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x
1
x
3
x
2
Figure 2: The catenoid given by revolving x
1
= cosh x
3
around the x
3
-axis.
Necks connecting parallel planes.
Figure 3: The Riemann examples: Parallel planes connected by necks.
axis and the separation between each pair of adjacent ends is constant (in fact
the surfaces are periodic). Locally, one can imagine connecting − 1 planes
by − 2 necks and add half of a catenoid to each of the two outermost planes,
possibly with some restriction on how the necks line up and on the separation
of the planes; see [FrMe], [Ka], [LoRo].
To illustrate how Theorem 0.3 below will be used in [CM6] where we give
the actual “pair of pants decomposition” observe that the catenoid can be de-
composed into two minimal annuli each with one exterior convex boundary and
one interior boundary which is a short simple closed geodesic. (See also [CM9]
for the “pair of pants decomposition” in the special case of annuli.) In the case
of the Riemann examples (see Figure 4), there will be a number of “pairs of
pants”, that is, topological disks with two subdisks removed. Metrically these
“pairs of pants” have one convex outer boundary and two interior boundaries
each of which is a simple closed geodesic. Note also that this decomposition
can be made by putting in minimal graphical annuli in the complement of the
domains (in R
3
) which separate each of the pieces; cf. Corollary 0.4 below.
Moreover, after the decomposition is made then every intersection of one of
the “pairs of pants” with an extrinsic ball away from the interior boundaries
is simply connected and hence the results of [CM3]–[CM5] apply there.
The next theorem is a kind of effective removable singularity theorem
for embedded stable minimal surfaces with small interior boundaries. It as-
serts that embedded stable minimal surfaces with small interior boundaries are
graphical away from the boundary. Here small means contained in a small ball
PLANAR DOMAINS
527
A “pair of pants” (in bold).
Graphical annuli (dotted) separate
the “pairs of pants”.
Figure 4: Decomposing the Riemann examples into “pair of pants” by cutting
along small curves; these curves bound minimal graphical annuli separating
the ends.
Stable Γ with ∂Γ ⊂ B
r
0
/4
∪ ∂B
R
.
Components of Γ in B
R/C
1
\ B
C
1
r
0
are graphs.
C
1
r
0
R
r
0
4
R
C
1
Figure 5: Theorem 0.3: Embedded stable annuli with small interior boundary
are graphical away from their boundary.
in R
3
(and not that the interior boundary has small length). This distinction
is important; in particular if one had a bound for the area of a tubular neigh-
borhood of the interior boundary, then Theorem 0.3 would follow easily; see
Corollary II.1.34 and cf. [Fi].
Theorem 0.3 (see Figure 5). Given τ>0, there exists C
1
> 1, so that
if Γ ⊂ B
R
⊂ R
3
is an embedded stable minimal annulus with ∂Γ ⊂ ∂B
R
∪B
r
0
/4
(for C
2
1
r
0
<R) and B
r
0
∩ ∂Γ is connected, then each component of B
R/C
1
∩
Γ \ B
C
1
r
0
is a graph with gradient ≤ τ.
Many of the results of this paper will involve either graphs or multi-valued
graphs. Graphs will always be assumed to be single-valued over a domain in
the plane (as is the case in Theorem 0.3).
Combining Theorem 0.3 with the solution of a Plateau problem of Meeks-
Yau (proven initially for convex domains in Theorem 5 of [MeYa1] and extended
to mean convex domains in [MeYa2]), we get (the result of Meeks-Yau gives
the existence of Γ below):
Corollary 0.4 (see Figure 6). Given τ>0, there exists C
1
> 1, so
that the following holds:
528 TOBIAS H. COLDING AND WILLIAM P. MINICOZZI II
Let Σ ⊂ B
R
⊂ R
3
with ∂Σ ⊂ ∂B
R
be an embedded minimal surface with
gen(Σ) = gen(B
r
1
∩ Σ) and let Ω be a component of B
R
\ Σ.
If γ ⊂ B
r
0
∩ Σ \ B
r
1
is noncontractible and homologous in Σ \ B
r
1
to a
component of ∂Σ and r
0
>r
1
, then a component
ˆ
Σ of Σ \ γ is an annulus and
there is a stable embedded minimal annulus Γ ⊂ Ω with ∂Γ=∂
ˆ
Σ.
Moreover, each component of (B
R/C
1
\ B
C
1
r
0
) ∩ Γ is a graph with gradient
≤ τ.
γ ⊂ Σ not contractible in Σ.
B
r
0
Stable annulus Γ.
Component Ω of B
R
\ Σ
where γ is not contractible.
Figure 6: Corollary 0.4: Solving a Plateau problem gives a stable graphical
annulus separating the boundary components of an embedded minimal annu-
lus.
Stability of Γ in Theorem 0.3 is used in two ways: To get a pointwise
curvature bound on Γ and to show that certain sectors have small curvature.
In Section 2 of [CM4], we showed that a pointwise curvature bound allows us
to decompose an embedded minimal surface into a set of bounded area and a
collection of (almost stable) sectors with small curvature. Using this, we see
that the proof of Theorem 0.3 will also give (if 0 ∈ Σ, then Σ
0,t
denotes the
component of B
t
∩ Σ containing 0):
Theorem 0.5. Given C, there exist C
2
,C
3
> 1, so that the following
holds:
Let 0 ∈ Σ ⊂ B
R
⊂ R
3
be an embedded minimal surface with connected
∂Σ ⊂ ∂B
R
.Ifgen(Σ
0,r
0
) = gen(Σ), r
0
≤ R/C
2
, and
sup
Σ\B
r
0
|x|
2
|A|
2
(x) ≤ C,(0.6)
then
Area(Σ
0,r
0
) ≤ C
3
r
2
0
.
The examples constructed in [CM13] show that the quadratic curvature
bound (0.6) is necessary to get the area bound in Theorem 0.5.
In [CM5] a strengthening of Theorem 0.5 (this strengthening is Theorem
III.3.1 below) will be used to show that, for limits of a degenerating sequence of
PLANAR DOMAINS
529
embedded minimal disks, points where the curvatures blow up are not isolated.
This will eventually give (Theorem 0.1 of [CM5]) that for a subsequence such
points form a Lipschitz curve which is infinite in two directions and transversal
to the limit leaves; compare with the example given by a sequence of rescaled
helicoids where the singular set is a single vertical line perpendicular to the
horizontal limit foliation.
To describe a neighborhood of each of the finitely many points, coming
from Theorem 0.1, where the genus concentrates (specifically to describe when
there is one component
˜
Σ
k,j
of genus > 0 in “(3)” of Theorem 0.1), we will
need in [CM6]:
Corollary 0.7. Given C, g, there exist C
4
,C
5
so that the following holds:
Let 0 ∈ Σ ⊂ B
R
⊂ R
3
be an embedded minimal surface with connected
∂Σ ⊂ ∂B
R
, r
0
<R/C
4
, and gen(Σ
0,r
0
) = gen(Σ) ≤ g.If
sup
Σ\B
r
0
|x|
2
|A|
2
(x) ≤ C,(0.8)
then
Σ is a disk and Σ
0,R/C
5
is a graph with gradient ≤ 1.
This corollary follows directly by combining Theorem 0.5 and theorem
1.22 of [CM4]. That is, we note first that for r
0
≤ s ≤ R, it follows from the
maximum principle (since Σ is minimal) and Corollary I.0.11 that ∂Σ
0,s
is con-
nected and Σ \ Σ
0,s
is an annulus. Second, theorem 0.5 bounds Area(Σ
0,R/C
2
)
and Theorem 1.22 of [CM4] then gives the corollary.
Theorems 0.3, 0.5 and Corollary 0.7 are local and are for simplicity stated
and proved only in R
3
although they can with only very minor changes easily
be seen to hold for minimal planar domains in a sufficiently small ball in any
given fixed Riemannian 3-manifold.
Throughout Σ, Γ ⊂ M
3
will denote complete minimal surfaces possibly
with boundary, sectional curvatures K
Σ
,K
Γ
, and second fundamental forms
A
Σ
, A
Γ
. Also, Γ will be assumed to be stable and have trivial normal bundle.
Given x ∈ M, B
s
(x) will be the usual ball in R
3
with radius s and center x.
Likewise, if x ∈ Σ, then B
s
(x) is the intrinsic ball in Σ. Given S ⊂ Σ and
t>0, let T
t
(S, Σ) ⊂ Σ be the intrinsic tubular neighborhood of S in Σ with
radius t and set
T
s,t
(S, Σ) = T
t
(S, Σ) \T
s
(S, Σ) .
Unless explicitly stated otherwise, all geodesics will be parametrized by ar-
clength.
We will often consider the intersections of various curves and surfaces with
extrinsic balls. We will always assume that these intersections are transverse
since this can be achieved by an arbitrarily small perturbation of the radius.
530 TOBIAS H. COLDING AND WILLIAM P. MINICOZZI II
I. Topological decomposition of surfaces
In this part we will first collect some simple facts and results about planar
domains and domains that are planar outside a small ball. These results will
then be used to show Theorem 0.1. First we recall an elementary lemma:
Lemma I.0.9 (see Figure 7). Let Σ be a closed oriented surface (i.e.,
∂Σ=∅) with genus g. There are transverse simple closed curves η
1
, ,η
2g
⊂
Σ so that for i<j
#{p | p ∈ η
i
∩ η
j
} = δ
i+g,j
.(I.0.10)
Furthermore, for any such {η
i
}, if η ⊂ Σ \ ∪
i
η
i
is a closed curve, then η
divides Σ.
η
1
η
2
η
3
η
4
Figure 7: Lemma I.0.9: A basis for homology on a surface of genus g.
Recall that if ∂Σ = ∅, then
ˆ
Σ is the surface obtained by replacing each
circle in ∂Σ with a disk. Note that a closed curve η ⊂ Σ divides Σ if and only
if η is homologically trivial in
ˆ
Σ.
Corollary I.0.11. If Σ
1
⊂ Σ and gen(Σ
1
) = gen(Σ), then each simple
closed curve η ⊂ Σ \ Σ
1
divides Σ.
Proof. Since Σ
1
has genus g = gen(Σ), Lemma I.0.9 gives transverse
simple closed curves η
1
, ,η
2g
⊂ Σ
1
satisfying (I.0.10). However, since η does
not intersect any of the η
i
’s, Lemma I.0.9 implies that η divides Σ.
Corollary I.0.12. If Σ has a decomposition Σ=∪
β=1
Σ
β
where the
union is taken over the boundaries and each Σ
β
is a surface with boundary
consisting of a number of disjoint circles, then
β=1
gen(Σ
β
) ≤ gen(Σ) .(I.0.13)
Proof. Set g
β
= gen(Σ
β
). Lemma I.0.9, gives transverse simple closed
curves
η
β
1
, ,η
β
2g
β
⊂ Σ
β
PLANAR DOMAINS
531
satisfying (I.0.10). Since Σ
β
1
∩ Σ
β
2
= ∅ for β
1
= β
2
, this implies that the rank
of the intersection form on the first homology (mod 2) of
ˆ
Σis≥ 2
β=1
g
β
.In
particular, we get (I.0.13).
In the next lemma, M
3
will be a closed 3-manifold and Σ
2
i
a sequence of
closed embedded oriented minimal surfaces in M with fixed genus g.
Lemma I.0.14. There exist x
1
, ,x
m
∈ M with m ≤ g and a subse-
quence Σ
j
so that the following hold:
• For x ∈ M \{x
1
, ,x
m
}, there exist j
x
,r
x
> 0 so that gen(B
r
x
(x)∩Σ
j
)=
0 for j>j
x
.
• For each x
k
, there exist R
k
,g
k
> 0,R
k
>R
k,j
→ 0 so that
m
k=1
g
k
≤ g
and for all j,
gen(B
R
k
(x
k
) ∩ Σ
j
)=g
k
= gen(B
R
k,j
(x
k
) ∩ Σ
j
) .
Proof. Suppose that for some x
1
∈ M and any R
1
> 0 we have infinitely
many i’s where
gen(B
R
1
(x
1
) ∩ Σ
i
)=g
1,i
> 0 .
By Corollary I.0.12, we have g
1,i
≤ g and hence there is a subsequence Σ
j
and
a sequence R
1,j
→ 0 so that for all j
gen(B
R
1,j
(x
1
) ∩ Σ
j
)=g
1
> 0 .(I.0.15)
By repeating this construction, we can suppose that there are disjoint points
x
1
, ,x
m
∈ M and R
k,j
> 0 so that for any k we have R
k,j
→ 0 and
gen(B
R
k,j
(x
k
) ∩ Σ
j
)=g
k
> 0 .
However, Corollary I.0.12 implies that for j sufficiently large
0 ≤ gen(Σ
j
\∪
k
B
R
k,j
(x
k
)) ≤ gen(Σ
j
) −
m
k=1
gen(B
R
k,j
(x
k
) ∩ Σ
j
) ≤ g −
m
k=1
g
k
.
(I.0.16)
In particular,
m
k=1
g
k
≤ g and we can therefore assume that
m
k=1
g
k
is max-
imal. This has two consequences:
• First, given x ∈ M \{x
1
, ,x
m
}, there exist r
x
> 0 and j
x
so that
gen(B
r
x
(x) ∩ Σ
j
) = 0 for j>j
x
.
• Second, for each x
k
, there exist R
k
> 0 and j
k
so that gen(B
R
k
(x
k
) ∩
Σ
j
)=g
k
for j>j
k
.
The lemma now follows easily.
532 TOBIAS H. COLDING AND WILLIAM P. MINICOZZI II
By Corollary I.0.12, each R
k
,R
k,j
from Lemma I.0.14 can (after going
to a further subsequence) be replaced by any R
k
,R
k,j
with R
k
≤ R
k
and
R
k,j
≥ R
k,j
. Similarly, each r
x
can be replaced by any r
x
≤ r
x
. This will be
used freely in the proof of Theorem 0.1 below.
Proof of Theorem 0.1. Let x
k
,g
k
,R
k
,R
k,j
and r
x
be from Lemma I.0.14.
We can assume that each R
k
> 0 is sufficiently small so that B
R
k
(x
k
) is es-
sentially Euclidean (e.g., R
k
< min{i
0
/4,π/(4k
1/2
)}). Part (1) follows directly
from Lemma I.0.14.
For each x
k
, we can assume that there are
k
and n
,k
so that:
• B
R
k
(x
k
) ∩ Σ
j
has components {Σ
k,j
}
1≤≤
k
with genus > 0.
• B
R
k,j
(x
k
) ∩ Σ
k,j
has n
,k
components with genus > 0.
We will use repeatedly that, by (1) and Corollary I.0.12, n
,k
is nonincreasing
if either R
k,j
increases or R
k
decreases. For each , k with n
,k
> 1, set
ρ
k,j
= inf{ρ>R
k,j
| #{components of B
ρ
(x
k
) ∩ Σ
k,j
} <n
,k
} .(I.0.17)
There are two cases. If lim inf
j→∞
ρ
k,j
= 0, then choose a subsequence Σ
j
with
ρ
k,j
→ 0; n
,k
decreases if we replace R
k,j
with any R
k,j
>ρ
k,j
. Otherwise,
set 2 ρ
k
= lim inf
j→∞
ρ
k,j
> 0 and choose a subsequence Σ
j
so that ρ
k,j
<ρ
k
;
k
increases if we replace R
k
with any R
k
≤ ρ
k
. In either case,
,k
(n
,k
− 1)
decreases. Since
,k
n
,k
≤ g (by Corollary I.0.12), repeating this ≤ g times
gives
0 <R
k
≤ R
k
and R
k,j
≤ R
k,j
→ 0 (as j →∞)
as well as a subsequence so that only one component
˜
Σ
k,j
of B
R
k,j
(x
k
) ∩ Σ
k,j
has genus > 0 (i.e., each new n
,k
= 1). By Corollary I.0.12 (and (1)) and the
remarks before the proof, Parts (1), (2), and (3) now hold for any r
k
≤ R
k
and
R
k,j
≤ r
k,j
→ 0.
Suppose that for some k, there exists j
k,
so that ∂Σ
k,j
has at least two
components for all j>j
k,
.ForR
k,j
≤ t ≤ R
k
, let Σ
k,j
(t) be the component
of B
t
(x
k
) ∩ Σ containing
˜
Σ
k,j
. Set
r
k,j
= inf{t>R
k,j
| #{components of ∂Σ
k,j
(t)} > 1} .(I.0.18)
There are two cases:
• If lim inf
j→∞
r
k,j
= 0, then choose a subsequence Σ
j
with r
k,j
→ 0.
By the maximum principle (since Σ is minimal) and Corollary I.0.11, a
component of (the new) ∂
˜
Σ
k,j
separates two components of ∂Σ
k,j
for any
r
k,j
→ 0 with r
k,j
>r
k,j
.
• On the other hand, if lim inf
j→∞
r
k,j
=2r
k
> 0, then choose a subse-
quence so that (the new) ∂Σ
k,j
is connected for any r
k
≤ r
k
.
PLANAR DOMAINS
533
After repeating this ≤ g times (each time either increasing R
k,j
or decreasing
R
k
), Part (4) also holds.
In [CM6] we will need the following (here, and elsewhere, if 0 ∈ Σ ⊂ R
3
,
then Σ
0,t
denotes the component of B
t
∩ Σ containing 0):
Proposition I.0.19. Let 0 ∈ Σ
i
⊂ B
S
i
⊂ R
3
with ∂Σ
i
⊂ ∂B
S
i
be a
sequence of embedded minimal surfaces with genus ≤ g<∞ and S
i
→∞.
After going to a subsequence,Σ
j
, and possibly replacing S
j
by R
j
and Σ
j
by
Σ
0,j,R
j
where R
0
≤ R
j
≤ S
j
and R
j
→∞, then
gen(Σ
j,0,R
0
) = gen(Σ
j
) ≤ g
and either (a) or (b) holds:
(a) ∂Σ
j,0,t
is connected for all R
0
≤ t ≤ R
j
.
(b) ∂Σ
j,0,R
0
is disconnected.
Proof. We will first show that there exists R
0
> 0, a subsequence Σ
j
,
and a sequence R
j
→∞with R ≤ R
j
≤ S
j
, such that (after replacing Σ
j
by
Σ
j,0,R
j
)
gen(Σ
j,0,R
0
) = gen(Σ
j
) ≤ g.
Suppose not; it follows easily from the monotonicity of the genus (i.e., Corollary
I.0.12) that there exists a subsequence Σ
j
and a sequence G
k
→∞such that
for all k there exists a j
k
so that for j ≥ j
k
g ≥ gen(Σ
j,0,G
k+1
) > gen(Σ
j,0,G
k
) ,(I.0.20)
which is a contradiction.
For each j, let R
0,j
be the infimum of R with R
0
≤ R ≤ R
j
where ∂Σ
j,0,R
is disconnected; set R
0,j
= R
j
if no such exists. There are now two cases:
• If lim inf R
0,j
< ∞, then, after going to a subsequence and replacing R
0
by lim inf R
0,j
+1, we are in (b) by the maximum principle.
• If lim inf R
0,j
= ∞, then we are in (a) after replacing R
j
by R
0,j
.
II. Estimates for stable minimal surfaces
with small interior boundaries
In this part we prove Theorem 0.3. That is, we will show that all embedded
stable minimal surfaces with small interior boundaries are graphical away from
the boundary. Here small means contained in a small ball in R
3
(and not that
the interior boundary has small length).
534 TOBIAS H. COLDING AND WILLIAM P. MINICOZZI II
II.1. Long stable sectors contain multi-valued graphs
In [CM3], [CM4] we proved estimates for the total curvature and area of
stable sectors. A stable sector in the sense of [CM3], [CM4] is a stable subset
of a minimal surface given as half of a normal tubular neighborhood (in the
surface) of a strictly convex curve. For instance, a curve lying in the boundary
of an intrinsic ball is strictly convex. In this section we give similar estimates for
half of normal tubular neighborhoods of curves lying in the intersection of the
surface and the boundary of an extrinsic ball. These domains arise naturally
in our main result and are unfortunately somewhat more complicated to deal
with due to the lack of convexity of the curves.
In this section, the surfaces Σ and Γ will be planar domains and, hence,
simple closed curves will divide the surface into two planar (sub)domains.
We will need some notation for multi-valued graphs. Let P be the univer-
sal cover of the punctured plane C \{0} with global (polar) coordinates (ρ, θ)
and set
S
θ
1
,θ
2
r,s
= {r ≤ ρ ≤ s, θ
1
≤ θ ≤ θ
2
} .
An N-valued graph Σ of a function u over the annulus D
s
\D
r
(see Figure 8) is
a (single-valued) graph (of u)overS
−N π,N π
r,s
(Σ
θ
1
,θ
2
r,s
will denote the subgraph
of Σ over S
θ
1
,θ
2
r,s
). The separation w(ρ, θ) between consecutive sheets is (see
Figure 8)
w(ρ, θ)=u(ρ, θ +2π) − u(ρ, θ) .(II.1.1)
x
3
-axis
u(ρ, θ +2π)
w
u(ρ, θ)
Figure 8: The separation w for a multi-valued graph in (II.1.1).
PLANAR DOMAINS
535
The main result of the next two sections is the following theorem (Γ
1
(∂)
is the component of B
1
∩ Γ containing B
1
∩ ∂Γ):
Theorem II.1.2 (see Figure 9). Given N, τ > 0, there exist ω>1, d
0
so that the following holds:
Let Γ be a stable embedded minimal annulus with ∂Γ ⊂ B
1/4
∪∂B
R
, B
1/4
∩
∂Γ connected, and R>ω
2
. Given a point z
1
∈ ∂B
1
∩ ∂Γ
1
(∂), then (after a
rotation of R
3
) either (1) or (2) below holds:
(1) Each component of B
R/ω
∩ Γ \ B
ω
is a graph with gradient ≤ τ.
(2) Γ contains a graph Γ
−Nπ,Nπ
ω,R/ω
with gradient ≤ τ and dist
Γ\Γ
1
(∂)
(z
1
, Γ
0,0
ω,ω
)
<d
0
.
B
1
z
1
B
ω
B
R/ω
Interior boundary B
1/4
∩ ∂Γ.
Γ contains a large “flat region” between
B
ω
and B
R
/ω
. Since Γ is embedded,
this either (1) closes up to give a graphical
annulus or
(
2
)
spirals to give an N -valued graph.
Figure 9: Theorem II.1.2: Embedded stable annuli with small interior bound-
ary contain either: (1) a graphical annulus, or (2) an N-valued graph away
from its boundary.
Note that if Γ is as in Theorem II.1.2 and one component of B
R/ω
∩Γ\B
ω
contains a graph over D
R/(2ω)
\ D
2ω
with gradient ≤ 1, then every component
of
B
R/(Cω)
∩ Γ \ B
Cω
is a graph for some C>1. Namely, embeddedness and the gradient estimate
(which applies because of stability) would force any nongraphical component
to spiral indefinitely, contradicting that Γ is compact. Thus it is enough to
find one component that is a graph. This will be used below.
We will eventually show in Section II.3 that (2) in Theorem II.1.2 does
not happen; thus every component is a (single-valued) graph. This will easily
give Theorem 0.3.
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