# On the waist and width inequality in complete 3-manifolds with positive
scalar curvature

Yevgeny Liokumovich  Department of Mathematics, University of Toronto, 40 St
George Street, Toronto, ON M5S 2E4, Canada<EMAIL_ADDRESS>and  Zhichao
Wang  Department of Mathematics, University of British Columbia, Vancouver, BC
V6T 1Z2, Canada<EMAIL_ADDRESS>
###### Abstract.

We show that a complete non-compact 3-manifold with scalar curvature bounded
below by a positive constant admits a singular foliation by surfaces of
controlled area and diameter.

##  1\. Introduction

In [Gro20]*§3.10, Width/Waist Conjecture Gromov conjectured that a complete
manifold with scalar curvature $R\geq 6$ admits a singular foliation by
surfaces of area and diameter bounded by a universal constant. This conjecture
was proved for compact 3-manifolds in [LM20]. In this paper the result is
generalized to complete non-compact 3-manifolds.

######  Theorem 1.1.

Let $M$ be a complete non-compact 3-manifold with scalar curvature $R\geq 6$.
Then there exists a proper Morse function $f:M\rightarrow\mathbb{R}$, such
that for every $t\in\mathbb{R}$ each connected component $\Gamma$ of
$f^{-1}(t)$ satisfies

$\displaystyle\mathrm{diam}_{M}(\Gamma)<18\pi;$
$\displaystyle\mathrm{Area}(\Gamma)\leq 33\pi.$

For compact manifolds it was proved in [LM20] that one can construct a
foliation by surfaces of controlled area and diameter and genus at most $2$
(the maximal Heegaard genus of a compact 3-manifold with positive scalar
curvature). We conjecture that it should also be possible to prove a similar
genus bound in Theorem 1.1.

Recall that the Urysohn $1$-width of a Riemannian three-manifold $M$ is the
infimum of real numbers $d\geq 0$ with the property that there exist a Morse
function $f:M\to\mathbb{R}$ such that for all $t\in\mathbb{R}$, each connected
component of $f^{-1}(t)$ has diameter bounded by $d$ from above. For three-
manifolds with $R\geq 6$ and zero first Betti number, Gromov-Lawson [Gromov-
Lawson-diameter] proved an upper bound for Urysohn 1-width by considering
level sets of the distance function to a fixed point; in [Gro20]*§3.10,
Property A, Gromov proved the Urysohn 1-width upper bound without the
topological conditions; in [CL20] Chodosh and Li proved the Urysohn 1-width
upper bound for $3$-dimensional mu-bubbles in $4$-manifolds with positive
scalar curvature. For closed manifolds, the work of Maximo and the first
author [LM20] implies a new proof of the Urysohn 1-width bound. Inspired by
their work, the second author with Zhu proved the Urysohn 1-width upper bound
for mean convex domains (possibly non-compact) with non-negative Ricci
curvature in [Wang-Zhu-Urysohn]. Urysohn width bounds were used in [CCL23] to
obtain topological classification results for four- and five-manifolds with
positive scalar curvature. As a direct corollary, Theorem 1.1 gives a new
proof of the Urysohn 1-width upper bound of three-manifolds with $R\geq 6$.

### Idea of the proof

Our proof relies on methods from [LM20], construction of $\mu$-bubbles and
ideas of Gromov and Song on construction of minimal hypersurfaces in non-
compact manifolds ([Gromov2014], [Song19], see also [ChLi20]).

We first cut the complete three-manifold $(M,g)$ along minimal surfaces with
index less than or equal to 1. As a result, if these surfaces are countable
(e.g. the metric $g$ is bumpy), then such a cutting process decomposes $M$
into countably many regions. Then by adapting an argument of Gromov
[Gromov2014] and Song [Song19], the interior of these regions does not contain
any closed minimal surfaces. These regions are said to be geometrically prime,
a generalization of a similar notion in the compact case [LM20].

The key step is to decompose the geometrically prime regions into countably
many small pieces with diameter and boundary area upper bounds; see Definition
3.1 for the concept of admissible regions.

For prime regions with an unstable boundary component, we will use
$\mu$-bubble techniques and constrained minimization to construct finitely
many surfaces with controlled size to decompose it as a union of an admissible
region and finitely many regions with exactly one mean convex boundary
component; see Lemma 3.3. By repeating this argument, we obtain the desired
decomposition for geometrically prime regions with unstable boundary
components.

For those geometrically prime regions whose boundaries are stable minimal
surfaces (or the boundary is empty) we use arguments in [Gromov2014] [Song19]
to obtain a two-sided surface with controlled size and non-vanishing mean
curvature. Then for the mean convex region, the argument in the last paragraph
gives the desired decomposition. For the mean concave region, we cut off an
admissible region so that its complement consists of finitely many connected
regions with the property that their boundary contains exactly one non-minimal
component (the others are stable minimal surfaces). Moreover, each non-minimal
boundary component has controlled size and non-vanishing mean curvature.
Observe that we have dealt with these cases. Thus, by repeating the process,
we obtain the desired decomposition.

It remains to construct a Morse foliation for each admissible regions. To do
this, we regard the region as a subdomain in geometrically prime regions and
then consider mean curvature flow starting on the mean convex surfaces. Note
that the diameter upper bound may not be preserved in the flow. To obtain the
diameter bound we intersect level sets of the flow with admissible regions and
make necessary surgeries so that they have regularity of level sets of a Morse
function, similarly to how it was done in [LM20] and [ChLi20]. Combining all
of the foliations together finishes the proof.

### Outline

This paper is organized as follows. In Section 2, we present some known
results. We first introduce the bumpy metric theorem in Section 2.1; the area
and diameter bounds for minimal surfaces with index less than or equal to one
and $\mu$-bubbles are provided in Section 2.2. Finally, in Section 2.3, we
introduce the concept of geometrically prime regions and a dichotomy
proposition by adapting arguments from [Gromov2014] [Song19]. Section 3 is
devoted to three decomposition lemmas. They are used to decompose
geometrically prime regions into compact domains of controlled diameter and
boundary area. In Section 4, we construct foliation in each compact region and
prove our main theorem.

### Acknowledgement

Y.L. would like to thank Davi Maximo for many useful conversations. Y.L. was
partially supported by NSERC Discovery Grant and Sloan Fellowship. Z.W. would
like to thank Professor Jingyi Chen and Professor Ailana Fraser for their
support and encouragement.

##  2\. Preliminary

###  2.1. Bumpy metrics

Let $M$ be a complete (possibly non-compact) Riemannian manifold. We say that
metric $g$ on $M$ is bumpy if there are no closed minimal submanifolds with
non-zero Jacobi fields. By [WhiteBumpy17]*Theorem 2.1 (see also
[WhiteBumpy91]*Theorem 2.1) for any metric $g$ on $M$ there exists a sequence
$g_{i}$ of smooth bumpy metrics on $M$ with $|g-g_{i}|_{C^{3}}\rightarrow 0$.
Hence, without any loss of generality we can assume that our metric is bumpy.
We will also use the bumpy metric theorem with partially fixed metrics.

######  Theorem 2.1 ([WhiteBumpy17]*Theorem 3.1).

Let $(N,\gamma_{0})$ be a complete Riemannian manifold. Let $U\subset N$ be an
open set. Consider the class $\Gamma$ of smooth metrics $\gamma^{\prime}$ on
$N$ such that $\gamma^{\prime}$ and $\gamma_{0}$ agree on $N\setminus U$. Then
a generic metric $\gamma\in\Gamma$ has the following property: if $\Sigma$ is
a closed, minimal immersed submanifold of $N$ and if each connected component
of $\Sigma$ intersects $U$, then $\Sigma$ has no nontrivial Jacobi fields.

###  2.2. Existence of surfaces with bounded size

In this part, we present the existence of surfaces with controlled area and
diameter. Those surfaces will be used to decompose the manifold into small
pieces.

The classical candidate surfaces for decomposition are minimal surfaces with
index less than or equal to 1. In the following result, the area estimates in
the second item are from Marques-Neves [Marques-Neves-Duke]*Proposition A.1
and the diameter bounds are due to Schoen-Yau [Schoen-Yau-diameter]*Theorem 1
(see also [Gromov-Lawson-diameter]).

######  Lemma 2.2 ([LM20]*Theorem 2.1 and Corollary 2.4).

Let $(M^{3},g)$ be a complete three-manifold and $\Sigma\subset M$ be a two-
sided closed embedded minimal surface. Suppose that $R\geq 6$ in a
neighbourhood of $\Sigma$.

  1. (1)

If $\Sigma$ is stable, then

$\mathrm{diam}_{\Sigma}\leq\frac{2\pi}{3},\quad\mathrm{Area}\leq\frac{4\pi}{3}.$

  2. (2)

If $\Sigma$ has index one, then

$\mathrm{diam}_{\Sigma}\leq\frac{4\pi}{3},\quad\mathrm{Area}\leq\frac{16\pi}{3}.$

Schoen-Yau [Schoen-Yau-diameter]*Theorem 1 also gives a local version for
surfaces-with-boundary.

######  Lemma 2.3 ([LM20]*Theorem 2.3).

Let $(M^{3},g)$ be a complete three-manifold and $\Sigma\subset M$ be an
embedded stable minimal surface with boundary. Suppose that $R\geq 6$ in a
neighbourhood of $\Sigma$. Then

$\mathrm{dist}_{\Sigma}(x,\partial\Sigma)\leq\frac{2\pi}{3}.$

This implies the following Frankel property.

######  Lemma 2.4.

Let $(N,\partial N,g)$ be a complete smooth Riemannian three-manifold with
smooth boundary and $R\geq 6$. Suppose that each connected component of
$\partial N$ is a closed surface with non-negative mean curvature. Suppose
that there are two connected component $\Sigma_{1}$ and $\Sigma_{2}$ that are
not stable minimal surfaces. Then there is a two-sided closed stable minimal
surface in $(N\setminus\partial N,g)$.

###### Proof.

Since $\Sigma_{i}$ ($i=1,2$) is not a stable minimal surface, then
$\Sigma_{i}$ is either non-minimal or minimal but unstable. In either case, we
can perturb it slightly to be strictly mean convex. Thus without loss of
generality, we assume that $\Sigma_{1}$ and $\Sigma_{2}$ are mean convex. Let
$\gamma$ be a curve connecting $\Sigma_{1}$ and $\Sigma_{2}$. Then by taking
the area minimizer among all surface that is homologous to $\Sigma_{1}$, there
is a stable two-sided minimal surface intersecting $\gamma$. Such a minimizer
must be compact because of Lemma 2.2. This completes the proof of Lemma 2.4. ∎

After cutting along minimal surfaces with index less than or equal to 1, we
will use the $\mu$-bubble technique to subdivide a manifold with positive
scalar curvature into submanifolds with boundary components of controlled
size.

For a subset $\mathcal{K}$ of a Riemannian manifold $(M,g)$ and $r>0$, let

$N_{r}^{g}(\mathcal{K})=\\{x\in
M:\operatorname{dist}_{(M,g)}(x,\mathcal{K})<r\\}.$

We will drop the superscript $g$ whenever it is clear which Riemannian metric
is being used. The following lemma is a consequence of arguments in Zhu
[Zhu20]*Theorem 1.1, Gromov [Gro20] and Chodosh-Li [CL20]*§6.2 (see also
[GroAspherical]).

######  Lemma 2.5.

Let $N$ be a manifold with boundary and $R\geq 6$. Suppose that each connected
component of $\partial N$ is a stable minimal surface. Then for each compact
region $\Omega$, there exists a compact region $\hat{\Omega}$ with

$\Omega\subset\hat{\Omega}\subset N_{\frac{4}{3}\pi}(\Omega)$

and for each connected component
$\Gamma\subset\partial\hat{\Omega}\setminus\partial N$, it is one of the
following cases:

  1. (1)

$\Gamma$ is diffeomorphic to $S^{2}$ and

$\mathrm{diam}_{N}\Gamma\leq\pi;\quad\mathrm{Area}(\Gamma)<2\pi;$

  2. (2)

$\Gamma$ is diffeomorphic to $\mathbb{RP}^{2}$ and

$\mathrm{diam}_{N}\Gamma\leq\pi;\quad\mathrm{Area}(\Gamma)<\pi;$

  3. (3)

$\Gamma$ is a free boundary disc and

$\operatorname{dist}(x,\partial\Gamma)\leq\pi\text{ for
}x\in\Gamma;\quad\mathrm{diam}_{N}\Gamma\leq\frac{8}{3}\pi\quad\mathrm{Area}(\Gamma)<\pi;$

###### Proof.

Let $\Omega_{0}$ be a small perturbation of
$N_{\frac{2}{3}\pi}(\Omega)=\\{x\in
N:\operatorname{dist}(x,\Omega)<\frac{2}{3}\pi\\}$, so that
$\partial\Omega_{0}$ is smooth. Let $d:N\rightarrow\mathbb{R}$ denote a signed
distance function from $\partial\Omega_{0}$,
$d(x)=\operatorname{dist}(x,\partial\Omega_{0})$ if $x\notin\Omega_{0}$ and
$d(x)=-\operatorname{dist}(x,\partial\Omega_{0})$ if $x\in\Omega_{0}$. Note
that $d$ is Lipschitz. Denote the interior of
$N_{\frac{4}{3}\pi}(\Omega)\setminus\Omega$ by $U$ and let
$\rho:\overline{U}\rightarrow\mathbb{R}$ be a $C^{0}$ perturbation of $d$ that
is smooth and satisfies $\rho(x)=-\frac{2}{3}\pi$ for $x\in\partial\Omega$,
$\rho(x)=\frac{2}{3}\pi$ for $x\in\partial N_{\frac{4}{3}\pi}(\Omega)$ and
$|\nabla\rho|<1+\varepsilon$ for some small $\varepsilon<\frac{1}{100}$.
Define a smooth function $h$ on $U$ by

$h(x):=(1+\varepsilon)\tan\Big{[}\frac{3}{4}\rho(x)\Big{]}.$

By [CL20]*Propositions 12 and 15 there exists a minimizer $\mathcal{K}$, which
minimizes

$\mathcal{A}^{h}(\mathcal{K}^{\prime}):=\mathcal{H}^{2}(\partial\mathcal{K}^{\prime}\setminus\partial
N)-\int_{U}(\chi_{\mathcal{K}^{\prime}}-\chi_{\Omega_{0}})h\,\mathrm{d}\mathcal{H}^{2}$

among all compact region $\mathcal{K}^{\prime}$ satisfying

$\Omega\subset\mathcal{K}^{\prime}\subset\\{x\in
N;\operatorname{dist}_{N}(x,\Omega)<\frac{4}{3}\pi\\}.$

Let $\Gamma$ be a connected component of $\partial\mathcal{K}\setminus\partial
N$. Then $\Gamma$ is a smoothly embedded prescribed mean curvature surface
with free boundary, with scalar mean curvature $H=h$ and meeting $\partial N$
orthogonally. The minimizing property gives that for all smooth functions
$\phi$ defined on $\Gamma$,

$\int_{\Gamma}|\nabla\phi|^{2}-(|A|^{2}+\mathrm{Ric}(\mathbf{n},\mathbf{n})+\langle\mathbf{n},\nabla
h\rangle)\phi^{2}\,\mathrm{d}\mathcal{H}^{2}\geq\int_{\partial\Gamma}A_{\partial
N}(\mathbf{n},\mathbf{n})\phi^{2}\,\mathrm{d}\mathcal{H}^{1},$

where $A$ and $A_{\partial N}$ are the second fundamental forms of $\Gamma$
and $\partial N$; $\mathbf{n}$ is the unit normal vector field of $\Gamma$.
Now let $\phi\equiv 1$. Recall that

$|A|^{2}+\mathrm{Ric}(\mathbf{n},\mathbf{n})=\frac{1}{2}(|A|^{2}+|H|^{2}+R)-K_{\Gamma}.$

Using $|A|^{2}\geq\frac{1}{2}|H|^{2}$ (by Cauchy inequality) and
$2\pi\chi(\Gamma)=\int_{\Gamma}K_{\Gamma}d\mathcal{H}^{2}-\int_{\partial\Gamma}A_{\partial
N}(\mathbf{n},\mathbf{n})d\mathcal{H}^{1}$ (by Gauss-Bonnet and minimality of
$\partial N$) we obtain

$2\pi\chi(\Gamma)\geq\int_{\Gamma}\frac{1}{2}|A|^{2}+\frac{1}{2}h^{2}+\langle\nabla
h,\mathbf{n}\rangle+\frac{1}{2}R\,\mathrm{d}\mathcal{H}^{2}\geq\int_{\Gamma}\frac{3}{4}h^{2}-|\nabla
h|+3\geq(\frac{9}{4}-\varepsilon)\mathrm{Area}(\Gamma)>2\mathrm{Area}(\Gamma),$

where $\chi(\Gamma)$ is the Euler characteristic of $\Gamma$; $H^{\partial N}$
is the mean curvature of $\partial N$. If $\Gamma$ is non-orientable, then
$\Gamma$ is diffeomorphic to $\mathbb{RP}^{2}$ and
$\mathrm{Area}(\Gamma)<\pi$. If $\Gamma$ is orientable and closed, then
$\Gamma$ is diffeomorphic to $S^{2}$ and $\mathrm{Area}(\Gamma)<2\pi$. If
$\Gamma$ is orientable and has non-empty boundary, then $\Gamma$ is a free
boundary disc and $\mathrm{Area}(\Gamma)<\pi$.

It remains to prove the diameter upper bound. By applying the same argument as
in [LZ18]*Proposition 2.2, we conclude that

  * •

if $\Gamma$ is closed, then $\mathrm{diam}_{N}\Gamma\leq\pi$;

  * •

if $\Gamma$ is a disc, then $\operatorname{dist}_{N}(x,\partial\Gamma)\leq\pi$
for $x\in\Gamma$.

Since each connected component of $\partial N$ is a stable minimal surface,
then by Lemma 2.2, we have that its diameter is bounded by $\frac{2}{3}\pi$.
If $\Gamma$ is a disc it then follows that
$\mathrm{diam}_{N}\Gamma\leq\frac{2}{3}\pi+2\pi=\frac{8}{3}\pi$.

Hence, Lemma 2.5 is proved. ∎

###  2.3. Geometrically prime regions

We will use the concept of geometrically prime regions, which was defined in
[LM20]*Definition 2.5. Note that the region here may be non-compact.

######  Definition 2.6.

A region $N$ is geometrically prime if

  1. (1)

$N\setminus\partial N$ does not contain any closed embedded minimal surfaces;

  2. (2)

each connected component of $\partial N$ is a compact minimal surface with
index less than or equal to 1.

######  Remark 2.7.

Let $N$ be a geometrically prime region with $R\geq 6$. We conclude that each
embedded closed two-sided surface $\Sigma\subset N\setminus\partial N$
separates $N$. Suppose not, then there exists an area minimizer $\Sigma$ in
the homology class. Then $\Sigma$ is compact by Lemma 2.3. This gives a
contradiction.

We use the dichotomy theorem about existence of a minimal surface or a mean
convex foliation from [Gromov2014] and [Song19]*Theorem 2.1 to prove the
following result.

######  Proposition 2.8.

Let $N$ be a (possibly non-compact) three-manifold with (possibly empty)
boundary, bumpy metric and $R\geq 6$. Suppose that all boundary components of
$N$ are stable minimal surfaces and that $N$ does not contain any closed one-
sided embedded minimal surfaces with stable double covers or two-sided stable
embedded minimal surfaces in its interior. Then there are two possibilities:

  1. (1)

$N\setminus\partial N$ contains a closed two-sided minimal surface of index
one;

  2. (2)

$N\setminus\partial N$ does not contain any closed embedded minimal surfaces.
Let $\Omega\subset N$ be a compact region with smooth boundary so that each
connected component of $\partial N$ is either contained in $\partial\Omega$ or
disjoint from it. In this case, there exists a connected, closed, two-sided,
embedded $C^{1,1}$ surface $\Sigma\subset
N\setminus(\Omega\setminus\partial\Omega)$ intersecting $\partial\Omega$ and
having mean curvature vector pointing towards $\Omega$.

###### Proof.

If $N$ is compact, then we can apply min-max theory to obtain an index 1 two-
sided embedded minimal surface in the interior of $N$ (see [Zhou-15-one-
parameter]*Theorem 1.1 and Remark 1.3 [KMN-catenoid-estimate]*Theorem 1.5).
Now we assume that $N$ is non-compact.

Suppose that $N\setminus\partial N$ does not admit an index $1$ two-sided
embedded closed minimal surface in its interior.

Let $\Omega\subset N$ be a compact region with smooth boundary so that each
connected component of $\partial N$ is either contained in $\partial\Omega$ or
disjoint from it. We will show that there exists a connected, closed, two-
sided, embedded surface $\Sigma\subset
N\setminus(\Omega\setminus\partial\Omega)$ intersecting $\partial\Omega$ and
having mean curvature vector pointing towards $\Omega$. We argue as in
[Song19]*Theorem 2.1. Let $\mathcal{K}\supset N_{3\pi}(\Omega)$ be a set with
smooth boundary; we can assume that each connected component of $\partial N$
is either contained in $\partial\mathcal{K}$ or disjoint from it. Define a new
metric $\widetilde{g}$ that coincides with $g$ outside of
$N^{g}_{\frac{\pi}{2}}(\partial\mathcal{K}\setminus\partial N)$ and so that
$\partial\mathcal{K}\setminus\partial N$ is mean convex with respect to
$\widetilde{g}$. By Theorem 2.1, we may assume that metric $\widetilde{g}$ is
bumpy. By the choice of $\widetilde{g}$ and Lemma 2.3, for each closed,
connected two-sided minimal surface $\Gamma\subset(\mathcal{K},\widetilde{g})$
of index less than or equal to one, it is either disjoint from $N_{r}(\Omega)$
or disjoint from $N_{\frac{\pi}{2}}^{g}(\partial\mathcal{K}\setminus\partial
N)$.

Consider mean curvature flow $\\{\partial\mathcal{K}_{t}\\}$ starting on
$\mathcal{K}_{0}=\mathcal{K}$ in $(\mathcal{K},\widetilde{g})$. By
[White2000]*Theorem 11.1, there exist two possibilities:

  1. (i)

there exists $t^{\prime}>0$, such that
$\partial\mathcal{K}_{t^{\prime}}\cap\partial\Omega$ is non-empty;

  2. (ii)

there exists set $B$, $\Omega\subset B\subset\mathcal{K}$, with $\partial B$ a
smooth stable minimal surface in $(\mathcal{K},\widetilde{g})$.

Consider the first possibility. Let $\hat{t}<t^{\prime}$ be the first time of
the flow touching $\Omega$. Now consider the constrained minimizing problem
$\mathcal{P}(\Omega,\mathcal{K};\widetilde{g})$ (see [Lin-
obstacle][ZhihanWang-obstacle]*Section 2.4 for the regularity): minimizing
$\mathrm{Area}(\partial\Omega^{\prime};\widetilde{g})$ among all
$\Omega^{\prime}$ so that $\Omega\subset\Omega^{\prime}\subset\mathcal{K}$ and
$\partial\Omega^{\prime}$ is smooth. Denote by $\hat{\Omega}$ a constrained
minimizer of $\mathcal{P}(\Omega,\mathcal{K};\widetilde{g})$. Clearly,
$\partial\hat{\Omega}\setminus\Omega$ is a stable minimal surface w.r.t.
$\widetilde{g}$. Since $\partial\mathcal{K}_{t}\setminus\partial N$ is mean
convex, then by Maximum Principle [White-MP-2010]*Theorem 1,
$\partial\mathcal{K}_{t}\setminus\partial N$ does not intersect
$\partial\hat{\Omega}\setminus\Omega$ for all $t\in[0,\hat{t})$. Moreover, we
have that $\hat{\Omega}\subset\mathcal{K}_{\hat{t}}$. In the next claim, we
will prove that $\partial\hat{\Omega}$ intersects $\partial\Omega$. Denote by
$\Sigma$ the connected component of $\partial\hat{\Omega}$ that intersect
$\Omega$. Then this is the desired surface.

######  Claim 1.

$\partial\hat{\Omega}$ intersects $\partial\Omega$.

###### Proof of Claim 1.

Suppose not, then $\partial\hat{\Omega}$ is a stable minimal surface w.r.t.
$\widetilde{g}$. Since $(N\setminus\partial N,g)$ has no two-sided stable
minimal surface, then each connected component of
$\partial\hat{\Omega}\setminus\partial N$ intersects
$N_{\frac{\pi}{2}}^{g}(\partial\mathcal{K}\setminus\partial N)$. By Lemma 2.3,
for each $x\in\partial\hat{\Omega}$,

$\operatorname{dist}_{N}(x,N_{\frac{\pi}{2}}^{g}(\partial\mathcal{K}\setminus\partial
N)\leq\frac{2\pi}{3}.$

It follows that

$\operatorname{dist}_{N}(\partial\hat{\Omega}\setminus\partial N,\Omega)\geq
3\pi-\frac{\pi}{2}-\frac{2\pi}{3}>\pi,$

which implies that $\hat{\Omega}\supset\\{x\in
N;\operatorname{dist}_{N}(x,\Omega)\leq\pi\\}.$ This contradicts the fact of
$\hat{\Omega}\subset\mathcal{K}_{\hat{t}}$. Then Claim 1 is proved. ∎

Suppose we are in the second possibility. We can then apply the min-max
construction to find a two-sided index $1$ minimal surface $S$. By the
diameter estimate in Lemma 2.2 and our assumption that there are no two-sided
index $1$ minimal surfaces in $(N,g)$, we have that $S$ does not intersect
$\Omega$. Let $\tilde{\mathcal{K}}^{1}$ denote a connected component of
$\mathcal{K}\setminus S$ that contains $\Omega$. We can perturb
$S\subset\partial\tilde{\mathcal{K}}^{1}$ to the inside using the first
eigenfunction of the stability operator to obtain set
$\mathcal{K}^{1}\subset\tilde{\mathcal{K}}^{1}$ with mean convex boundary. We
can now apply mean curvature flow to $\mathcal{K}^{1}$ and arrive at the same
two possibilities (i) and (ii) as before. If we are in situation (i), we
obtain mean convex surface $\Sigma$ as above. If we are in situation (ii), we
apply the min-max construction again. Since the metric is bumpy, we can
encounter possibility (ii) only finitely many times.

So far we know that mean convex surface $\Sigma$ exists. Suppose $\Omega$
contains an embedded minimal surface $\Sigma^{\prime}$ in its interior. By our
assumption $\Sigma^{\prime}$ is either two-sided and unstable or one-sided
with unstable double cover. Consider manifold with boundary
$\Omega^{\prime\prime}$ obtained as metric completion of
$\Omega\setminus\Sigma^{\prime}$. Then by Lemma 2.4 at least one connected
component of $\Omega^{\prime\prime}$ (and hence $\Omega$) contains a stable
two-sided minimal surface in its interior, contradicting the assumption of the
theorem. Since this is true for all compact domains $\Omega$, we conclude that
possibility (2) of the theorem holds. ∎

The next theorem asserts that we can decompose $M$ into geometrically prime
regions. We will say that a countable collection of smooth connected
3-manifolds with boundary $\\{U_{i}\\}$ is a _decomposition_ of a 3-manifold
$M$ if there exists a countable collection of disjoint embedded closed
surfaces $\\{\Sigma_{j}\\}$ in $M$, such that there exists a Riemannian
isometry from $M\setminus\bigcup_{j}\Sigma_{j}$ to the union of interior of
$\\{U_{i}\\}$. (For example, $M=\mathbb{RP}^{3}$ and $\\{U\\}$ is a
decomposition of $M$ consisting of one element $U$ equal to the metric
completion of an open ball obtained by removing a non-trivial embedded
$\mathbb{RP}^{2}\subset M$.) By a slight abuse of notation we will identify
the interiors of $U_{i}$ and the corresponding isometric subsets of $M$.

######  Theorem 2.9.

Let $M$ be a (possibly non-compact) manifold with (possibly empty) boundary,
bumpy metric and $R\geq 6$. There exists a decomposition $\\{N_{i}\\}$ of $M$
into countably many geometrically prime submanifolds $N_{i}$.

###### Proof.

Since the metric is bumpy there exists a countable maximal collection of
disjoint minimal surfaces $\\{\Sigma_{i}\\}$, such that each $\Sigma_{i}$
satisfies one of the following three possibilities: $\Sigma_{i}$ two-sided and
stable; $\Sigma_{i}$ is two-sided and index $1$; or $\Sigma_{i}$ is one-sided
and has a stable double cover. We claim that
$M\setminus\bigsqcup_{i}\Sigma_{i}$ is the desired collection of geometrically
prime submanifolds.

Let $N$ be the metric completion of a connected component of
$M\setminus\bigsqcup_{i}\Sigma_{i}$. Then we finish the proof by considering
the following two possibilities.

  * •

If all boundary component of $\partial N$ are stable, or if $M=N$ (and the set
$\\{\Sigma_{i}\\}$ is empty), then by Proposition 2.8, $N\setminus\partial N$
does not contain any closed minimal surface, and hence $N$ is geometrically
prime.

  * •

If $\partial N$ has a boundary component that is a minimal surface of index
$1$, we now prove the conclusion by contradiction. Suppose on the contrary
that there exists a closed embedded minimal surface $\Gamma$ in
$N\setminus\partial N$. Then by assumption, $\Gamma$ is either unstable or
one-sided and has an unstable double cover. In either case, we cut $N$ along
$\Gamma$ and let $\widetilde{N}$ be the metric completion. Then
$\partial\widetilde{N}$ contains two unstable minimal surfaces of
$\widetilde{N}$. By Lemma 2.4, $\widetilde{N}\setminus\partial\widetilde{N}$
contains a two-sided stable minimal surface. This gives a contradiction.

∎

##  3\. Decomposition of geometrically prime regions

Let $M$ be three dimensional Riemannian manifolds with $R\geq 6$. In this
section, $N$ is always a geometrically prime region in $M$. We will divide
such an $N$ into compact regions with controlled diameter and boundary area.
More precisely, we will divide $N$ into subsets satisfying the following
definition.

######  Definition 3.1.

Let $N\subset M$ be a geometrically prime region, $R\geq 6$. Then a compact
region $\Omega\subset N$ is admissible if

  1. (1)

$\partial\Omega$ is closed;

  2. (2)

each connected component $\Gamma^{\prime}$ of $\partial\Omega$ satisfies

$\mathrm{diam}_{N}\Gamma^{\prime}\leq 6\pi;$

moreover,

$\mathrm{Area}(\Gamma^{\prime})\leq 4\pi\text{ if $\Gamma^{\prime}$ is mean
concave};\quad\mathrm{Area}(\Gamma^{\prime})\leq\frac{16}{3}\pi\text{ if
$\Gamma^{\prime}$ is mean convex};$

  3. (3)

one connected component $\Gamma$ of $\partial\Omega$ is weakly mean convex and
the others are weakly mean concave; moreover,

$\mathrm{Area}(\partial\Omega)\leq\frac{32}{3}\pi;$

  4. (4)

for each $x\in\Omega$ the distance to the mean convex component $\Gamma$ of
$\partial\Omega$ satisfies

$\operatorname{dist}_{N}(x,\Gamma)<6\pi.$

In particular, we have $\mathrm{diam}_{N}(\Omega)<18\pi$.

######  Proposition 3.2.

Let $N$ be a geometrically prime region. Then $N$ can be decomposed into
countably many admissible regions $\\{\Omega_{j}\\}$.

To prove Proposition 3.2, we will need three lemmas about existence of
separating surfaces of controlled size and mean curvature.

###  3.1. Decomposition of mean convex regions

The first lemma concerns existence of a mean convex surface of controlled size
in a subset of $N$ with mean convex boundary. It gives that for a subdomain of
a geometrically prime region with a mean convex boundary component (which is
the unique non-minimal boundary component) one can cut off an admissible
region near the mean convex boundary.

Let $(N,g)$ be a Riemannian manifold with smooth boundary. In this section,
for any compact set $\mathcal{K}\subset N$, we let

$N_{r}(\mathcal{K})=\\{x\in N:\operatorname{dist}_{N}(x,\mathcal{K})<r\\}.$

######  Lemma 3.3.

Let $N\subset M$ be a geometrically prime region. Let $\Sigma$ be a two-sided
connected surface with non-vanishing mean curvature. Then $\Sigma$ separates
$N$ into two connected components. Denote by $\Omega$ the one that has mean
convex boundary, and suppose that $dist(x,\Sigma)>\frac{x}{\Sigma}$ for some
$x\in\Omega$. Then there exists a compact connected domain
$\hat{\Omega}\subset\Omega$ with $\Sigma\subset\partial\hat{\Omega}$ and
satisfying the following.

  1. (1)

$\partial\hat{\Omega}\setminus\partial\Omega$ is a closed and embedded surface
in $N$ and has mean curvature vector pointing away from $\Sigma$;

  2. (2)

$\operatorname{dist}_{N}(\partial\hat{\Omega}\setminus\partial\Omega,\Sigma)\geq\frac{\pi}{4}$;

  3. (3)

$\sup\\{\operatorname{dist}_{N}(x,\Sigma);x\in\hat{\Omega}\\}<3\pi$;

  4. (4)

Each connected component
$\Gamma^{\prime}\subset\partial\hat{\Omega}\setminus\Sigma$ satisfies
$\mathrm{Area}(\Gamma^{\prime})<\frac{7}{3}\pi$ and
$\mathrm{diam}_{N}\Gamma^{\prime}<6\pi$;

  5. (5)

$\mathrm{Area}(\partial\hat{\Omega}\setminus\Sigma)<\mathrm{Area}(\Sigma)$.

In particular, $\hat{\Omega}$ is admissible provided that
$\mathrm{Area}(\Sigma)\leq\frac{16}{3}\pi$ and $\mathrm{diam}_{N}\Sigma\leq
6\pi$.

###### Proof.

Since $N$ is geometrically prime, then $N\setminus\partial N$ does not contain
minimal surfaces. Note that $\Sigma=\partial\Omega\setminus\partial N$ is mean
convex. Since $\Sigma$ is mean convex as a boundary component of $\Omega$,
then by Lemma 2.4, each connected component of $\Omega\cap\partial N$ is a
stable minimal surface. Then by Lemma 2.2, such a component has area bounded
by $\frac{4}{3}\pi$ and diameter bounded by $\frac{2}{3}\pi$. By Lemma 2.5,
there exists a compact region

$\Omega^{\prime}\subset\\{x\in\Omega;\operatorname{dist}_{N}(x,\Sigma)<\frac{4}{3}\pi\\}$

with $\Sigma\subset\partial\Omega^{\prime}$ and so that each connected
component
$\Gamma^{\prime}\subset\partial\Omega^{\prime}\setminus\partial\Omega$ is
either an $S^{2},\mathbb{RP}^{2}$ with

$\mathrm{Area}(\Gamma^{\prime})<2\pi,\quad\mathrm{diam}_{N}(\Gamma^{\prime})\leq\pi,$

or a free boundary disc with

$\mathrm{Area}(\Gamma^{\prime})<\pi,\quad\mathrm{diam}_{N}(\Gamma^{\prime})\leq\frac{8}{3}\pi.$

Denote by $\\{\mathcal{B}_{j}\\}$ the connected components of
$\partial\Omega^{\prime}\setminus\partial\Omega$. Note that each
$\mathcal{B}_{j}$ separates $\Omega$. Let $E_{j}$ denote the metric completion
of the connected component of $\Omega\setminus\mathcal{B}_{j}$ that does not
contain $\Sigma$.

We can take a set $\widetilde{\Omega}$,
$N_{\frac{2}{3}\pi+\frac{1}{4}\pi}(\Omega^{\prime})\cap\Omega\subset\widetilde{\Omega}\subset
N_{\frac{5}{3}\pi}(\Omega^{\prime})\cap\Omega$, so that
$\partial\widetilde{\Omega}\setminus\partial\Omega$ is a smooth surface.
Moreover, we perturb $\widetilde{\Omega}$ so that each connected component of
$\partial N$ is either contained in $\partial\widetilde{\Omega}$ or disjoint
from it. Let $\mathcal{C}_{j}$ denote the union of connected components of
$\partial\widetilde{\Omega}\setminus\partial\Omega$ that lie in $E_{j}$. It
follows that for all $\mathcal{C}_{i}$ and $\mathcal{B}_{j}$,

(3.1)
$\operatorname{dist}_{N}(\mathcal{C}_{j},\mathcal{B}_{i})\geq\frac{2}{3}\pi+\frac{1}{4}\pi.$

$\Sigma$
---
$\Omega^{\prime}$
---
$\mathcal{B}_{j}$
---
$\mathcal{C}_{j}$
---
$\mathcal{C}_{j}$
---
$\mathcal{C}_{j}$
---
$\mathcal{C}_{j}$
---
$\hat{\Sigma}$
---
$\hat{\Sigma}$
---
$\hat{\Sigma}$
---
$\hat{\Sigma}$
---
Figure I. Cutting a mean convex region

Let $\hat{\Sigma}$ be an area minimizer among all surfaces in
$\widetilde{\Omega}$ that are homologous to $\Sigma$; see Figure I. Then
$\hat{\Sigma}$ is a $C^{1,1}$ weakly mean concave surface (see [Lin-
obstacle][ZhihanWang-obstacle]*Section 2.4 for the regularity). Moreover,
$\hat{\Sigma}\setminus(\cup_{j}\mathcal{C}_{j})$ is a stable minimal surface.
Thus for each $p\in\hat{\Sigma}\setminus\partial N$, one has

$\operatorname{dist}_{N}(p,\cup_{j}\mathcal{C}_{j})\leq\frac{2}{3}\pi.$

This together with (3.1) implies that $\hat{\Sigma}$ does not intersect
$\mathcal{B}_{j}$. Let $\hat{\Omega}$ be the connected component of
$\Omega\setminus\hat{\Sigma}$ that contains $\Sigma$. Then for each connected
component of $\partial\hat{\Omega}\setminus\Sigma$, it is either a closed
embedded minimal surface or a mean concave surface intersecting
$\cup\mathcal{C}_{j}$. Observe that for the first case, it is a connected
component of $\partial N$. Thus it suffices to prove the diameter and area
upper bounds for the second case.

Let $\Gamma^{\prime}$ be a connected component of $\hat{\Sigma}$ that
intersects $E_{j}$. Since $\Gamma^{\prime}$ does not intersect
$\mathcal{B}_{j}$ we have that $\Gamma^{\prime}\subset E_{j}$. Moreover,

$\Gamma^{\prime}\subset E_{j}\cap\widetilde{\Omega}.$

It follows that

$\mathrm{diam}_{N}\Gamma^{\prime}<\mathrm{diam}_{N}\mathcal{B}_{j}+\frac{5}{3}\pi+\frac{5}{3}\pi\leq
6\pi.$

The area upper bound for $\partial\widetilde{\Omega}\setminus\Sigma$ follows
from $\mathrm{Area}(\hat{\Sigma})<\mathrm{Area}(\Sigma)$. The area upper bound
for each connected component $\Gamma^{\prime}$ of
$\hat{\Sigma}\setminus\partial\Omega$ follows by comparing the area of
$\Gamma^{\prime}$ to the area of a connected component $\mathcal{B}_{E_{j}}$
of $\partial E_{j}$ that separates $\Gamma^{\prime}$ from $\Sigma$. Recall
that $\mathcal{B}_{E_{j}}$ is either a boundary component of
$\partial\Omega^{\prime}$ of area $<2\pi$, or a union of a disc in
$\partial\Omega^{\prime}\setminus\partial\Omega$ and part of a component in
$\partial\Omega$ of area $<\pi+\frac{4}{3}\pi<\frac{7}{3}\pi$.

Moreover,

$\operatorname{dist}_{N}(\Gamma^{\prime},\Sigma)\geq\inf_{i}\operatorname{dist}_{N}(\mathcal{C}_{i},\Sigma)-\frac{2}{3}\pi>\frac{\pi}{4}.$

The third item comes from the fact of $\hat{\Omega}\subset\widetilde{\Omega}$.
This completes the proof of Lemma 3.3. ∎

Given a geometrically prime region $N$ with one boundary component of Morse
index $1$, it is straightforward to decompose it into admissible regions using
Lemma 3.3 repeatedly. But how can we decompose a geometrically prime region
that only has stable boundary components (or no boundary at all)? The next two
lemmas deal with this situation.

In the second lemma we consider a domain $\Omega\subset N$ with two non-
minimal boundary components. Suppose that one of them is mean convex and the
other is mean concave. We will show that $\Omega$ can be divided by a
collection of closed surfaces into finitely many domains with mean convex
boundaries and a compact region. Moreover, all of these surfaces have
uniformly bounded area and diameter.

######  Lemma 3.4.

Let $N$ be a geometrically prime region of $M$ with $R\geq 6$. Let
$\Omega_{1}\subset\Omega_{0}\subset N$ be two regions so that
$\partial\Omega_{i}\setminus\partial N$ is a closed connected surface
$\Sigma_{i}$ and let $\Omega=\Omega_{0}\setminus\Omega_{1}$. Suppose that
$\Sigma_{i}$ is mean convex as part of $\partial\Omega_{i}$ for $i=0,1$.
Denote by

$d_{0}=\sup\\{\operatorname{dist}_{N}(x,\Sigma_{0});x\in\Sigma_{1}\\}.$

Then there exists a compact connected domain $\hat{\Omega}\subset\Omega$ with
$\partial\hat{\Omega}$ containing $\Sigma_{0},\Sigma_{1}$ and satisfying the
following.

  1. (1)

$\partial\hat{\Omega}\setminus\partial\Omega$ is a closed and embedded surface
in $N$ and has mean curvature vector pointing away from $\Sigma_{0}$;

  2. (2)

$\sup\\{\operatorname{dist}_{N}(x,\Sigma_{0});x\in\hat{\Omega}\\}<d_{0}+3\pi$;

  3. (3)

Each connected component
$\Gamma^{\prime}\subset\partial\hat{\Omega}\setminus(\Sigma_{0}\cup\Sigma_{1})$
satisfies $\mathrm{Area}(\Gamma^{\prime})<\frac{7}{3}\pi$ and
$\mathrm{diam}_{N}\Gamma^{\prime}<6\pi$;

  4. (4)

$\mathrm{Area}(\partial\hat{\Omega}\setminus(\Sigma_{0}\cup\Sigma_{1}))<\mathrm{Area}(\Sigma_{0})$.

In particular, $\hat{\Omega}$ is admissible provided that $d_{0}\leq 3\pi$ and
$\mathrm{Area}(\Sigma_{i})<4\pi$, $\mathrm{diam}_{N}\Sigma_{i}\leq 6\pi$,
$i=0,1$.

###### Proof.

The argument is similar to Lemma 3.3 with minor modifications. Apply Lemma 2.5
to the compact set $N_{d_{0}}(\Sigma_{0})\cap\Omega$; after intersecting with
$\Omega$ we obtain a compact region $\Omega^{\prime}$, such that

$N_{d_{0}}(\Sigma_{0})\cap\Omega\subset\Omega^{\prime}\subset
N_{d_{0}+\frac{4\pi}{3}}(\Sigma_{0})\cap\Omega,$

so that each connected component
$\Gamma^{\prime}\subset\partial\Omega^{\prime}\setminus\partial\Omega$ is
either an $S^{2},\mathbb{RP}^{2}$ with

$\mathrm{Area}(\Gamma^{\prime})<2\pi,\quad\mathrm{diam}_{N}(\Gamma^{\prime})\leq\pi.$

or a free boundary disc with

$\mathrm{Area}(\Gamma^{\prime})<\pi,\quad\mathrm{diam}_{N}(\Gamma^{\prime})\leq\frac{8}{3}\pi.$

Denote by $\\{\mathcal{B}_{j}\\}$ the connected component of
$\partial\Omega^{\prime}\setminus\partial\Omega$. Note that $\mathcal{B}_{j}$
separates $\Omega_{0}$. Denote by $E_{j}$ the connected component of
$\Omega_{0}\setminus\mathcal{B}_{j}$ that does not contain $\Sigma_{0}$.
Observe that each connected component of $\partial N$ has diameter bounded by
$\frac{2}{3}\pi$. We can take a set $\widetilde{\Omega}$,
$N_{\frac{2}{3}\pi+\frac{1}{4}\pi}(\Omega^{\prime})\cap\Omega_{0}\subset\widetilde{\Omega}\subset
N_{\frac{5}{3}\pi}(\Omega^{\prime})\cap\Omega_{0}$, so that
$\partial\widetilde{\Omega}\setminus\partial\Omega$ is a smooth surface
(possibly with boundary in $\partial\Omega_{0}$). Let

$\mathcal{C}_{j}=E_{j}\cap\partial\widetilde{\Omega}\setminus\partial\Omega_{0},\quad\widetilde{\mathcal{C}}=\Omega_{1}\cap\partial\widetilde{\Omega}\setminus\partial\Omega_{0}.$

Note that $\mathcal{C}_{j}$ and $\widetilde{\mathcal{C}}$ may have boundary on
$\partial\Omega_{0}$. It follows that for all $\mathcal{C}_{i}$ and
$\mathcal{B}_{j}$,

(3.2)
$\operatorname{dist}_{N}(\mathcal{C}_{j},\mathcal{B}_{i})\geq\frac{2}{3}\pi+\frac{1}{4}\pi,\quad\operatorname{dist}_{N}(\widetilde{\mathcal{C}},\Sigma_{1})\geq\frac{2}{3}\pi+\frac{1}{4}\pi.$

$\Sigma_{1}$
---
$\Sigma_{0}$
---
$\Omega_{1}$
---
$\mathcal{B}_{j}$
---
$\hat{\Sigma}$
---
$\Omega$
---
$\widetilde{\mathcal{C}}$
---
$\mathcal{C}_{j}$
---
$\mathcal{C}_{j}$
---
$\hat{\Sigma}$
---
$\hat{\Sigma}$
---
$\hat{\Sigma}$
---
Figure II. Cutting the region bounded by two given surfaces

Let $\hat{\Sigma}$ be the area minimizer among all surfaces in
$\widetilde{\Omega}$ that are homologous to $\Sigma_{0}$; see Figure II. Then
$\hat{\Sigma}$ is a $C^{1,1}$ weakly mean concave surface (see [Lin-
obstacle][ZhihanWang-obstacle]*Section 2.4 for the regularity). Moreover,
$\hat{\Sigma}\setminus(\cup_{j}\mathcal{C}_{j}\cup\widetilde{\mathcal{C}})$ is
a stable minimal surface with respect to the normal vector field pointing away
from $\Sigma_{0}$. Since $\hat{\Sigma}$ is a stable minimal surface outside
$\cup\mathcal{C}_{j}\cup\mathcal{C}$, then by Lemma 2.3, for each
$p\in\hat{\Sigma}$ lying in the interior of $\Omega_{0}$, one has

$\operatorname{dist}_{N}(p,\cup_{j}\mathcal{C}_{j}\cup\widetilde{\mathcal{C}})\leq\frac{2}{3}\pi.$

Thus by (3.2), $\hat{\Sigma}\setminus\partial\Omega_{0}$ does not intersect
$\Sigma_{1}$ or the interior of $\mathcal{B}_{j}$. Let $\hat{\Omega}$ be the
connected component of $\Omega\setminus\hat{\Sigma}$ that contains
$\Sigma_{0}$. Then for each connected component of
$\partial\hat{\Omega}\setminus\partial\Omega$, it is either a closed embedded
minimal surface or a weakly mean concave surface intersecting
$\cup\mathcal{C}_{j}$. Observe that for the first case, it is a connected
component of $\partial N$. The second item comes from the fact of
$\hat{\Omega}\subset\widetilde{\Omega}$. The last item follows from that
$\mathrm{Area}(\hat{\Sigma})<\mathrm{Area}(\Sigma)$. The area and diameter
estimates for each connected component of
$\partial\hat{\Omega}\setminus(\Sigma_{0}\cup\Sigma_{1})$ follows from the
same argument in Lemma 3.3. This completes the proof of Lemma 3.4. ∎

Let $\Sigma$ be a two-sided surface that separates $N$ into two subsets
$\Omega$ and $N\setminus\Omega$ and assume the mean curvature vector of
$\Sigma$ is pointing away from $\Omega$. Recall that by Proposition 2.8, there
exists another two-sided surface in $\Omega$ with mean curvature vector
pointing towards $\Sigma$. Assuming that $N$ has $R\geq 6$ we give an
improvement: the new surface has controlled area and diameter.

######  Lemma 3.5.

Let $N$ be a geometrically prime region of $M$ with $R\geq 6$. Let
$\Omega\subset N$ be a non-compact region so that
$\Sigma:=\partial\Omega\setminus\partial N$ is a closed connected embedded
mean concave surface and $\partial N\cap\partial\Omega$ is a collection of
stable minimal surfaces. Then there exists a connected closed embedded surface
$\Sigma_{1}\subset\Omega$ such that

  1. (1)

$\Sigma_{1}$ has mean curvature vector pointing towards $\Sigma$;

  2. (2)

$\mathrm{diam}_{N}\Sigma_{1}<5\pi$ and
$\mathrm{Area}(\Sigma_{1})<\frac{8\pi}{3}$;

  3. (3)

$\operatorname{dist}_{N}(\Sigma,\Sigma_{1})\geq\frac{\pi}{3}$,
$\sup\\{\operatorname{dist}_{N}(x,\Sigma);x\in\Sigma_{1}\\}<3\pi.$

###### Proof.

Note that by Lemma 2.2, each connected component of $\partial N$ satisfies

(3.3)
$\mathrm{diam}_{N}\leq\frac{2\pi}{3},\quad\mathrm{Area}\leq\frac{4\pi}{3}.$

By Lemma 2.5, there exists a compact region $\Omega^{\prime}$,

(3.4) $N_{\frac{\pi}{3}}(\Sigma)\cap\Omega\subset\Omega^{\prime}\subset
N_{\frac{5\pi}{3}}(\Sigma)\cap\Omega,$

so that each connected component
$\Gamma^{\prime}\subset\partial\Omega^{\prime}\setminus\partial N$ is one of
the following cases:

  * •

$\Gamma^{\prime}$ is an $S^{2}$ or $\mathbb{RP}^{2}$ and

$\mathrm{Area}(\Gamma^{\prime})<2\pi,\quad\mathrm{diam}_{N}(\Gamma^{\prime})\leq\pi;$

  * •

$\Gamma^{\prime}$ is a disc and

$\mathrm{Area}(\Gamma^{\prime})<\pi,\quad\mathrm{dist}_{\Gamma^{\prime}}(x,\partial\Gamma^{\prime})\leq\pi,\quad\forall\,x\in\Gamma^{\prime}.$

Combining with (3.3), we have that for each connected component of
$\partial\Omega^{\prime}\setminus\Sigma$, it is either a smooth surface which
is diffeomorphic to $\mathbb{S}^{2}$ or $\mathbb{RP}^{2}$ with

$\mathrm{Area}(\Gamma^{\prime})<2\pi,\quad\mathrm{diam}_{N}(\Gamma^{\prime})\leq\pi;$

or the union of a free boundary disc $D$ and part of a connected component of
$\partial N$ with

$\mathrm{Area}<\pi+\frac{4\pi}{3}=\frac{7\pi}{3},\quad\mathrm{diam}_{N}\leq
2\pi+\frac{2\pi}{3}=\frac{8\pi}{3}.$

Note that $D$ has boundary on a minimal surface with diameter bounded by
$\frac{2\pi}{3}$ (see Lemma 2.2). Then given $\delta>0$, by modifying non-
smooth boundary of $\partial\Omega^{\prime}$ slightly, we can take a region
$\widetilde{\Omega}$ so that

  * •

$\Omega^{\prime}\subset\widetilde{\Omega}\subset\\{x\in\Omega;\operatorname{dist}_{N}(x,\Omega^{\prime})<\frac{2\pi}{3}+\delta\\}$;

  * •

$\partial\widetilde{\Omega}\setminus\partial N$ is a smooth surface;

  * •

each connected component of $\partial\widetilde{\Omega}$ satisfies

$\mathrm{Area}<\frac{8\pi}{3},\quad\mathrm{diam}_{N}<3\pi.$

Using (3.4), we can take $\delta>0$ sufficiently small so that

(3.5)
$\sup_{x\in\widetilde{\Omega}}\operatorname{dist}_{N}(x,\Sigma)<\frac{7\pi}{3}.$

Denote by $\widetilde{\Omega}_{0}$ the connected component of
$\widetilde{\Omega}$ that contains $\Sigma$. By Proposition 2.8, there exists
a two-sided closed surface $\mathcal{C}\subset
N_{20\pi}(\widetilde{\Omega})\setminus N_{10\pi}(\widetilde{\Omega})$ with
mean curvature vector pointing towards $\widetilde{\Omega}$. Clearly,
$\mathcal{C}$ does not intersect $\Sigma$. Thus, $\mathcal{C}$ is either
contained in $\Omega$ or disjoint from $\Omega$. Recall that $\Sigma$ is mean
concave and $N$ is geometrically prime. Then by Lemma 2.4, we conclude that
$\mathcal{C}\subset\Omega$. Now we take a large connected, compact region
$\mathcal{K}$ satisfying

  * •

$\partial\mathcal{K}\setminus\partial N$ is a closed surface;

  * •

$\\{x\in
N;\operatorname{dist}_{N}(x,\widetilde{\Omega})\leq\operatorname{dist}_{N}(\mathcal{C},\widetilde{\Omega})+\mathrm{diam}_{N}\mathcal{C}+100\pi\\}\subset\mathcal{K}$.

$\Sigma$
---
$\partial\Omega^{\prime}$
---
$D\subset\partial\Omega^{\prime}$
---
$S$
---
$\Sigma_{1}$
---
$\mathcal{C}$
---
$\partial\mathcal{K}$
---
$\partial\mathcal{K}$
---
$\partial\mathcal{K}$
---
$\partial\mathcal{K}$
---
Figure III. Another mean convex surface with bounded size.

Let $\widetilde{\mathcal{K}}$ be the metric completion of connected component
of $\mathcal{K}\setminus(\mathcal{C}\cup\widetilde{\Omega}_{0})$ that contains
$\Sigma$ and $\mathcal{C}$. Denote by
$S:=\partial\widetilde{\mathcal{K}}\cap\partial\widetilde{\Omega}_{0}$; see
Figure III.

######  Claim 2.

$S$ is connected.

###### Proof of Claim 2.

Suppose not, let $S_{1}$ and $S_{2}$ be two connected components. Then they
can be joined by a curve in $\widetilde{\mathcal{K}}$. By definition,
$S_{i}\subset\widetilde{\Omega}_{0}$, $i=1,2$. Since $\widetilde{\Omega}_{0}$
is connected, then $S_{1}$ and $S_{2}$ can be joined by a curve in
$\widetilde{\Omega}_{0}$. Thus we conclude that there is a simple closed curve
intersecting $S_{1}$ with algebraic intersection number one. This contradicts
that $S_{1}$ is two-sided and separates $N$ by Remark 2.7. ∎

So far, we have proved that $\partial\widetilde{\mathcal{K}}\setminus\partial
N$ has at least two connected components: one is $\mathcal{C}$ (mean convex)
and the other is $S\subset\partial\widetilde{\Omega}_{0}$. Now we consider the
constrained minimizer among all closed surfaces that are homologous to $S$ in
$\widetilde{\mathcal{K}}$. Denote by $\hat{\Sigma}$ the minimizer. Clearly,
$\hat{\Sigma}$ is a stable minimal surface (with boundary) in the interior of
$\widetilde{\mathcal{K}}$.

######  Claim 3.

$\hat{\Sigma}$ intersects $S$.

###### Proof of Claim 3.

Denote by
$\mathfrak{C}_{0}:=\partial\widetilde{\mathcal{K}}\setminus(\mathcal{C}\cup
S\cup\partial N)$; that is, $\mathfrak{C}_{0}$ is the non-minimal boundary of
$\widetilde{\mathcal{K}}$ except $\mathcal{C}$ and $S$. Since
$\widetilde{\mathcal{K}}$ has no closed minimal surfaces, then for each
connected component of $\hat{\Sigma}$, it is either one of $\partial N$ or
intersect $S\cup\mathfrak{C}_{0}$. Suppose that $\hat{\Sigma}$ does not
intersect $S$. Then $\hat{\Sigma}\setminus\partial N$ lies in a neighbourhood
of $\mathfrak{C}_{0}$ by Lemma 2.3. This contradicts $\hat{\Sigma}$ being in
the homology class of $S$. ∎

Let $\Sigma_{1}$ be the connected component of $\hat{\Sigma}$ that intersects
$S$. Then $\Sigma_{1}$ intersects $\partial\widetilde{\Omega}\setminus\partial
N$. Moreover, $\Sigma_{1}\setminus\widetilde{\Omega}$ is a stable minimal
surface. Then by Lemma 2.3, for any $x,y\in\Sigma_{1}$,

$\operatorname{dist}_{N}(x,y)\leq\operatorname{dist}_{N}(x,S)+\operatorname{dist}_{N}(y,S)+\mathrm{diam}_{N}S<5\pi.$

By the minimizing property of $\hat{\Sigma}$, we have

$\mathrm{Area}(\Sigma_{1})\leq\mathrm{Area}(S)<\frac{8\pi}{3}.$

It remains to prove the last item. Clearly,

$\operatorname{dist}_{N}(\Sigma,\Sigma_{1})\geq\operatorname{dist}_{N}(\Sigma,S)\geq\frac{\pi}{3}.$

By (3.5) and Lemma 2.3, for $x\in\Sigma_{1}$,

$\operatorname{dist}_{N}(x,\Sigma)\leq\operatorname{dist}_{N}(x,S)+\sup_{y\in
S}\operatorname{dist}_{N}(y,\Sigma)<3\pi.$

This completes the proof of Lemma 3.5. ∎

###  3.2. Proof of decomposition of geometrically prime regions into
admissible regions

Now we can use the three lemmas above to prove Proposition 3.2.

###### Proof of Proposition 3.2.

We first prove the lemma for $N$ so that each connected component of $\partial
N$ is a stable minimal surface. The proof can be divided into two steps.

Step I: There exists a sequence of embedded surfaces $\\{\Sigma_{j}\\}_{j\geq
1}$ decomposing $N$ into $\\{K_{j}\\}_{j\geq 0}$ so that for $j=1,\cdots$,

  * •

$\partial K_{j-1}\setminus\partial N=\Sigma_{j-1}\cup\Sigma_{j}$
($\Sigma_{0}:=\emptyset$);

  * •

in $K_{j-1}$, $\Sigma_{j-1}$ is mean concave and $\Sigma_{j}$ is mean convex;

  * •

$\mathrm{diam}_{N}\Sigma_{j}<5\pi$ and
$\mathrm{Area}(\Sigma_{j})<\frac{8\pi}{3}$;

  * •

$\operatorname{dist}_{N}(\Sigma_{j},\Sigma_{j+1})\geq\frac{\pi}{3}$ and
$\sup\\{\operatorname{dist}_{N}(x,\Sigma_{j});x\in\Sigma_{j+1}\\}<3\pi$ for
$j\geq 1$.

###### Proof of Step I.

Lemma 3.5 applies to $\Sigma=\emptyset,\Omega=N$ to obtain a two-sided
embedded closed mean convex surface $\Sigma_{1}$ so that

  * •

$\Sigma_{1}$ is mean convex for a choice of normal vector fields;

  * •

$\mathrm{diam}_{N}\Sigma_{1}<5\pi$ and
$\mathrm{Area}(\Sigma_{1})<\frac{8\pi}{3}$.

Then $\Sigma_{1}$ separates $N$ into two regions by Remark 2.7; denote by
$K_{0}$ the one so that $\Sigma_{1}$ is mean convex. Then we construct
$\Sigma_{j}$ inductively.

Suppose that we have $\Sigma_{j}$ and $K_{j-1}$. Note that $\Sigma_{i}$ is the
common boundary of $K_{i-1}$ and $K_{i}$, $i=1,\cdots,j-1$. It follows that

$\partial(N\setminus\cup_{i=0}^{j-1}K_{i})\setminus\partial N=\Sigma_{j}.$

Then we apply Lemma 3.5 again for $\Omega=N\setminus\cup_{i=0}^{j-1}K_{i}$ and
$\Sigma=\Sigma_{j}$ to get a two-sided embedded closed mean convex surface
$\Sigma_{j+1}$ so that

  * •

$\Sigma_{j+1}$ has mean curvature vector pointing towards $\Sigma_{j}$;

  * •

$\mathrm{diam}_{N}\Sigma_{j+1}<5\pi$ and
$\mathrm{Area}(\Sigma_{j+1})<\frac{8\pi}{3}$;

  * •

$\operatorname{dist}_{N}(\Sigma_{j+1},\Sigma_{j})\geq\frac{\pi}{3}$,
$\sup\\{\operatorname{dist}_{N}(x,\Sigma_{j});x\in\Sigma_{j+1}\\}<3\pi$.

Hence Step I is finished by letting $K_{j}$ be the region bounded by
$\Sigma_{j}$ and $\Sigma_{j+1}$. ∎

Then by Lemma 3.4 (with $d_{0}<3\pi$ therein), there exists a compact region
$\hat{K}_{j}\subset K_{j}$ so that

  * •

$\partial\hat{K}_{j}\setminus\partial N$ is a closed embedded surface
containing $\Sigma_{j}$ and $\Sigma_{j+1}$;

  * •

$\partial\hat{K}_{j}\setminus\partial K_{j}$ is mean concave;

  * •

$\sup\\{\operatorname{dist}_{N}(x,\Sigma_{j+1});x\in\hat{K}_{j}\\}<3\pi+3\pi$;

  * •

$\mathrm{Area}(\partial\hat{K}_{j}\setminus(\Sigma_{j}\cup\Sigma_{j+1}))<\mathrm{Area}(\Sigma_{j+1})$.

Here the third item together with the diameter bound of $\Sigma_{j+1}$ implies
that $\mathrm{diam}_{N}(\hat{K}_{j})<17\pi$; the last item gives that

$\mathrm{Area}(\partial\hat{K}_{j}\setminus\Sigma_{j+1})<\mathrm{Area}(\Sigma_{j+1})+\mathrm{Area}(\Sigma_{j})<\frac{16\pi}{3}.$

Recall that by Remark 2.7, each two-sided embedded surface separates $N$. Then
$K_{j}\setminus\hat{K}_{j}$ consists of finitely many connected components.
Moreover, for each connected component $K^{\prime}$, $\partial
K^{\prime}\setminus\partial N$ is a connected mean convex surface
$\Gamma^{\prime}$ satisfying

$\mathrm{diam}_{N}\Gamma^{\prime}<6\pi,\quad\mathrm{Area}(\Gamma^{\prime})<\mathrm{Area}(\Sigma_{j+1})\leq\frac{8\pi}{3}.$

Let $\Omega$ be one of the connected components of $K_{j}\setminus\hat{K}_{j}$
or $K_{0}$. Denote by $B=\partial\Omega\setminus\partial N$. Then $B$ is a
connected mean convex surface with

$\mathrm{diam}_{N}B<6\pi,\quad\mathrm{Area}(B)<\frac{8\pi}{3}.$

It suffices to divide $\Omega$ into compact regions as required.

Step II: For each $\Omega$ as above, it can be divided into countably many
admissible compact regions $\\{R_{i}\\}$.

###### Proof of Step II.

We prove it by induction. By Lemma 3.3, there exists a compact connected
region $R_{1}$ containing $B$ so that

  * •

$\partial R_{1}\setminus\partial\Omega$ is a closed embedded surface in $N$
and has mean curvature vector pointing away from $\Sigma$;

  * •

$\operatorname{dist}_{N}(\partial
R_{1}\setminus\partial\Omega,B)\geq\frac{\pi}{4}$, and
$\sup\\{\operatorname{dist}_{N}(x,B);x\in R_{1}\setminus\partial\Omega\\}\leq
3\pi$;

  * •

each connected component of $\partial R_{1}\setminus\partial\Omega$ has
diameter less than $6\pi$;

  * •

$\mathrm{Area}(\partial R_{1}\setminus B)<\mathrm{Area}(B)<\frac{8\pi}{3}$.

Observe that for each connected component $\Omega^{\prime}$ of
$\Omega\setminus R_{1}$,
$B^{\prime}:=\partial\Omega^{\prime}\setminus\partial\Omega$ is a connected
mean convex surface and

$\mathrm{diam}_{N}B^{\prime}<6\pi,\quad\mathrm{Area}(B^{\prime})<\frac{8\pi}{3}.$

Thus we can repeat the above argument. This finishes Step II. ∎

We now consider the case when one connected component $\Sigma$ of $\partial N$
has index one. Then by Lemma 2.2,

$\mathrm{diam}_{N}\Sigma\leq\frac{4}{3}\pi,\quad\mathrm{Area}(\Sigma)\leq\frac{16}{3}\pi.$

Since $\Sigma$ is unstable, then there exists a neighborhood
$\mathcal{N}_{\delta}$ of $\Sigma$ and a diffeomorphism
$\phi:\Sigma\times[0,\delta)\rightarrow\mathcal{N}_{\delta}$ so that
$\phi(\Sigma\times\\{t\\})$ has mean curvature vector pointing away from
$\Sigma$. It follows that

$\mathrm{Area}(\phi(\Sigma\times\\{\delta\\}))<\frac{16}{3}\pi,\quad\mathrm{diam}_{N}<2\pi.$

Thus $\mathcal{N}_{\delta}$ is an admissible region. Let $K$ be the metric
completion of $N\setminus\mathcal{N}_{\delta}$. Then by Lemma 3.3, there
exists an admissible region $\hat{K}$ containing
$\phi(\Sigma\times\\{\delta\\})$ so that the metric completion of
$K\setminus\hat{K}$ consists of finitely many regions $\\{\hat{\Omega}_{j}\\}$
satisfying

  * •

$\partial\hat{\Omega}_{j}\setminus\partial N$ is a closed embedded surface
having mean curvature vector pointing away from
$\phi(\Sigma\times\\{\delta\\})$;

  * •

$\mathrm{diam}_{N}(\partial\hat{\Omega}_{j}\setminus\partial N)<6\pi$ and
$\mathrm{Area}(\partial\hat{\Omega}_{j}\setminus\partial
N)<\frac{7\pi}{3}<\frac{8\pi}{3}$.

Then applying the argument in Step II, each $\\{\hat{\Omega}_{j}\\}$ can be
divided into countably many admissible regions. Therefore, we conclude that
$N$ can be divided into admissible regions. ∎

##  4\. Foliation of admissible and prime regions and proof of Theorem 1.1

By modifying the mean curvature flow with surgery it was proved in [LM20] that
compact geometrically prime regions admit a singular foliation with bounded
area. Similarly we will obtain foliations of admissible regions by singular
surfaces with area bounds.

######  Definition 4.1.

Let $A\subset\mathbb{R}$ be a closed interval or a ray $[0,\infty)$. Given a
set $U\subset M$, we say that a family of closed surfaces
$\\{\Sigma_{t}\\}_{t\in A}$ is a Morse foliation of $U$ if there exists a
proper Morse function $f:U\rightarrow A$ with $f^{-1}(t)=\Sigma_{t}$, $t\in
A$.

Below we recall some basic results about Morse foliations of subsets of a
Riemannian manifold.

######  Lemma 4.2.

Let $M^{3}$ be a Riemannian manifold and $\Sigma\subset M$ be a smooth closed
surface in $M$. For every $\varepsilon>0$ there exists
$\delta(\Sigma,\varepsilon)>0$, such that $U=N_{\delta}(\Sigma)$ admits a
Morse foliation $\\{\Sigma_{t}\\}_{t\in[0,1]}$ with $\Sigma_{0}=\partial U$,
$\Sigma_{1}=\\{p\\}$ for a point $p$ in the interior of $U$. Moreover, we have
$\mathrm{Area}(\Sigma_{t})\leq 2\mathrm{Area}(\Sigma)+\varepsilon$.

###### Proof.

Observe that for an arbitrarily small $r>0$ we can choose $\delta>0$
sufficiently small, so that $U$ is an interval bundle over $\Sigma$,
$\pi:U\rightarrow\Sigma$, and for every $p\in\Sigma$ we have that
$\pi^{-1}(B_{r}(p))$ is $(1+\frac{\sqrt{\varepsilon}}{1000})$-bilipschitz
diffeomorphic to $B_{r}^{\mathbb{R}^{2}}(0)\times[-\delta,\delta]$. In the
construction below we will use local coordinates $(x,s)$, $x\in\Sigma$,
$t\in[-\delta,\delta]$, on $U$.

Fix a Morse function $g:\Sigma\rightarrow[0,1]$ and consider a family of
surfaces with boundary $\\{\Sigma^{\prime}_{t}\\}_{t\in[0,1]}$ defined by
$\Sigma^{\prime}_{t}=(\pi\circ g)^{-1}(t)$. Observe that for sufficiently
small $\delta>0$ we have that
$\mathrm{Area}(\Sigma^{\prime}_{t})<\frac{\varepsilon}{4}$. Also, we can
slightly modify the family near critical points of $g$ to turn it into a
family of level sets of a Morse function on $U$. Indeed, if $x$ is a critical
point of $g$ of index $0$ or $2$ with critical value $g(x)=t^{\prime}$, then
in the neighbourhood of $\pi^{-1}(x)$ surfaces $\Sigma^{\prime}_{t}$ look like
a family of cylinders $\\{C_{t}\times[-\delta,\delta]\\}$. It is
straightforward to modify this family to a family of concentric spheres around
point $(x,0)$ as $t\rightarrow t^{\prime}$. Similarly, if $x$ is a critical
point of $g$ of index $1$, then surfaces $\Sigma^{\prime}_{t}$ near $(x,0)$
look like a family $\\{H_{t}\times[-\delta,\delta]\\}$ where $H_{t}$’s are
hyperbolas. Creating a small neck that connects two connected components of
$H_{t}\times[-\delta,\delta]$ and opening up the neck we modify this family of
surfaces in the neighbourhood of $\pi^{-1}(x)$ to a family of level sets of
Morse function with a cricial point of Morse index $1$.

After these modifications we obtain a Morse function $f:U\rightarrow[0,1]$
with critical points in the interior of $U$ and the area of fibers bounded by
$\frac{\varepsilon}{2}$. We can now apply [ChLi20]*Lemma 4.1 to obtain the
desired Morse foliation of $U$. ∎

######  Lemma 4.3.

Let $\Omega$ be an admissible region. Suppose $S_{0}$ is a closed (possibly
empty) subset of $\partial\Omega$ and $S_{1}=\partial\Omega\setminus S_{0}$.
Then for every $\varepsilon>0$ there exists a Morse foliation
$\\{\Sigma_{t}\\}$ of $\Omega$, such that

  * •

if $S_{0}$ is empty, then $\Sigma_{0}=\\{p\\}$ is a point in the interior of
$\Omega$ and $\Sigma_{1}=S_{1}$;

  * •

if $S_{0}$ is non-empty, then $\Sigma_{0}=S_{0}$ and $\Sigma_{1}=S_{1}$;

and $\mathrm{Area}(\Sigma_{t})\leq
3\mathrm{Area}(\partial\Omega)+\varepsilon<33\pi$.

###### Proof.

Let $U$ denote a small tubular neighbourhood of $\partial\Omega$ in $\Omega$
and $\Omega^{\prime}=\Omega\setminus U$. Note that each connected component
$U_{i}$ of $U$ can be foliated by a family of surfaces $\\{\Gamma^{i}_{t}\\}$,
where $\Gamma^{i}_{0}=\Gamma^{i}$ is a connected component of $\partial\Omega$
and $\mathrm{Area}(\Gamma^{i}_{t})\leq\mathrm{Area}(\Gamma^{i})+\delta$ for
$\delta>0$ that can be chosen arbitrarily small by making neighbourhood $U$
small enough.

Consider $\Omega$ as a subset of $M$ and define Mean Curvatue Flow with
surgery starting on the mean convex boundary component $S$ of
$\partial\Omega$. As in [LM20] we can modify this family of surfaces to obtain
a Morse function $f^{\prime}:\Omega\rightarrow\mathbb{R}$ with the area of
fibers at most $\mathrm{Area}(S)+\delta$. (Note that some of the fibers may
not be closed surfaces and may have boundary in $\partial\Omega\setminus S$.)
We can also assume that $f^{\prime}$ is a Morse function when restricted to
$\Omega^{\prime}$ and $\partial\Omega^{\prime}$. By [ChLi20]*Lemma 4.1 there
exists a Morse foliation $\\{\Sigma^{\prime}_{t}\\}$ of $\Omega^{\prime}$
starting at a point $\Sigma^{\prime}_{0}=\\{p\\}$ in the interior of
$\Omega^{\prime}$, ending on $\Sigma^{\prime}_{1}=\partial\Omega^{\prime}$ and
satisfying
$\mathrm{Area}(\Sigma^{\prime}_{t})\leq\mathrm{Area}(S)+\mathrm{Area}(\partial\Omega^{\prime})+2\delta$.

If $S_{0}$ is empty then we can extend $\\{\Sigma^{\prime}_{t}\\}$ to $U$ to
obtain the desired Morse foliation of $\Omega$.

Suppose $S_{0}$ is not empty. Let $\tilde{U}$ denote the union of connected
components of $U$ that intersect $S_{0}$ and
$\\{\tilde{\Gamma}_{t}^{0}\\}_{t\in[0,1]}$ denote the foliation of $\tilde{U}$
by surfaces isotopic to $S_{0}$, as defined above. Let
$\\{\tilde{\Gamma}_{t}^{1}\\}_{t\in[0,1]}$ denote the corresponding foliation
of $U\setminus\tilde{U}$. Let $V$ denote a subset of $\tilde{U}$ bounded by
$\tilde{\Gamma}_{\frac{1}{2}}^{0}$ and $\tilde{\Gamma}_{1}^{0}$. By Lemma 4.2
we can define a Morse foliation $\\{\tilde{\Sigma}_{t}\\}_{t\in[0,1]}$ of $V$
that starts on $\tilde{\Gamma}_{\frac{1}{2}}^{0}\cup\tilde{\Gamma}_{1}^{0}$
and terminates at a point in the interior of $V$.

We can now define the desired Morse foliation of $\Omega$ by setting

$\Sigma_{t}=\left\\{\begin{aligned} &\tilde{\Gamma}_{t}^{0}&\text{ for
$t\in[0,\frac{1}{4})$},\\\
&\Sigma^{\prime}_{4t-1}\cup\tilde{\Gamma}_{t}^{0}&\text{ for
$t\in[\frac{1}{4},\frac{1}{2})$},\\\
&\tilde{\Sigma}_{4t-2}\cup\tilde{\Gamma}_{2t-1}^{1}&\text{ for
$t\in[\frac{1}{2},\frac{3}{4})$},\\\ &\tilde{\Gamma}_{2t-1}^{1}&\text{ for
$t\in[\frac{3}{4},1]$}.\end{aligned}\right.$

It is straightforward to check that with the definition above we have
$\Sigma_{0}=S_{0}$, $\Sigma_{1}=S_{1}$ and the area bound is satisfied. ∎

###### Proof of Theorem 1.1.

Let $M$ be a complete $3$-manifold with scalar curvature $R\geq 6$. We perturb
the metric on $M$ to a bumpy metric. We will construct the desired Morse
foliation for the perturbed metric and note that the diameter and area bounds
for surfaces in the folation will be worse by at most $\varepsilon$ for the
original metric, where $\varepsilon>0$ depends on the perturbation and can be
chosen arbitrarily small.

By Theorem 2.9 and Proposition 3.2 there exists a decomposition of $M$ into
countably many admissible regions $\mathcal{U}=\\{\Omega_{i}\\}$. Define graph
$G$, where each vertex $V_{\Omega_{i}}$ corresponds to an admissible region
$\Omega_{i}$ and two vertices $V_{\Omega_{i}}$ and $V_{\Omega_{j}}$, $i\neq
j$, are connected by an edge if and only if the inclusions of $\Omega_{i}$ and
$\Omega_{j}$ share a common boundary component in $M$.

Choose an admissible region $\Omega_{0}\in\mathcal{U}$. Let
$i_{\Omega_{0}}:\Omega_{0}\rightarrow M$ denote the inclusion of $\Omega_{0}$
into $M$ that is a Riemannian isometry in the interior of $\Omega_{0}$. Let
$S_{1}(\Omega_{0})$ denote the image of the boundary components of
$\Omega_{0}$ on which $i_{\Omega_{0}}$ is a 2-to-1 map (for example, if
$i_{\Omega_{0}}$ sends the interior of $\Omega_{0}$ onto the interior of
$\mathbb{RP}^{3}$ with some balls and nontrivial $\mathbb{RP}^{2}$ removed).
By Lemma 4.3, for a sufficiently small $\delta>0$, there exists a Morse
function $f:\Omega_{0}\setminus
N_{\delta}(\partial\Omega_{0})\rightarrow[0,1-\delta]$, such that $f^{-1}(0)$
is a point in the interior of $\Omega_{0}$ and
$f^{-1}(1)=\partial(\Omega_{0}\setminus N_{\delta}(\partial\Omega_{0}))$ and
the fibers of $f$ satisfy the desired area and diameter bounds. We can extend
$f$ to a Morse foliation of $i_{\Omega_{0}}(\Omega_{0})$,
$f:i_{\Omega_{0}}(\Omega_{0})\rightarrow[0,1]$, by applying Lemma 4.2 in the
$\delta$-neighbourhood of $S_{1}(\Omega_{0})$ and foliating the rest of
$i_{\Omega_{0}}(N_{\delta}(\partial\Omega_{0}))$ by level sets of the distance
function.

Let $V_{n}$ denote the collection of all vertices in graph $G$ that are at a
graph distance $n$ from vertex $V_{\Omega_{0}}$ and let $U_{n}$ denote closure
of the union of the corresponding regions in $M$. Suppose, by induction, that
we have defined a Morse function $f:\bigcup_{k=0}^{n-1}U_{k}\rightarrow[0,n]$
with fibers of the function satisfying the desired area and diameter bounds
and, moreover, we have that $f(x)=n$ for
$x\in\partial\bigcup_{k=0}^{n-1}U_{k}$.

For each admissible region $\Omega^{\prime}$ corresponding to a vertex in
$V_{n}$ we decompose $\partial\Omega^{\prime}$ into four closed subsets. Let
$S_{0}(\Omega^{\prime})=\partial
i_{\Omega^{\prime}}({\Omega^{\prime}})\cap\partial U_{n-1}$,
$S_{1}(\Omega^{\prime})$ denote the connected components of
$i_{\Omega^{\prime}}(\partial\Omega^{\prime})$ on which $i_{\Omega^{\prime}}$
is a double covering, $S_{2}(\Omega^{\prime})=\partial
i_{\Omega^{\prime}}({\Omega^{\prime}})\cap\partial(U_{n}\setminus
i_{\Omega^{\prime}}({\Omega^{\prime}}))$, and $S_{3}(\Omega^{\prime})=\partial
i_{\Omega^{\prime}}({\Omega^{\prime}})\setminus(S_{0}(\Omega^{\prime})\cup
S_{1}(\Omega^{\prime})\cup S_{2}(\Omega^{\prime}))$. Note that $S_{1}$,
$S_{2}$ and $S_{3}$ may be empty.

For a sufficiently small $\delta>0$ we can apply Lemma 4.3 to define $f$ on
$\bigcup i_{\Omega}(\Omega)\setminus N_{\delta}(S_{1}(\Omega)\cup
S_{2}(\Omega)\cup S_{3}(\Omega))$, where the union is over all sets
$\Omega\in\mathcal{U}$ correspond to vertices in $V_{n}$, mapping it onto
$[n,n+1-\delta]$, so that it agrees with the inductive definition of $f$ on
$f^{-1}(n)$ and $f^{-1}(n+1-\delta)=\partial\bigcup
N_{\delta}(S_{1}(\Omega)\cup S_{2}(\Omega)\cup S_{3}(\Omega))$. We can then
apply Lemma 4.2 to extend $f$ on $\bigcup N_{\delta}(S_{1}(\Omega)\cup
S_{2}(\Omega))$, mapping it to $[n+1-\delta,n+1]$. Finally, the set $\bigcup
N_{\delta}(S_{3}(\Omega))\cap\Omega$ can be foliated by surfaces isotopic to
$\bigcup S_{3}(\Omega)$, giving us a Morse foliation with $f=n+1$ on
$\partial\bigcup_{k=0}^{n}U_{k}$. This finishes the inductive construction of
$f$.

It follows from the definition of admissible regions and the area and diameter
estimates above that for each connected component $\Sigma$ of $f^{-1}(x)$ we
have $\mathrm{Area}(\Sigma)<33\pi$. Since each surface is contained inside an
admissible region we have that the extrinsic diameter satisfies
$\mathrm{diam}_{M}\Sigma<18\pi$. ∎

## References


# Lumped-Parameter Modeling and Control for Robotic High-Viscosity Fluid
Dispensing in Additive Manufacturing

William van den Bogert1, James Lorenz1, Xili Yi2, Nima Fazeli2, and Albert J.
Shih1 1 William van den Bogert, James Lorenz, and Albert J. Shih are with the
Mechanical Engineering Department at the University of Michigan, MI, USA
<EMAIL_ADDRESS>Xili Yi and Nima Fazeli are with the
Robotics Department at the University of Michigan, MI, USA
<EMAIL_ADDRESS>
###### Abstract

In this paper, we present a novel flow model and compensation strategy for
high-viscosity fluid deposition that yields high quality parts in the face of
large transient delays and nonlinearity. Robotic high-viscosity fluid
deposition is an essential process for a broad range of manufacturing
applications including additive manufacturing, adhesive and sealant
dispensing, and soft robotics. However, high-viscosity fluid deposition
without compensation can lead to poor part quality and defects due to large
transient delays and complex fluid dynamics. Our computationally efficient
model is well-suited to real-time control and can be quickly calibrated and
our compensation strategy leverages an iterative Linear-Quadratic Regulator to
compute compensated deposition paths that can be deployed on most dispensing
systems, without additional hardware. We demonstrate the improvements provided
by our method when 3D printing using a robotic manipulator.

##  I Introduction

The capability for precise robotic deposition of high-viscosity fluid is a key
enabler for industries such as automotive manufacturing, tissue engineering,
and soft robotics. When applied to 3D printing, this technology, known as
direct ink writing (DIW) [1], enables the production of high-quality flexible
parts with complex geometries. The high viscosity of the fluid allows a
printed shape to remain after deposition. However, this same viscosity leads
to printing defects that require mitigation through compensation. The main
contributions of this paper are a model that is quickly and easily
generalizable for most pump-based high-viscosity fluid deposition systems, a
calibration method for this model, and a demonstration of this model in a
universally implementable open-loop control scheme without the need for
additional hardware such as sensors.

Our contributions enable DIW systems to approach the precision of more common
3D printing technologies. The traditional Fused Deposition Modeling 3D
printing process typically controls flow behavior in a naive open-loop manner,
where a motor position command controls the deployment of thermoplastic
material. However, when the naive control is applied to DIW, transient effects
due to viscosity-driven impedance produce imprecision in deposition that
causes part defects such as the stringing found in Fig. 1.

Figure 1: Robotic 3D printing with model-based flow compensation. A silicone
3D printer head held by a robot arm printing two separated objects. Panel (a)
shows the resultant print with untuned naive planning where we observe
significant stringing and part defects due to over-extruded silicone. In
comparison, Panel (b) shows the result of our approach with model-based flow
compensation where the defect-free objects are separated cleanly. Panel (c)
shows the robot 3D printing silicone.

Our work fills a gap in high-viscosity fluid deposition research with regards
to describing the transient flow dynamics of high-viscosity fluids within the
positive displacement pump (PDP) systems common to DIW. To model fluid flow
behaviors of various PDP systems, research groups have developed control
schemes using analytical solutions of physical laws [2] [3], physics-based
finite element simulations [4] [5] [6], and data-driven modeling approaches
[5] [7]. However, these approaches are either too costly in numerical
computation time for control applications, or do not consider transient
effects in PDPs. To bridge this research gap, we propose a model of the fluid
system dynamics that can predict transient effects of high-viscosity fluid
flows in PDP systems, paired with an open-loop control paradigm to compensate
for deposition errors during 3D printing. We demonstrate a significant
improvement in fluid deposition precision when we apply this compensation to
the trajectories for our robotic 3D printer, as shown in Fig. 1.

Figure 2: Positive displacement pump (PDP) system. Two progressive cavity
pumps extrude A and B parts of the two-part liquid silicone. A and B parts are
mixed via static mixing elements, and mixture is deposited through the nozzle.
Here, $\bm{g}$ represents the direction of gravity.

###  I-A Problem Statement

The PDP system we use for fluid deposition is shown in Fig. 2. We assume the
fluids in reservoirs A and B share the same physical parameters, and that both
motors receive the same input. Here, the control input is the motor speed
signal $u(t)$. Motor speed is proportional to the intended rate of
displacement of the progressing cavities in the PDPs. This input produces the
nozzle outlet flow rate $q(t)$, and both $u(t)$ and $q(t)$ are in units of
volumetric flow rate. While naive flow rate control assumes otherwise, we have
observed that $u(t)\neq q(t)$ $\forall t$ (see Section IV for examples). For a
given time horizon $n$, our goal is to solve for the trajectory $u(t)$ such
that a prescribed $q(t)$ is achieved. We work with the discretized
trajectories $u_{k}$ and $q_{k}$ over time steps $k=\\{1,...,n\\}$. Let
$\mathcal{M}$ denote the model mapping the trajectories $\bm{u}_{n\times 1}$
onto $\bm{q}_{n\times 1}$:

$\bm{q}_{model}=\mathcal{M}(\bm{u})$ (1)

This model $\mathcal{M}$ is an approximation of the true map $\mathcal{R}$
representing the complex nonlinear fluid dynamics. Our first contribution is
developing a parameteric model $\mathcal{M}$ that is i) amenable to real-time
control and ii) sufficiently closely approximates $\mathcal{R}$:

$\mathcal{M}^{*}=\arg\min_{\mathcal{M}}|\mathcal{R}(\bm{u})-\mathcal{M}(\bm{u})|$
(2)

For open-loop control, we do not have access to $\mathcal{R}$, and instead use
the model map $\mathcal{M}$ to find the optimal input trajectory $\bm{u}^{*}$
which produces the prescribed output trajectory $\bm{q}_{ref}$:

$\bm{u}^{*}=\arg\min_{\bm{u}}||\mathcal{M}(\bm{u})-\bm{q}_{ref}||$

subject to the constraints of the PDP pump and robot end-effector motion. The
second contribution of this work is a strategy that leverages our proposed
model to compute this optimal open-loop control.

##  II Related Works

Baranovskii and Artemov have proposed generalized solutions for optimal
control of nonlinear-viscous fluid models based on Navier-Stokes [2], though
their specific analytic solutions demand simplifications specific to the fluid
system. Hapanowicz’s flow-resistance model [3] is only applicable to quasi-
steady cylindrical flows and does not account for transient pump dynamics.
Froehlich and Kemmetmüller provide a computationally efficient control
strategy [8], but it is designed only for valved systems found in injection
molding machines.

To control the fluid flow of PDP systems, researchers instead developed
simplified models with closed-loop controllers and numerical solving methods
in place of analytical physics-based solutions of complex fluid models.
Josifovic et al use computational fluid dynamics to model the quasi-step input
of a valved pump system [4], and Wang et al combine pressure sensing equipment
and PID optimization to correct errant flow behavior [9]. Numerical solution
methods have also been proposed for viscous flow control of screw-pump [10],
helical gear pump [11], and peristaltic pump systems [12]. Though seldom used
in fluid dynamics controllers, a data-based lumped parameter model was used by
Fresia and Rundo [5] and Zardin et al [7] to compensate pulsing fluid dynamics
generated by variable displacement pumps and vane pumps, respectively.
However, because each group derived their analytical flow models based on the
geometry of their specific PDP system, their utility is limited because none
are generalizable to the broad spectrum of PDP systems.

Flow models for progressive cavity pumps are relatively unexplored in academic
research, due to the significant improvement in accuracy that this system
offers over alternative types of PDPs. Fisch et al [13] and Franchin et al
[14] note a strong improvement in precision fluid deposition in 3D printing
applications enabled by progressive cavity pump-controlled extrusion DIW
systems. As a result, research on models for fluid interaction with
progressive cavity PDPs has been limited to basic feedback control [15] and
numerical computational fluid dynamics methods to model specific behaviors of
the progressive cavity pump system, including stator wear over time [6],
inertial flow effects and viscous losses between the stator and rotor [16],
and backflow of progressive cavity pumps [17]. While these models predict
unsteady flow in local regions of the progressive cavity systems (e.g. flow
between the stators and rotors), none of these models provide a comprehensive
control model of the entire fluid dispensing system required for our robotic
3D printing device, and all prior models require either additional hardware to
close the control loop or computationally expensive solution methods which
prohibit real-time compensation.

##  III Methods

Our approach consists of two main components: In Section III-A we propose a
novel computationally efficient high-viscosity fluid model that is amenable to
real-time control and error compensation, and in Section III-B we propose an
optimal control formulation and solution method that leverages the model to
rapidly compute open-loop control signals.

###  III-A Modeling of High-Viscosity Fluid Deposition

Our goal is to design $\mathcal{M}$ such that it i) is amenable to real-time
control and ii) sufficiently closely approximates $\mathcal{R}$ in Eq. (2),
which represents the complex nonlinear fluid dynamics and interaction with
both pump and mixer. We model the entire PDP system as a 3 degree of freedom
coupled linear dynamic system. Our model is a representation of fluid flow
impedance in the form of elasticity $k_{j}$, damping $c_{j}$, and inertia
$m_{j}$ within both the pump ($j=1$) and mixer ($j=2$), as well as inertia
$m_{f}$ of the fluid that sits between them. The model’s mass-spring-damper
system structure can be seen in Fig. 3.

Figure 3: The high-viscosity fluid, pump and mixer are simplified to a 3
degree of freedom coupled linear dynamical system. Our model is a
representation of fluid flow impedance in the pump ($j=1$) and mixer ($j=2$)
in the form of elasticity $k_{j}$, damping $c_{j}$, and inertia $m_{j}$, as
well as inertia $m_{f}$ of the fluid.

When coupled with the initial state of the system, our map $\mathcal{M}$
introduced in Eq. (1) can take the form of a dynamics function $f$ that
predicts the next state vector $\bm{x}$, of which $q$ and its time derivative
are elements:

$\bm{x}_{k+1}=f(\bm{x}_{k},u_{k})$ (3)

The dynamics function defined in Eq. (3) is necessary for predictive modeling.
The state vector $\bm{x}$ for the dynamics is defined using the 2 internal
degrees of freedom $x_{1}$ and $x_{2}$ and the output degree of freedom $q$:

$\bm{x}=\left[{\begin{array}[]{cccccc}x_{1}&x_{2}&q&\dot{x_{1}}&\dot{x_{2}}&\dot{q}\end{array}}\right]^{T}$

The system can be described with state-space form, with input $u$ and output
$q$:

$\bm{x}_{k+1}=\mathbb{A}\bm{x}_{k}+\mathbb{B}u_{k}$
$q_{k}=\mathbb{C}\bm{x}_{k}$

where the matrices $\mathbb{A}$, $\mathbb{B}$ and $\mathbb{C}$ are as follows:

$\mathbb{A}_{s}=\left[{\begin{array}[]{cccccc}0&0&0&1&0&0\\\ 0&0&0&0&1&0\\\
0&0&0&0&0&1\\\
-\frac{k_{1}}{m_{1}}&0&0&-\frac{c_{1}}{m_{1}}&0&\frac{c_{1}}{m_{1}}\\\
0&-\frac{k_{2}}{m_{2}}&0&0&-\frac{c_{2}}{m_{2}}&\frac{c_{2}}{m_{2}}\\\
0&0&0&\frac{c_{1}}{m_{f}}&\frac{c_{2}}{m_{f}}&-\frac{c_{1}+c_{2}}{m_{f}}\\\
\end{array}}\right]$ $\mathbb{A}=\mathbb{I}_{6\times 6}+\mathbb{A}_{s}\Delta
t$
$\mathbb{B}=\left[{\begin{array}[]{cccccc}0&0&0&\frac{k_{1}}{m_{1}}&0&0\end{array}}\right]^{T}\Delta
t$
$\mathbb{C}=\left[{\begin{array}[]{cccccc}0&0&1&0&0&0\end{array}}\right]^{T}$

We identify values of the seven parameters shown in Fig. 3 by using a
gradient-descent scheme. Specifically, given measured data from the real
system, we optimize the parameters so that the model response best-fits the
data. The cost function $c$ for this optimization problem is calculated for a
given time horizon as a function of the parameters $\bm{\phi}$, where the data
is sampled with the same period as the simulation time-step $\Delta t$:

$c(\bm{\phi})=\sum_{k=1}^{n}\frac{1}{q_{k,meas}+q_{b}}(q_{k,model}(\bm{\phi})-q_{k,meas})^{2}$
(4)
$\bm{\phi}=\left[{\begin{array}[]{cccccccc}k_{1}&c_{1}&m_{1}&m_{f}&k_{2}&c_{2}&m_{2}\end{array}}\right]^{T}$

where $q_{k,meas}$ is the measured data for the time with index $k$,
$q_{k,model}$ is the model flow rate for the time with index $k$, and $q_{b}$
is a biasing term. For each iteration of optimization $i$ the model is solved
and the gradient of the cost computed. Then, the parameters are updated
according to that gradient:

$\bm{\phi}_{i+1}=\bm{\phi}_{i}-\frac{h}{c(\bm{\phi}_{i})}\frac{\partial{c(\bm{\phi_{i}})}}{\partial\bm{\phi}_{i}}$

where $h$ is an adjustable increment. We set this optimization to stop once
the cost changes by less than 0.1% from one iteration to the next.

###  III-B Control of High-Viscosity Fluid Deposition

With this model in hand, we desire the input profile $u$ for a given time
horizon such that the nozzle flow rate tracks the desired $q$. We construct a
quadratic convex cost function $b$ for this optimization:

$b_{k}=(\bm{x}_{k}-\bm{x}_{k}^{ref})^{T}\mathbb{Q}(\bm{x}_{k}-\bm{x}_{k}^{ref})+\bm{u}_{k}^{T}\mathbb{R}_{k}\bm{u}_{k}$
(5)

where $\bm{x}_{k}$ is the current state, $u_{k}$ is the current input,
$\bm{x}_{k}^{ref}$ is the desired state at the time with index $k$.

In our specific case, where our input only has a single degree of freedom,
$\mathbb{R}_{k}$ and $\bm{u}_{k}$ reduce to scalars, so Eq. (5) can be
rewritten:

$b_{k}=(\bm{x}_{k}-\bm{x}_{k}^{ref})^{T}\mathbb{Q}(\bm{x}_{k}-\bm{x}_{k}^{ref})+R_{k}u_{k}^{2}$

where $\mathbb{Q}$ is a diagonal matrix containing the weightings for tracking
error of the six states. Since we only are trying to track the output $q$, the
third state, all other members of the diagonal can be set to zero:

$\mathbb{Q}_{6\times 6}=diag(0,0,\xi,0,0,0)$

With the linear dynamics of our model, the cost function defined in Eq. 5
could ordinarily be minimized analytically within the bounds of linear
quadratic regulator (LQR) theory. However, we have observed that the reference
trajectory is tracked more effectively in reality if the weighting on
actuation effort $R_{k}$ is significantly reduced ($r_{1}<r_{2}$) once the
input $u_{k}$ falls below a certain negative value $u_{th}$, which occurs when
the pumps are run backwards in an effort to quickly reduce the output flow
rate to zero when commanded as such:

$R_{k}=\left\\{\begin{array}[]{ll}r_{1}&\quad u_{k}<u_{th}\\\ r_{2}&\quad
u_{k}\geq u_{th}\end{array}\right.$

Since the impact of actuation effort on cost changes depending on the
magnitude of the input, we must use an iterative LQR (iLQR) solver [18] for
this optimization problem. Each iteration $i$ occurs in two parts. The first
is a backward pass in which the controller gains $\mathbb{K}_{k}$ are computed
using the value function $\mathbb{V}_{k}$ at the next time step, which is then
updated for the current time step:

$\mathbb{K}_{k}=(R_{k}+\mathbb{B}^{*T}\mathbb{V}_{k+1}\mathbb{B}^{*})^{-1}\mathbb{B}^{*T}\mathbb{V}_{k+1}\mathbb{A}^{*}$
$\mathbb{P}_{k}=\mathbb{A}^{*}+\mathbb{B}^{*}\mathbb{K}_{k}$
$\mathbb{V}_{k}=\mathbb{Q}^{*}_{k}+\mathbb{K}_{k}^{T}R_{k}\mathbb{K}_{k}+\mathbb{P}_{k}^{T}\mathbb{V}_{k+1}\mathbb{P}_{k}$

where $\mathbb{A}^{*}$, $\mathbb{B}^{*}$, and $\mathbb{Q}^{*}_{k}$ are the
homogenized versions of $\mathbb{A}$, $\mathbb{B}$, and $\mathbb{Q}$
respectively:

$\mathbb{A}^{*}=\left[{\begin{array}[]{cc}\mathbb{A}&\mathbb{0}_{6\times 1}\\\
\mathbb{0}_{1\times 6}&1\\\
\end{array}}\right],\quad\mathbb{B}^{*}=\left[{\begin{array}[]{c}\mathbb{B}\\\
0\\\ \end{array}}\right]$

$\mathbb{Q}^{*}_{k}=\left[{\begin{array}[]{cc}\mathbb{Q}&\mathbb{Q}(\bm{x}_{k}-\bm{x}_{k}^{ref})\\\
(\bm{x}_{k}-\bm{x}_{k}^{ref})^{T}\mathbb{Q}&(\bm{x}_{k}-\bm{x}_{k}^{ref})^{T}(\bm{x}_{k}-\bm{x}_{k}^{ref})\\\
\end{array}}\right]$

The second part is a forward pass in which the input $u_{k}$ is updated
according to the controller gains, and the state $\bm{x}_{k}$ is updated
according to the dynamics with the input:

$u_{k}^{i+1}=u_{k}^{i}+\mathbb{K}_{k-1}(\bm{x}^{i+1}_{k-1}-\bm{x}^{i}_{k-1})$
$\bm{x}_{k+1}^{i+1}=\mathbb{A}\bm{x}_{k}^{i+1}+\mathbb{B}u_{k}^{i+1}$

We set the controller optimization to stop if the cost changes by less than
0.1% from one iteration to the next.

##  IV Results

###  IV-A Experimental Setup and Data Collection

Figure 4: Experiment setup for printing. The whole robotic printing system
consists of the robot arm, printer head, and adjustable build plate. The
printer head includes a custom printer frame, pressurized silicone reservoirs,
PDP, and a nozzle.

Our experimental setup can be seen in Fig. 4. We used a 7DOF Kuka LBR 14 R820
robot arm equipped with a customized print head alongside an adjustable build
plate fixed to the base of the world frame. The customized print head consists
of 3d-printed printer frame, pressurized reservoirs, PDPs, mixer, and nozzle.
The silicone material used for 3D printing is Dow Corning 121 Structural
Glazing Sealant. The stepper motors in the PDPs are controlled by a Raspberry
Pi 4 Model B using integration with ROS, and the silicone reservoirs are
connected to a high-pressure air source limited at 100 psi by a pressure
regulator.

Figure 5: Visualization of binary segmentation measurement method for flow
rate data. RGB images of both top and side views of printed beads are
converted to HSV color space. We then segment them with a saturation threshold
to get the flow rate measurement.

We collected all data for model training and verification, as well as control
verification, by segmenting side and top view images of a bead printed at
constant velocity $v$ while varying commanded flow rate $u$, the result of
which is shown in Fig. 5. With the scale in Fig. 5, we obtained a map between
pixels and physical distance. Assuming that the cross-section of the printed
bead is always rectangular, we calculated the cross-sectional area $A$ at each
time step $k$. The nozzle flow rate is then:

$q_{k,meas}=A_{k}v$

Figure 6: The results of training the lumped-parameter fluid deposition model
show a more accurate prediction for lower magnitude flow rates. This is by
design, using our selection of the $q_{b}$ term in Eq. (4). Due to the real
non-linear and specifically non-Newtonian nature of the silicone, a linear
model cannot accurately predict across the entirety of a wide range of flow
rate magnitudes.

###  IV-B Model Training

To implement the model training algorithm described in Methods, we used values
of $h=0.1$ for the step increment, $q_{b}=0.1$ mm${}^{3}/$s for the cost
function bias term, $\Delta{t}=0.01$ s for the simulation time step. The data
seen in Fig. 6 was sampled at this time step before the cost was computed.

We trained the model on an input command profile consisting of 18 square
pulses of different magnitudes, both negative and positive, as seen in Fig. 6.
This was to anticipate the necessary distribution for compensating step-up and
step-down flow rate profiles at the nozzle. The optimization returned the
parameters shown in Table I.

TABLE I: Parameters found as a result of model training $k_{1}$ | $c_{1}$ | $m_{1}$ | $m_{f}$ | $k_{2}$ | $c_{2}$ | $m_{2}$
---|---|---|---|---|---|---
0.827 | 18.784 | 1.195 | 1.930 | 9.930 | 5.458 | 0.954

With these parameters defined, the model exhibits the behavior shown in Fig.
6. The trained model can be used to predict the data used in training, with
the $q_{b}$ term chosen to ensure that lower magnitude flow rates are more
accurately predictable. Higher magnitudes are thus less predictable. Fig. 7
shows the ability of the trained model to predict unseen data, consisting of
the flow rate input and output during the targeted compensation of this paper.

###  IV-C Controller Optimization

To implement the iLQR solver described in the Methods section, we used values
of $\xi=100$ for the tracking weight, $r_{1}=40$, $r_{2}=400$, $u_{th}=-2$
mm${}^{3}/$s for the actuation effort weighting, and the time step was changed
to $\Delta{t}=0.0005$ s.

Using these values, the iLQR solved for the input command seen in Fig. 7 and
Fig. 8. Fig. 8 shows the desired output flow profile, a series of four square
pulses with a magnitude of 2.4 mm3/s. These correspond to the four dashes
shown in Fig. 8, that would ideally be deposited as the nozzle travels from
left to right. The naive open-loop control will involve these four square
pulses being sent as an untuned input to the pumps. When we do this, however,
the intention of depositing four discrete dashes is lost, due to the high
impedance of deposition. When we use the iLQR-tuned input, the four dashes
become much more distinct. The plot in Fig. 8 shows the flow rate vs. time
response of the system provided with the untuned, and then iLQR-tuned input.
The root-mean-square error between these respective responses and the desired
response can be seen in Table II, showing an improvement of over 50% when the
compensation is applied.

Figure 7: Prediction of unseen data by the lumped-parameter model. With the
parameters from trained model, the model is able to predict unseen data,
consisting of the flow rate input and output from the compensation validation.

Figure 8: Comparison of model-based LQR control of fluid dispensing. Top pictures visualize the ideal, untuned, and iLQR-tuned fluid output profile. The bottom plot compares the input and output for them. The iLQR-tuned fluid output is closer to ideal output. TABLE II: RMSE between desired output and measured output Output | RMSE
---|---
Untuned response | 1.057
iLQR-tuned response | 0.522

###  IV-D Robotic Additive Manufacturing

To validate the iLQR-tuned compensation during 3D printing, we devised
printing tasks that would showcase misplaced material as a result of the
parasitic effects when uncompensated. Trajectory for each printing task is
generated with Matlab R2020b, and each waypoint in the trajectory includes the
print head nozzle position in 3 cartesian coordinates and its corresponding
flow rate. The print head is considered to only translate but not rotate. The
robot and printing system communicate over a Robot Operating System (ROS)
network where the robot sequentially executes waypoint while adjusting the
flow rate with the Raspberry Pi connected to the network.

Figure 9: Robotic printing results. With untuned planning (a), over-extruded
silicone was deposited between and inside the hollow truncated cones. With
iLQR-tuned planning (b), the objects are separated cleanly.

We fabricated two printing samples with the same waypoints and target shape,
that of two hollow truncated cones, separated by 40 mm between the central
axes. One sample used the naive untuned flow rate, and the other used the
iLQR-tuned flow rate. The printing results are shown in Fig. 9.

We segmented the top and side view RGB images of the printing results,
producing binary masks of printed material, which can be seen in Fig. 10. We
also produced a target binary mask (also in Fig. 10) that represents the ideal
placement of material, derived from the geometry of the mesh that we converted
to waypoints for printing. To quantify the printing success, we calculated an
intersection-over-union (IoU) score using the formula:

$\text{IoU}(\mathbb{S})=\frac{\mathbb{S}\cap{\mathbb{T}}}{\mathbb{S}\cup{\mathbb{T}}}$

where $\mathbb{S}\in\\{0,1\\}_{y\times{z}}$ is the matrix representation of
the binary mask in question (top or side view, iLQR-tuned or untuned) and
$\mathbb{T}\in\\{0,1\\}_{y\times{z}}$ is the matrix representation of the
associated target binary mask (top or side view). The results shown in Fig. 10
demonstrate that iLQR-tuned printing has 0.030 and 0.163 higher IoU score than
untuned printing when examining the top and side views respectively,
demonstrating a more intentional placement of material.

Figure 10: The segmentation and resulting intersection-over-union measurement,
quantifying the success of the open-loop LQR control scheme over the naive
control scheme.

##  V Discussion

In this paper, we demonstrate how after quickly identifying just seven
parameters that define our lumped-parameter fluid model, and applying the
model to the iLQR flow control paradigm, we experimentally validated the
efficacy of our approach with deposition tests using our robotic 3D printer.
Misplaced material during the print was significantly reduced due to the
application of our model-based compensation. Fig. 9(a) shows that with the
untuned input, over-extruded silicone was deposited between the two hollow
truncated cones, as well as inside the hollow region of one of the cones. In
Fig. 9(b), the two objects are separated cleanly when the iLQR-tuned input is
applied. Beyond additive manufacturing, the results in Fig. 8 show that the
model-based compensation increases the precision of material placement along a
single trajectory. This improvement has the potential to shrink the margin of
error for robotic deposition of sealant and adhesive for manufacturing.

There are a few limitations with this work. Fig. 6 shows the model predicting
less accurately for higher flow rate magnitudes. We selected the $q_{b}$ term
so that the flow rates around the magnitude necessary for printing ($\sim$2–10
mm${}^{3}/$s) would be better predicted by the model. The prediction of unseen
data within this range can be seen in Fig. 7, and it is satisfactory for the
purposes of this paper. Due to our use of a linear model, the non-Newtonian
behavior of the silicone cannot be predicted. Since iLQR can be easily used
with nonlinear systems [18], one approach for mitigating this limitation would
be to use a more expressive model, such as a neural network, as a
representation of the fluid dynamics. This may solve the eccentricities of the
predictions seen in Fig. 7.

One motivation to keep our fluid deposition model computationally inexpensive
is to prepare for the future implementation of closed-loop control using such
a model alongside vision-based feedback. Closed-loop control will provide
further improvements in precise material placement, as our model cannot
completely describe the real system. A more comprehensive control scheme will
also take into account the dynamics of the robotic arm, optimizing the nozzle
trajectory alongside the flow rate so that the transient delay still present
in the tuned output (see Fig. 7) will not be a permanent barrier to precise
material placement. Through this we will achieve even further improvements
than those demonstrated in this paper.

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