exp_{p}(s\mathbf{u})\qquad(s\in[-a,0]),$
$\displaystyle\beta(s)$
$\displaystyle=\exp_{q}(s\mathbf{v})\qquad(s\in[0,b]).$

Then

$\displaystyle d(\alpha(-a),\beta(b))$ $\displaystyle\geq d(X,Y)$
$\displaystyle=2R$ $\displaystyle\geq x(p)+y(p)-\delta$
$\displaystyle=a+b-\delta.$

Thus $d(\mathbf{u},\mathbf{v})<\epsilon$ by choice of $\eta$; see Proposition
11. ∎

######  Corollary 15.

Let $\epsilon$ and $\eta$ be as in Theorem 14. Suppose that

$\displaystyle d(p_{1},p_{2})\leq\eta/2,$
$\displaystyle|x(p_{i})-R|\leq\eta/2,$
$\displaystyle\mathbf{u}_{i}\in\mathcal{X}(p_{i}),$
$\displaystyle\mathbf{v}_{i}\in\mathcal{Y}(p_{i}).$

for $i=1,2$. Then $d(\mathbf{u}_{1},\mathbf{u}_{2})<2\epsilon.$ and
$d(\mathbf{v}_{1},\mathbf{v}_{2})<2\epsilon$.

###### Proof.

By Theorem 14,

$d(\mathbf{u}_{i},\mathbf{v}_{1})<\epsilon$

for $i=1,2$. Thus $d(\mathbf{u}_{1},\mathbf{u}_{2})<2\epsilon.$ Similarly,
$d(\mathbf{v}_{1},\mathbf{v}_{2})\leq 2\epsilon$. ∎

######  Corollary 16.

Let $p\in S$. Then there is a unique vector $\mathbf{u}=\mathbf{u}(p)$ in
$\mathcal{X}(p)$, and $\mathbf{u}(p)$ is also the unique vector in
$\mathcal{Y}(p)$. Furthermore, $\mathbf{u}(p)$ depends continuously on $p\in
S$.

The following theorem describes the function $x(\cdot)$ near a point $p$ where
$x(p)$ and $y(p)$ are both very close to $R$. Roughly speaking, it says that
if $\mathbf{u}\in\mathcal{X}(p)$ at such a point $p$, then $u-u(p)$ behaves
(near $p$) like a $C^{1}$ function whose gradient at $p$ is $\mathbf{u}$.

######  Theorem 17.

Suppose that $p_{i}\in N$, that $\mathbf{u}_{i}\in\mathcal{X}(p_{i})$, and
that

$|x(p_{i})-R|+|y(p_{i})-R|\to 0.$

Let $L_{i}:\mathbf{R}^{m+1}\to\operatorname{Tan}(N,p_{i})$ be a linear
isometry such that

$|L\mathbf{e}_{1}-\mathbf{u}_{i}|\to 0.$

Let $r_{i}\to 0$, and define

$F_{i}:B_{i}=B^{m+1}(0,r/r_{i})\mathbf{R}^{m+1}\to N$

by

$F_{i}(v)=\exp_{p_{i}}L(r_{i}v),$

where $r=\frac{1}{2}\operatorname{inj}(N)$.

Let $g_{i}$ be the metric on $B_{i}$ such that $F_{i}$ maps a curve of
$g_{i}$-length $s$ to a curve of length $r_{i}s$

Define $x_{i}:B_{i}\to\mathbf{R}$ by

$x_{i}(v)=\frac{x(F_{i}(v))-x(F_{i}(0))}{r_{i}}=\frac{x(F_{i}(v))-x(p_{i})}{r_{i}}.$

Then $x_{i}(\cdot)$ converges uniformly on compact sets to the function

$\displaystyle\tilde{x}:\mathbf{R}^{m+1}\to\mathbf{R},$
$\displaystyle\tilde{x}(v)=v\cdot\mathbf{e}_{1}.$

###### Proof.

Note that $g_{i}$ converges smoothly to the Euclidean metric. The function
$x_{i}$ is $1$-Lipschitz with respect to $g_{i}$ and $x_{i}(0)=0$. Thus, after
passing to a subsequence, the functions $x_{i}$ converge uniformly on compact
sets to a $1$-Lipschitz function $\tilde{x}$.

Consider a $q\in\mathbf{R}^{m+1}$. If $q\in B_{i}$, let
$\mathbf{u}_{i}\in\mathcal{X}(F_{i}(q))$, and let $\mathbf{w}_{i}$ be the unit
vector such that

$DF_{i}(q)\mathbf{w}_{i}=r_{i}\mathbf{u}_{i}.$

Then

$x_{i}(\exp_{q,i}{s\mathbf{w}_{i}})=x_{i}(q)+s\qquad\text{for $s\leq 0$},$

where $\exp_{q,i}$ is the exponential map at $q$ with respect to the metric
$g_{i}$. Now $\mathbf{w}_{i}\to\mathbf{e}_{1}$ (by Corollary 15), so

$\exp_{q,i}(s\mathbf{w}_{i})\to\exp_{q}(s\mathbf{e}_{1})=q+s\mathbf{e}_{1},$

and thus

(13) $\tilde{x}(q+s\mathbf{e}_{1})=\tilde{x}(q)+s$

for all $q$ and all $s\leq 0$. Replacing $q$ by $q-s\mathbf{e}_{1}$, we see
that

$\tilde{x}(q)=\tilde{x}(q-s\mathbf{e}_{1})+s,$

or

$\tilde{x}(q-s\mathbf{e}_{1})=\tilde{x}(q)-s.$

Thus (13) holds for all $q$ and all $s\in\mathbf{R}$. Since $\tilde{x}(0)=0$
and since $\tilde{x}$ is $1$-Lipschitz, it follows that

$\tilde{x}(v)\equiv v\cdot\mathbf{e}_{1}.$

(If this is not clear, see Lemma 19 below.) ∎

######  Remark 18.

In Theorem 17, let $\mathbf{v}_{i}\in\mathcal{Y}(p_{i})$. By Theorem 14,
$|\mathbf{v}_{i}-\mathbf{u}_{i}|\to 0$. Thus (reversing the roles of $X$ and
$Y$), we see from Theorem 17 that the functions

$y_{i}(v):=\frac{y(F_{i}(v))-y(p_{i})}{r_{i}}$

converge uniformly on compact sets to the function

$\displaystyle\tilde{y}:\mathbf{R}^{m+1}\to\mathbf{R},$
$\displaystyle\tilde{y}(v)=-v\cdot\mathbf{e}_{1}.$

The following lemma was used at the end of the proof of Theorem 17.

######  Lemma 19.

Suppose that $\mathbf{u}$ is a unit vector in $\mathbf{R}^{m+1}$ and that
$f:\mathbf{R}^{m+1}\to\mathbf{R}$ is a $1$-Lipschitz function such that

$f(s\mathbf{u})=s$

for all $s\in\mathbf{R}$. Then $f(v)=v\cdot\mathbf{u}$ for all $v$.

###### Proof.

If $r>0$, then for $p$ in the ball $B((s-r)\mathbf{u},r)$,

$\displaystyle f(p)$ $\displaystyle\leq f((s-r)\mathbf{u})+r$
$\displaystyle=s.$

Letting $r\to\infty$ shows that $f\leq s$ on the halfspace
$\\{v:v\cdot\mathbf{u}\leq s\\}$. Thus $f(v)\leq v\cdot\mathbf{u}$ for all
$v$.

Likewise (using the ball $B((s+r)\mathbf{u},r)$), we see that $f(v)\geq
v\cdot\mathbf{u}$ for all $v$. ∎

##  6\. The Interpolation Theorem

In this section, we prove the Interpolation Theorem that was used in the proof
of the Avoidance Theorem 8.

For $0<\rho<R$, let

$\displaystyle K(\rho)=\\{p:d(p,X\cup Y)\geq\rho\\}.$

Let $h^{\rho}:K(\rho)\to[0,1]$ be a harmonic function with boundary values

$h^{\rho}(p)=\begin{cases}0&\text{if $x(p)=\rho$},\\\ 1&\text{if
$y(p)=\rho$}.\end{cases}$

Existence of $h^{\rho}$ can be proved in a variety ways, such as the Perron
method as described in [GT]*§2.8. Note that that the boundary values can be
prescribed because $K(\rho)$ satisfies an exterior sphere condition; see
Remark 13.

######  Lemma 20.

For every $\epsilon>0$, there is a $\delta$ with the following property. If

$\displaystyle\rho>R-\delta,$ $\displaystyle h^{\rho}(p)\in[1/3,2/3],$
$\displaystyle d(p,\partial K(\rho))<\delta,$
$\displaystyle\mathbf{u}\in\mathcal{X}(p),$

then

  1. (1)

$|x(p)-R|+|y(p)-R|<\epsilon$.

  2. (2)

$\nabla h^{\rho}(p)\neq 0$.

  3. (3)

$d(\mathbf{w}(p),\mathbf{u})<\epsilon$, where

$\mathbf{w}(p)=\frac{\nabla h^{\rho}(p)}{|\nabla^{\rho}h(p)|}.$

###### Proof.

Suppose that $0<\rho_{i}<R$, that $\rho_{i}\to R$, that $p_{i}\in
K(\rho_{i})$, that $\mathbf{u}_{i}\in\mathcal{X}(p_{i})$, and that

$\displaystyle h(p_{i})\in[1/3,2/3],$ $\displaystyle d(p_{i},\partial
K(\rho_{i}))\to 0.$

It suffices to show that

$\nabla^{\rho_{i}}h(p_{i})\neq 0$

for large $i$, and that

$\displaystyle|x(p_{i})-R|+|y(p_{i})-R|\to 0,$ $\displaystyle
d(\mathbf{w}_{i},\mathbf{u}_{i})\to 0,$

where

$\mathbf{w}_{i}=\frac{\nabla h^{\rho_{i}}(p_{i})}{|\nabla
h^{\rho_{i}}(p_{i})|}.$

Let

$\eta_{i}=\operatorname{dist}(p_{i},\partial K(\rho_{i})).$

Thus $\eta_{i}\to 0.$

Let $L_{i}:\mathbf{R}^{m+1}\to\operatorname{Tan}(N,p_{i})$ be a linear
isometry such that $L_{i}\mathbf{e}_{1}=\mathbf{u}_{i}$.

Let $B_{i}=B^{m+1}(0,r/\eta_{i})$ (where
$r=\frac{1}{2}\operatorname{inj}(N)$), and let define $F_{i}:B_{i}\to N$ by

$F_{i}(v)=\exp_{p_{i}}(\eta_{i}v).$

We endow $B_{i}$ with the metric $g_{i}$ such that if $\Gamma$ is a curve in
$B_{i}$ with $g_{i}$-length $s$, then its image under $F_{i}$ is a curve of
length $\eta_{i}s$.

Let $K_{i}=F_{i}^{-1}(K)$.

By Theorem 17 and Remark 18, the $K_{i}$ converge (perhaps after passing to a
subsequence) to a limit $K^{\prime}$ of the form

$K^{\prime}=\\{p:a\leq p\cdot\mathbf{e}_{1}\leq b\\},$

where $-\infty\leq a\leq b\leq\infty$. Since

$\eta_{i}=d(p_{i},K^{c}),$

we see that $d(0,(K^{\prime})^{c})=1$. Thus $a\in[-\infty,-1]$ and
$b\in[1,\infty]$, and either $a=-1$ or $b=1$.

Note that the $g_{i}$-harmonic functions $h\circ F_{i}:B_{i}\to[0,1]$
converge, perhaps after passing to a further subsequence, to a harmonic
function $h^{\prime}:K^{\prime}\to[0,1]$ such that

$\displaystyle h^{\prime}(p)$ $\displaystyle=0\qquad\text{if
$p\cdot\mathbf{e}_{1}=a$},$ $\displaystyle h^{\prime}(p)$
$\displaystyle=1\qquad\text{if $p\cdot\mathbf{e}_{1}=b$},$

and such that

(14) $h^{\prime}(0)=c\in[1/3,2/3].$

If $b=\infty$, then $a=-1$, and thus $h^{\prime}\equiv 0$, contrary to (14).
Likewise, if $a=-\infty$, then $b=1$, and thus $h^{\prime}\equiv 1$, contrary
to (14). Consequently, $a$ and $b$ are both finite, so

$h^{\prime}(p)=\frac{p\cdot\mathbf{e}_{1}-a}{b-a}.$

and therefore

$\frac{\nabla h^{\prime}}{|\nabla h^{\prime}|}=\mathbf{e}_{1}.$

The assertions of the theorem follow immediately. ∎

######  Corollary 21.

For every $\epsilon>0$, there is a $\delta>0$ with the following property. If

$\displaystyle R-\delta<\rho<R,$ $\displaystyle p,q\in K(\rho),$
$\displaystyle d(p,q)<\delta,$ $\displaystyle d(p,\partial K(\rho))<\delta,$
$\displaystyle h^{\rho}(p),h^{\rho}(q)\in[1/3,2/3],$

Then

$d(\mathbf{w}(p),\mathbf{w}(q))<\epsilon.$

###### Proof.

Let $\mathbf{u}_{p}\in\mathcal{X}(p)$ and $\mathbf{u}_{q}\in\mathcal{X}(q)$.
By Lemma 20, we can, by choosing $\delta$ small, ensure that

$d(\mathbf{w}(p),\mathbf{u}_{p})+d(\mathbf{w}(q),\mathbf{u}_{q})<\epsilon/2.$

By Corollary 15, we can choose $\delta$ small enough that

$d(\mathbf{u}_{p},\mathbf{u}_{q})<\epsilon/2.$

Thus $d(\mathbf{w}(p),\mathbf{w}(q))<\epsilon$. ∎

######  Theorem 22 (Interpolation Theorem).

There is a closed region $\Omega$ in $N$ with the following properties:

  1. (1)

$\Omega$ is a $C^{1}$ manifold-with-boundary.

  2. (2)

$\Omega$ contains $\\{p:d(p,X)\leq R\\}$ and is disjoint from
$\\{p:d(p,Y)<R\\}$.

Furthermore, $\Sigma:=\partial\Omega$ is uniformly $C^{1}$ near $\partial K$
in the following sense. For every $\epsilon>0$, there is a $\delta>0$ such
that if $p,q\in\Sigma$, if $d(p,\partial K)<\delta$, and if $d(p,q)<\delta$,
then

$d(\nu(p),\nu(q))<\epsilon,$

where $\nu(\cdot)$ is the unit normal to $\Sigma$ that points out from
$\Omega$.

###### Proof.

Let $0<\rho_{i}<R$ with $\rho_{i}\to R$ and let $h^{\rho_{i}}$ be as the
harmonic function defined at the beginning of §6. By passing to a subsequence,
we can assume that $h^{\rho_{i}}$ converges locally uniformly in $K\setminus
S$ to a harmonic function such that

$h(p)=\begin{cases}0&\text{if $x(p)=R$ and $y(p)>R$},\\\ 1&\text{if $y(p)=R$
and $x(p)>R$}.\end{cases}$

Let $c\in[1/3,2/3]$ be a regular value of $h$ and also of each $h^{\rho_{i}}$.
Thus

$\Omega_{i}:=\\{h^{\rho_{i}}\leq c\\}\cup\\{x(\cdot)\leq\rho_{i}\\}$

is a $C^{1}$ manifold-with-boundary, the boundary being

$\Sigma_{i}:=\\{p:h^{\rho_{i}}(p)=c\\}.$

By passing to a further subsequence, we can assume that $\Omega_{i}$ and
$\Sigma_{i}$ converge to closed sets $\Omega$ and $\Sigma$. (Actually,
convergence holds without passing to a further subsequence, but we do not need
that fact.) Clearly, $\Sigma\setminus S=h^{-1}(c)$, and

(15) $h^{-1}(c)\subset\Sigma\subset h^{-1}(c)\cup S.$

Note that each path from a point $p$ with $x(p)<R$ to a point $q$ with
$y(q)<R$ must cross $\Sigma^{i}$ for all sufficiently large $i$ and thus must
also cross $\Sigma$. It follows that $S\subset\Sigma$ and therefore (by (15))
that

$\Sigma=h^{-1}(c)\cup S.$

The convergence of $\Sigma_{i}\setminus S$ to $\Sigma\setminus S$ is smooth
with multiplicity $1$ since $c$ is a regular value of $h$. Note that

$\nu_{i}=\nabla h^{\rho_{i}}/|\nabla h^{\rho_{i}}|$

is the unit normal to $\Sigma_{i}$ that points out of $\Omega_{i}$.

######  Claim 5.

If $p_{i}\in\Sigma_{i}$ converges to $p\in\Sigma$, then $\nu_{i}(p)$ converges
to a limit $\nu(p)$.

###### Proof of Claim 5.

If $p\in\Sigma\setminus S$, then this is true since $c$ is a regular value of
$h$. (In this case, $\nu(p)=\nabla h(p)/|\nabla h(p)|$.)

Now suppose that $p\in S$. By Corollary 16, there is a unique $\mathbf{u}(p)$
in $\mathcal{X}(p)$. Let $\mathbf{u}_{i}\in\mathcal{X}(p_{i})$. Then
$d(\nu_{i}(p_{i}),\mathbf{u}_{i})\to 0$ by Lemma 20, and
$\mathbf{u}_{i}\to\mathbf{u}(p)$ by Corollary 15, so $\nu_{i}(p)$ converges to
$\mathbf{u}(p)$. Thus Claim 5 is proved. ∎

Thus $\Sigma_{i}$ converges to $\Sigma$ in $C^{1}$ with multiplicity $1$. (To
see that the multiplicity is $1$, note that if the convergence were multi-
sheeted at a point $p\in\Sigma$, then on some of the sheets, $\nu_{i}$ would
converge to $\nu(p)$,and, on the other sheets, $\nu_{i}$ would converge to
$-\nu(p)$.)

Finally, the $(\epsilon,\delta)$ bounds on $\Sigma$ follow from the
corresponding bounds on the $\Sigma_{i}$. ∎

##  7\. Separating Flows

Suppose $t\in[0,\infty)\mapsto M(t)$ is an $m$-dimensional integral Brakke
flow in $N$. We say that $M(\cdot)$ is a separating flow provided:

  1. (1)

$M(0)$ is the Radon measure associated to an embedded, $C^{1}$ manifold
$\Sigma$.

  2. (2)

$\Sigma$ is the boundary of a region $\Omega$.

  3. (3)

if $p$ and $q$ are points in $N\setminus\Sigma$ with $p\in\Omega$ and
$q\in\Omega^{c}$, then any path in $N\times[0,\infty)$ from $(p,0)$ to $(q,0)$
must intersect the spacetime support $\mathbf{M}$ of $M(\cdot)$.

  4. (4)

The flow $M(\cdot)$ has the local regularity property of [white-local].

######  Theorem 23.

Suppose that $t\in[0,\infty)\mapsto M_{i}(t)$ is a sequence of separating
Brakke flows that converge to a Brakke flow $M(\cdot)$. Suppose also that
$\operatorname{spt}M(0)$ is a $C^{1}$ submanifold, and that
$\operatorname{spt}M_{i}(0)$ converges in $C^{1}$ with multiplicity $1$ to
$\operatorname{spt}M(0)$. Then $M(\cdot)$ is a separating flow.

The theorem follows easily from the fact that $\mathbf{M}_{i}$ converges to
$\mathbf{M}$ (which follows, for example, from Brakke’s clearing out lemma),
and from the local regularity theory in [white-local].

######  Theorem 24.

Let $\Omega$ be a region in $N$ such that $\Omega$ is an $(m+1)$-dimensional
manifold-with-boundary of class $C^{1}$. Let $\Sigma=\partial\Omega$. Then
there is a separating flow $M(\cdot)$ such that $M(0)$ is the Radon measure
associated to $\Sigma$.

This follows easily from elliptic regularization [ilmanen-elliptic] and
Theorem 23. For completeness, we sketch the proof below

###### Proof.

By passing to the double cover, we can assume that $N$ is oriented. Consider
first the case when $\Sigma$ is compact. Let $\tilde{g}$ denote the product
metric on $N\times[0,\infty)$. For $\lambda>0$, let $A^{\lambda}$ be an
$(m+1)$-chain in $N\times[0,\infty)$ that minimizes mass with respect to the
Riemannian metric

$g^{\lambda}_{ij}(x,z):=e^{-(2\lambda/m)z}\tilde{g}_{ij}(x,z)\qquad(x\in
N,z\in[0,\infty))$

among all $(m+1)$-dimensional locally integral currents $A$ with boundary
$[\Sigma]$. Note that if $p$ is in the interior of $\Omega$ and and if $q$ is
in $\Omega^{c}$, then $(p,0)$ and $(q,0)$ lie in different components of
$W:=(N\times[0,\infty))\setminus\operatorname{spt}A^{\lambda}$. It follows
that $(p,t)$ and $(q,t)$ lie in different components of $W$ for small $t>0$.

Now let $A^{\lambda}(t)$ be the image of $\Sigma$ under translation by
$(x,z)\mapsto(x,z-\lambda t)$, followed by restriction to $N\times(0,\infty)$.
For $t\geq 0$, let $M^{\lambda}(t)$ be the Radon measure on
$N\times(0,\infty)$ associated to $A^{\lambda}(t)$. Then $M^{\lambda}(\cdot)$
is a separating Brakke flow in $N\times(0,\infty)$. By the theory of elliptic
regularization [ilmanen-elliptic], the flows $M^{\lambda}(\cdot)$ converge
(after passing to a subsequence) as $\lambda\to\infty$ to a limit Brakke flow
$M^{\infty}(\cdot)$, and there is $m$-dimensional Brakke flow $M(\cdot)$ in
$N$ such that $M^{\infty}(\cdot)$ is obtained from taking the product of
$M(\cdot)$ with $(0,\infty)$. By Theorem 23, the flow $M^{\infty}(\cdot)$ is
separating. Thus $M(\cdot)$ is separating.

For the general case of a noncompact $\Sigma$, we can find a sequence of
compact $\Sigma_{i}$ converging to $\Sigma$ in $C^{1}$ with multiplicity $1$.
By the compact case of Theorem 24, there is a separating flow $M_{i}(\cdot)$
associated to $\Sigma_{i}$. By passing to a subsequence, we can assume that
the $M_{i}(\cdot)$ converge to an integral Brakke flow $M(\cdot)$. By Theorem
23, $M(\cdot)$ is also a separating flow. ∎

######  Lemma 25.

Suppose $t\in[0,\infty)\mapsto M(t)$ is a separating flow in
$\mathbf{R}^{m+1}$ such that $M(0)$ is a hyperplane $\Sigma$ with multiplicity
$1$. Then $M(t)=M(0)$ for all $t\geq 0$.

###### Proof.

Shrinking sphere barriers show that the spacetime support $\mathbf{M}$ of
$M(\cdot)$ is contained in $\Sigma\times[0,\infty)$. By the separating
property, $\mathbf{M}$ must be all of $\Sigma\times[0,\infty)$. Since the flow
is an integral Brakke flow, it follows each $M(t)$ is $\Sigma$ with
multiplicity $1$. ∎

## References


# Red- and blue-detuned magneto-optical trapping
with liquid crystal variable retarders

B. Piest<EMAIL_ADDRESS>(corresponding author) Institut für
Quantenoptik, Gottfried Wilhelm Leibniz Universität, Welfengarten 1, 30167
Hannover, Germany.  V. Vollenkemper  Institut für Quantenoptik, Gottfried
Wilhelm Leibniz Universität, Welfengarten 1, 30167 Hannover, Germany.  J. Böhm
Institut für Quantenoptik, Gottfried Wilhelm Leibniz Universität, Welfengarten
1, 30167 Hannover, Germany.  A. Herbst  Institut für Quantenoptik, Gottfried
Wilhelm Leibniz Universität, Welfengarten 1, 30167 Hannover, Germany.  E. M.
Rasel  Institut für Quantenoptik, Gottfried Wilhelm Leibniz Universität,
Welfengarten 1, 30167 Hannover, Germany.

###### Abstract

We exploit red- and blue-detuned magneto optical trapping (MOT) of 87Rb
benefitting from a simplified setup based on liquid crystal variable retarders
(LCVR). To maintain the trapping forces when switching from a red- to a blue-
detuned MOT, the circularity of the cooling beams needs to be reversed. LCVRs
allow fast polarization control and represent compact, simple and cost-
efficient components which can easily be implemented in existing laser
systems. This way, we achieve a blue-detuned type-II MOT for 87Rb atoms with
sub-Doppler temperatures of 44.5 µK. The phase space density is increased by
more than two orders of magnitude compared to the standard red-detuned type-I
MOT. The setup can readily be transferred to any other system working with
87Rb.

††preprint: AIP/123-QED

##  I Introduction

Trapped cold atoms with high phase space densities (PSD) are widely used in a
wealth of experiments in atomic, optical and molecular physics, cold chemistry
and nano physics. Typical applications range from inertial sensing Geiger  _et
al._ (2020), optical frequency standards Ludlow  _et al._ (2015) to
ultrasensitive isotope trace analysis Lu and Müller (2010) or bright sources
for ion beams Steele  _et al._ (2017). The most commonly used approach to
generate cold atoms is the use of a so-called type-I MOT Phillips: (1998)
although the achievable PSD in conventional MOTs is constrained by atom
collisions Prentiss  _et al._ (1988), photon reabsorption Walker, Sesko, and
Wieman (1990) and the lack of sub-Doppler cooling forces in magnetic
fieldsWalhout  _et al._ (1992). Among various approachesKetterle  _et al._
(1993); Radwell, Walker, and Franke-Arnold (2013), blue-detuned type-II MOTs
circumvent these limitations by enabling sub-Doppler forces even in non-zero
magnetic fields.
MOTs can be realized by using either type-I or type-II transitions which refer
to the involved atomic substates of the idealized two-level system. In type-I
MOTs, the total angular momentum of the excited state $F^{\prime}$ is larger
compared to that of the ground state $F$, thus satisfying $F^{\prime}=F+1$.
Type-II MOTs in contrast make use of transitions where $F^{\prime}=F$ or
$F^{\prime}=F-1$ (see fig. 1 for 87Rb). When using blue-detuned light fields
in a type-II MOT, the sub-Doppler cooling forces are substantially enhanced
albeit capturing and cooling atoms with higher temperatures is not feasible
Devlin and Tarbutt (2016). Thus, to efficiently access the sub-Doppler cooling
range, a pre-cooled atom ensemble loaded via a standard type-I MOT is required
Jarvis  _et al._ (2018). Furthermore, the polarization of the light fields or
the magnetic field gradients have to be reversed when switching between the
two MOT types to maintain the position depend trapping forces which requires
sophisticated optical setups Devlin and Tarbutt (2016); Jarvis  _et al._
(2018).
Here, we demonstrate a low expenditure method based on switching a
conventional red-detuned MOT to a blue-detuned type-II MOT using LCVRs. With
this marginally altered setup we increase the PSD of a 87Rb type-I MOT by more
than two orders of magnitude. We also show the feasibility of cooling and
trapping of 41K or 39K in a type-II MOT, although no improvement compared to
type-I MOTs is observed.

Figure 1: Frequency detunings used for the type-I and type-II MOT,
respectively. The experimentally determined ideal frequency detunings are
$\delta_{\mathrm{I},1}=-11\,$MHz, $\delta_{\mathrm{II},1}=46\,$MHz and
$\delta_{\mathrm{II},2}=22\,$MHz. ECDL-1 (ECDL-2) are indicated by solid
(dashed) arrows.

Due to its simplicity, the setup is especially suitable for compact devices
such as mobile atom sensors Hinton  _et al._ (2017); Bidel  _et al._ (2018) or
space-borne experiments Rudolph  _et al._ (2015); Elsen  _et al._ (2021); Frye
_et al._ (2021) and can easily and cost-efficiently be implemented in existing
setups.

(a)

(b)

Figure 2: (a) Setup of laser module and fiber splitter with respective fiber
outputs a-d. TA: Tapered amplifier, FS: fiber splitter, AOM: acousto-optical
modulator. The red (blue) arrows indicate the direction of polarization for
the red- (blue-) detuned MOT. (b) Setup of the experimental chamber with the
atom chip and external beam collimators. Shown are the four MOT laser beams
and the atom chip. The beams a and b are reflected on the chip surface which
forms a six-beam MOT configuration.

##  II Setup

The experimental setup shown in fig. 2 has initially been designed to operate
only type-I 87Rb MOTs. The experimental chamber features an atom chip housed
in a vacuum system and external Helmholtz coils to generate the required
magnetic fields in quadrupole configuration. Four collimators are fixed around
the vacuum chamber to form the MOT light fields, two of them being reflected
on the atom chip surface Folman  _et al._ (2000); Wildermuth  _et al._ (2004).
The details of the system are described in Elsen  _et al._ (2021). The laser
light is delivered via polarization maintaining (PM) optical fibers to the
respective collimators where it passes a quarter-waveplate to provide circular
polarization. It is generated by two home-built narrow linewidth external
cavity diode lasers (ECDL 1 and ECDL 2 in fig. 2 (a)) which operate initially
at the cooling ($\ket{F=2}\rightarrow\ket{F^{\prime}=3}$) and repumping
transitions ($\ket{F=1}\rightarrow\ket{F^{\prime}=2}$) of 87Rb at 780.242 nm
and 780.246 nm, respectively. The light is amplified in two single tapered
amplifiers with an optical output power of up to 1 W. Subsequently, each beam
passes an acousto-optic modulator that is used to control the intensities of
the respective light fields. A small fraction is split off to provide light
for spatially resolved absorption and fluorescence detection. The transmitted
parts used for atom cooling pass an LCVR (Thorlabs LCC1111-B) and are coupled
into a PM fiber-optic splitter. LCVRs consist of a transparent birefringent
polymer crystal which molecules align according to an applied AC voltage. Due
to the adjustable change of refraction index, they act as waveplates with
tunable retardance and fixed orientation. Here, they are regulated by a
square-wave voltage at a frequency of 2 kHz with variable amplitude which
allows to continuously rotate the polarization of the incident light. The
controller switches between two predefined amplitudes $V1$ and $V2$ by an
external trigger. The amplitudes are chosen in order to provide two orthogonal
output polarizations indicated by blue (green) arrows in fig. 2 (a). This
allows to make use either of the fast or slow polarization mode of the PM
fibers. In combination with the quarter-waveplates after the fiber splitter,
this allows to switch between left- and right-circular polarization of the MOT
beams.
The switching time has been measured in an external setup to be 0.6 ms from
$V1$ to $V2$ and 35 ms from $V2$ to $V1$. In principle, this time can be
shortened by using additional heating of the LCVRs or by using ultrafast but
usually high-cost Pockels cells. Since the detection light is not affected by
the LCVRs, only the fast switching direction is used during an experimental
sequence. Thus, no further gain from shorter switching times is expected for
the here-discussed experiments. The long-term stability of the retardance is
kept below $0.02\lambda$ over more than 100 weeks dat  which obviates any
further adjustments of amplitude voltages.
This setup allows to reverse the polarization for all beams inside the MOT
when switching from the red- to the blue-detuned MOT with only two LCVRs. It
additionally avoids any mechanical moving parts and is based on cost-efficient
components.

##  III Experiments

(a)

(b)

(c)

Figure 3: (a) Comparison of atom number over loading time for both MOTs.
Errorbars denote the $1\sigma$ statistical uncertainty over three consecutive
experimental runs. Transfer efficiency from type-I MOT to type-II MOT results
in $(84.3\pm 6.5)\,\%$. (b) Measurement of ensemble size in dependence of time
of flight after release from the respective MOTs in two spatial directions
$\sigma_{x}$ and $\sigma_{y}$ (see fig. 2 for coordinate system definition)
after a loading time of $500\,$ms. Errorbars denote three standard deviations
of three consecutive experimental runs for better visibility. The parameter
$\sigma$ is extracted for each direction with a 1D-Gaussian fit function. The
expansion is fitted with a linear model (solid lines). The type-II MOT
generates lower temperatures than the type-I MOT as revealed by the lower
expansion rate. (c) Fluorescence images of a red-detuned (blue-detuned) MOT
after free expansions of 5 ms (15 ms). The detected image of the red-detuned
MOT exceeds the size of the CCD sensor which deforms its Gaussian envelope
along x-direction leading to the observed deviation of the type-I MOT
expansion data.

In the following we will discuss our measurements to determine the feasibility
of a type-II MOT using the setup presented above and compare our results with
the case where we only use a type-I MOT.
For the type-I MOT the lasers ECDL 1 and 2 drive the transitions shown in fig.
1. The cooling laser is red detuned with $\delta_{I,1}=-11\,$MHz to the type-I
cooling transition, the repumper is in resonance with the transition. The
cooling and repumping beams operate at peak intensities of
$I_{I,1}=18.0\,$mW/cm2 and $I_{I,2}=2.0\,$mW/cm2 for each beam at the position
of the atoms, respectively. During the type-I MOT the LCVRs are operated with
the amplitude $V1$. After the loading time $T$, the type-I MOT is switched
off, followed by a 2 ms long period to stabilize the laser frequencies to the
blue detuned cooling transitions shown in fig. 1. The acousto-optical
modulators are switched off during this period to avoid unintended light
forces on the atoms. Also, the LCVRs are turned to $V2$ to reverse the light
polarization at the atom position. Subsequently, the lasers ECDL 1 and 2 drive
the type-II transitions $\ket{F=2}\rightarrow\ket{F^{\prime}=2}$ and
$\ket{F=1}\rightarrow\ket{F^{\prime}=1}$ with detunings of
$\delta_{\mathrm{II},1}=46\,$MHz and $\delta_{\mathrm{II},2}=22\,$MHz,
respectively. Notably the "repumper" transition
$\ket{F=1}\rightarrow\ket{F^{\prime}=1}$ contributes the main part to the
light forces Devlin and Tarbutt (2016). Thus the peak intensity of ECDL 2 is
increased to its maximum value of $I_{II,2}=8.2\,$mW/cm2 while the peak
intensity of ECDL 1 is slightly decreased to $I_{II,1}=13.2\,$mW/cm2 per beam.
In the type-II MOT, the atoms are trapped and further cooled for 20 ms.
Subsequently, the MOT light and magnetic fields are switched off and the atoms
are released into free fall. With a 20 µs light pulse resonant to the
transition $\ket{F=1}\rightarrow\ket{F^{\prime}=2}$, all atoms remaining in
$\ket{F=1}$ are pumped into the cycling detection transition
$\ket{F=2}\rightarrow\ket{F^{\prime}=3}$. The atoms are detected via
absorption and fluorescence imaging using the fiber output port shown in fig.
2(a). Here, absorption imaging is primarily used for atom number detection and
fluorescence imaging for expansion measurements in x- and y-direction.

##  IV Results

We characterize the properties of the blue-detuned type-II MOT by determining
its transfer efficiency from a type-I MOT and the achievable PSD. In fig.
3(a), the atom number over the (type-I) MOT loading time $T$ is shown for both
MOTs in comparison. The transfer efficiency between the type-I and type-II MOT
results in $(84.3\pm 6.5)\,\%$ averaged over all loading times. According to
simulations of the cooling forces Jarvis  _et al._ (2018), atoms below a
critical velocity of $v\approx 4.5\,$m/s are expected to be trapped in the
type-II MOT. Given the temperature of the type-I MOT of 1.4 mK, the transfer
efficiency should be close to 100 % based on a Maxwell-Boltzmann distribution
of the atoms in the type-I MOT. The $1/e$-lifetime of the type-II MOT has been
measured to be $(512\pm 17)\,$ms and accounts for a loss of 3.8 % of the
initially trapped atoms during the $20\,$ms type-II MOT period. We attribute
the remaining loss fraction to atoms which escape the trapping volume of the
type-II MOT during the $2\,$ms switching phase where the ensemble freely
expands. Faster frequency jumps could therefore reduce these losses.
Fig. 3(b) shows time of flight (ToF) measurements of the Gaussian ensemble
width in two spatial directions $\sigma_{x}$ and $\sigma_{y}$ of both MOTs for
a loading time of $500\,$ms. Here, the MOT light and the quadrupole field are
switched off, allowing the ensemble to expand freely until detection. The
ensemble size in the xy-plane is detected via fluorescence detection. The
expansion of an ensemble containing atoms with mass $m$ is related to its
temperature via $T=m\sigma_{v}^{2}/k_{B}$. The parameter $\sigma_{v}^{2}$ is
given by a least-square fit of the function
$\sigma^{2}(t)=\sigma^{2}(t=0)+\sigma_{v}^{2}t^{2}$. For the red-detuned
type-I MOT, a geometric mean temperature of
$\overline{T}_{\mathrm{I}}=(1.33\pm 0.03)\,$mK is determined. In the type-II
MOT, the ensemble temperature is further reduced to
$\overline{T}_{\mathrm{II}}=(44.5\pm 0.9)\,$µK which is below the Doppler
temperature limit $T_{D}\approx 140\,$µK Steck (2001) and indicates efficient
sub-Doppler cooling forces. Notably, the ensemble temperature is further
reduced for smaller atom densities which is attributed to reduced photon
rescattering Hillenbrand, Foot, and Burnett (1994); Townsend  _et al._ (1995).
For example, a temperature of 19.2 µK is reached for a type-II MOT with
$2.2\cdot 10^{8}$ atoms.
The data obtained from the TOF measurement allows us to calculate the PSD of
the ensemble by $\rho=n\cdot\lambda^{3}_{dB}$. Here, $n$ is the initial peak
density of the atom cloud right after release and

$\lambda_{dB}=\sqrt{\frac{2\pi\hbar^{2}}{mk_{B}T}}$

denotes the thermal de-Broglie wavelength. The peak density is evaluated using
$n=N/((\sqrt{2\pi})^{3}\sigma_{x}\sigma_{y}\sigma_{z})$ with the total atom
number $N$ and the initial Gaussian widths $\sigma_{x,y,z}$ immediately after
release. Here, the z-direction is accessible via absorption detection.
These equations lead to densities and PSDs of the ensembles after the type-I
MOT ($n_{\mathrm{I}}$, $\rho_{\mathrm{I}}$) and after the type-II MOT
($n_{\mathrm{II}}$, $\rho_{\mathrm{II}}$) of

$\displaystyle n_{\mathrm{I}}$ $\displaystyle=(6.8\pm 0.2)\cdot
10^{9}\,\mathrm{cm^{-3}}$ $\displaystyle\rho_{\mathrm{I}}$
$\displaystyle=(9.2\pm 0.4)\cdot 10^{-10}$ $\displaystyle n_{\mathrm{II}}$
$\displaystyle=(1.2\pm 0.1)\cdot 10^{10}\,\mathrm{cm^{-3}}$
$\displaystyle\rho_{\mathrm{II}}$ $\displaystyle=(2.7\pm 0.1)\cdot 10^{-7}.$

The PSD of the type-II MOT exceeds that of the type-I MOT by more than two
orders of magnitude.
The setup also provides an atom source and laser system for cooling and
manipulation of 41K and 39K which have most spectroscopic features of 87Rb in
commonTiecke (2019). In contrast to 87Rb, they exhibit narrow hyperfine
splittings of D2-transitions, which complicates laser cooling of 41K and 39K
in standard type-I MOTs.
By adapting the above described changes in the potassium laser system, the
feasibility of a blue-detuned type-II MOT for cooling of 41K and 39K is
explored. For both isotopes, cooling and trapping are only achieved if solely
the polarization of the blue-detuned repumping light field is reversed. The
cooling laser remains in the red-detuned type-I configuration. The experiments
show neither higher densities nor lower temperatures for the mixed type-I/II
MOT, indicating no advantages compared to the pure type-I MOT. This is
attributed to additional heating due to simultaneously driven type-I
transitions within the narrow level splitting of potassium.

##  V Conclusion

The setup and measurements presented in this paper demonstrate the feasibility
of a blue-detuned type-II MOT by utilizing LCVRs for fast polarization
control. This allows for a low-cost implementation in existing experimental
setups without the need of additional lasers or beam collimators.
By switching from red-detuned type-I to blue-detuned type-II transitions with
reversed polarizations, 85% of the 87Rb atoms trapped in the type-I MOT were
transferred into the type-II MOT. TOF measurements show that a geometric mean
temperature of $\overline{T}_{\mathrm{II}}=44.5\,$µK of the trapped was
reached. This is well below the Doppler temperature $T_{D}\approx 140\,$µK
which typically constitutes the lower temperature limit within a type-I MOT.
Furthermore, the PSD of the type-II MOT is shown to be increased by more than
two orders of magnitude compared to the type-I MOT. Among other methods like
dark spontaneous-force MOTs Ketterle  _et al._ (1993); Radwell, Walker, and
Franke-Arnold (2013), blue-detuned type-II MOTs offer an alternative and easy-
to-implement route to effectively increase the PSD of trapped ensembles.
A type-II MOT combines the advantages of a sub-Doppler cooled optical molasses
with the convenient position control of a trapped ensemble. Hence, type-II
MOTs might also help to improve the mode matching between the MOT and
consecutive stages of trapping and cooling, in particular for magnetic or
optical dipole traps. Accordingly, center-of-mass oscillations of transferred
ensembles due to positional mismatch could be reduced to a minimum compared to
a free-falling optical molasses.
LCVRs prove to be a simple and compact device to efficiently improve the atom
cooling performance. This applies especially for setups with rigorous
constraints on space, power consumption and weight like experiments on mobile
Hinton  _et al._ (2017); Bidel  _et al._ (2018) or microgravity platforms
Elsen  _et al._ (2021); Rudolph  _et al._ (2015); Frye  _et al._ (2021).

###### Acknowledgements.

We would like to thank Thijs Wendrich and Wolfgang Bartosch for electronic
support. We are grateful to Maike Lachmann for her experimental support and
comments on this manuscript.
This work is supported by the DLR Space Administration with funds provided by
the Federal Ministry for Economic Affairs and Energy (BMWi) under grant number
DLR 50WP1431, and is funded by the Deutsche Forschungsgemeinschaft (DFG,
German Research Foundation) under Germany’s Excellence Strategy – EXC-2123
QuantumFrontiers - 390837967.

## Data availability

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

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