Compact Embedded Minimal Surfaces of

Positive Genus Without Area Bounds

BRIAN DEANDepartment of Mathematics, Johns Hopkins University, 3400 N. Charles Street, Baltimore,MD 21218, U.S.A. e-mail: bdean@math.jhu.edu

(Received: 13 August 2002; accepted 23 October 2002)

Abstract. Let M3 be a three-manifold (possibly with boundary). We will show that, for any

positive integer g, there exists an open nonempty set of metrics on M (in the C2-topologyon the space of metrics on M) for each of which there are compact embedded stable minimalsurfaces of genus g with arbitrarily large area. This extends a result of Colding and Minicozzi,

who proved the case g ¼ 1.

Mathematics Subject Classifications. 53A10, 53C42.

Key words. differential geometry, minimal surfaces, stability.

Introduction

Throughout this paper, we use the C2-topology on the space of metrics on a mani-

fold. Our main result is the following theorem.

THEOREM 1. Let M3 be a three-manifold ð possibly with boundaryÞ, and let g be a

positive integer. There exists an open nonempty set of metrics on M for each of which

there are compact embedded minimal surfaces of genus g with arbitrarily large area. In

fact, these can be chosen to be stable, i.e., with Morse index zero.

Although the theorem ensures that there are ‘many’ metrics for which we can

embed compact genus g minimal surfaces of arbitrarily large area, the result is false

for a large class of metrics. Namely, a result of Choi and Wang (see [CW]) asserts

that for any metric in which M has Ricci curvature bounded below by a positive

constant, there is an upper bound on the area of compact embedded minimal surfaces

of genus g, depending on g and the lower bound for RicM.

Colding and Minicozzi (see [CM1]) have already proved Theorem 1 for g ¼ 1. In

Section 1, we will prove the theorem for g ¼ 2, with an argument borrowing heavily

from the genus one case. The theorem will then be extended easily to genus greater

than two in Section 2.

It remains an open question as to whether or not the theorem remains valid for

genus zero, i.e., embedded minimal 2-spheres.

Geometriae Dedicata 102: 45–52, 2003. 45# 2003 Kluwer Academic Publishers. Printed in the Netherlands.

1. The Genus 2 Case

Let S2 denote the standard genus two surface. This has fundamental group

p1ðS2Þ ¼ hx1; y1; x2; y2 jx1y1x�11 y�1

1 x2y2x�12 y�1

2 i

where x1 and x2 are freely homotopic to meridians of the two handles, where the

meridians have the same orientation, and y1 and y2 are freely homotopic to lines

of latitude of the two handles, where the lines of latitude have the same orientation.

Let O2 be a solid genus two surface with a solid genus two surface and two solid tori

removed, where the solid tori lie in the same handle of the ambient genus two sur-

face. This can be pictured as in Figure 1, where the top and bottom of the picture

are identified.

The fundamental group of O2 is

p1ðO2Þ ¼ ha; b; c1; d1; c2; d2 j ad1a�1d�1

1 ; bd1b�1d�1

1 ; c1d1c�11 d�1

1 c2d2c�12 d�1

2 i

where the generators are as follows:

(i) a and b are freely homotopic to meridians of the two removed solid tori (clock-

wise rotation around the two removed solid tori in Figure 1).

(ii) c1 is freely homotopic to a meridian of the handle of the removed solid genus

two in the same handle of the ambient solid genus two as the removed solid tori

(clockwise rotation around the left handle of the removed solid genus two in

Figure 1).

Figure 1. O2. The top and bottom of the picture are identified.

46 BRIAN DEAN

(iii) d1 is freely homotopic to a line of latitude of the left handle of the ambient solid

genus two in Figure 1.

(iv) c2 is freely homotopic to a meridian of the removed solid genus two in the other

handle from the meridian which is freely homotopic to c1 (clockwise rotation

around the right handle of the removed solid genus two in Figure 1).

(v) d2 is freely homotopic to a line of latitude of the right handle of the ambient

solid genus two in Figure 1, with the same orientation as the line of latitude

which is freely homotopic to d1.

Before we give the proof of Theorem 1 for the genus two case, we need the follow-

ing proposition, whose proof is inspired by a calculation in [Es].

PROPOSITION 1. Let On be a compact Riemannian manifold with boundary and

dimension n5 3. Then, the set of metrics on O in which O is strictly mean convex is

open and nonempty.

Proof. The set of such metrics is clearly open, by the definition of strictly mean

convex. To show it is nonempty, let g be any metric on O, and let ~gg ¼ e2fg be a

metric conformally related to g. Let fe1; . . . ; eng be a framing for O so that gij ¼ dijand en is the unit normal to @O in g (and therefore, e�fen is the unit normal to @Oin ~gg). Fix a point p 2 @O, and choose coordinates fx1; . . . ; xng at p so that, at p,

ei ¼ @=@xi for all i. Then, the second fundamental form of @O in g at p is given by,

for i; j ¼ 1; . . . ; n� 1,

hij ¼ gðHei ej; enÞ ¼Xnk¼1

gðGkijek; enÞ ¼ Gn

ij;

and the mean curvature of @O in g at p is given by

h ¼1

n� 1

Xn�1

i; j¼1

gijhij ¼1

n� 1

Xn�1

i; j¼1

dijGnij ¼

1

n� 1

Xn�1

i¼1

Gnii

The second fundamental form of @O in ~gg at p is given by, for i; j ¼ 1; . . . ; n� 1,

~hhij ¼ ~ggð ~HHeiej; e�fenÞ ¼ e2fgð ~HHei ej; e

�fenÞ ¼Xnk¼1

e fgð ~GGkijek; enÞ ¼ e f ~GGn

ij:

Now,

~GGn

ij ¼1

2

Xnl¼1

ð ~ggjl;i þ ~ggil; j � ~ggij;lÞ ~ggln;

where, for example,

~ggjl;i ¼@

@xi~ggjl

COMPACT MINIMAL SURFACES WITHOUT AREA BOUNDS 47

So, at p,

~GGnij ¼

1

2e�2f

Xnl¼1

ð ~ggjl;i þ ~ggil; j � ~ggij;lÞgln

¼1

2e�2f

Xnl¼1

ð ~ggjl;i þ ~ggil; j � ~ggij;lÞ dln

¼1

2e�2fð ~ggjn;i þ ~ggin; j � ~ggij;nÞ

¼1

2e�2f 2

@f

@xie2fgjn þ e2fgjn;i þ 2

@f

@xje2fgin þ e2fgin; j

��

�2@f

@xne2fgij � e2fgij;n

�

¼1

2ðgjn;i þ gin; j � gij;nÞ þ

@f

@xidjn þ

@f

@xjdin �

@f

@ndij

¼ Gnij �

@f

@ndij;

since i; j < n, where @f=@n is the normal derivative of f with respect to the unit normal

en. Therefore, we have

~hhij ¼ e f ~GGnij ¼ e fGn

ij � e f@f

@ndij ¼ e fGn

ij �@

@nðe fÞdij:

The mean curvature of @O in ~gg at p is then given by

~hh ¼1

n� 1

Xn�1

i; j¼1

~ggij ~hhij

¼1

n� 1

Xn�1

i; j¼1

e�2fdij e fGnij �

@

@nðe fÞdij

� �

¼1

n� 1

Xn�1

i¼1

e�fGnii � e�2f @

@nðe fÞ

� �

¼ e�f 1

n� 1

Xn�1

i¼1

Gnii

!� e�2fe f

@f

@n

¼ e�f h�@f

@n

� �

We have shown that this relation holds at an arbitrarily chosen point of @O, and so it

holds everywhere on @O since all quantities involved are tensorial. Let m be the

minimum of h on @O, which exists since @O is compact. Choose f so that

@f=@n < m everywhere on @O and f � 0 outside a small tubular neighborhood around

@O. Then, ~gg ¼ g except for a small tubular neighborhood around @O, and ~hh > 0

everywhere on @O, so ~gg is a metric in which O is strictly mean convex. This completes

the proof of Proposition 1. &

48 BRIAN DEAN

To prove Theorem 1, we will also need the following lemma.

LEMMA 1. Let N3 be a compact Riemannian manifold, and let fMng N be a

sequence of stable, compact, connected, embedded minimal surfaces without boundary

such that the following conditions hold:

ðiÞ there exists a constant C1 > 0 such that AreaðMnÞ4C1 for all n.

ðiiÞ there exists a constant C2 > 0 such that supMnjAnj

2 4C2; for all n, where An is the

second fundamental form of Mn.

Then, a subsequence of fMng converges to a compact, connected, embedded minimal

surface without boundary M N of finite multiplicity.

Proof. Take a finite covering fBrð yjÞg of N so that fBr=2ð yjÞg is still a covering of

N. Then, by [CM2], a subsequence of fMng converges in each Br=2ð yjÞ to a lamination

M with minimal leaves. By taking a diagonal subsequence, we have a subsequence of

fMng, which we still call fMng, converging to M everywhere. M is clearly minimal,

and it is embedded by the maximum principle.

We claim that the number of leaves of Mn in each Brð yjÞ which intersect Br=2ð yjÞ

has an upper bound which is uniform in n and j. Let Gn; j be any such leaf. Then,

there exists xj 2 Gn; j \ @Br=2ð yjÞ. So, Br=2ðxjÞ Brð yjÞ, and by monotonicity of area,

there exists a constant C > 0 so that

AreaðGn; j \ Br=2ðxjÞÞ5Cr

2

� �2

:

So, each Gn; j is of at least some fixed positive area, and so the area bound ði Þ gives an

upper bound for the number of such leaves which is uniform in n and j. We can take

a subsequence so that the number of leaves of Mn is the same in each Br=2ð yjÞ for all

n; j. Then, the limit M must have finite multiplicity, although the multiplicity may be

different in each connected component of M. We have shown that M is a surface,

since there are a fixed number of charts in each Br=2ð yjÞ. The diameter of M is boun-

ded, since M is covered by finitely many balls Br=2ð yjÞ \M. So, M is compact. M is

without boundary since each Mn is without boundary.

It remains to show that M is connected, which would imply that Mn ! M with

fixed finite multiplicity. Suppose M is not connected, and let A and B be distinct

connected components of M. Then, E ¼ distðA;BÞ > 0. Let

R ¼ x 2 NjE3< distðx;AÞ <

2E3

� :

So, R is disjoint from both A and B. Since Mn ! M, for large enough n we have

Mn \ A 6¼ ; and Mn \ B 6¼ ;, but Mn \ R ¼ ;, contradicting the connectedness of

Mn. So, M is connected.

Therefore, Mn converges to a compact, connected, embedded minimal

surface without boundary M N of finite multiplicity. This completes the proof of

Lemma 1. &

COMPACT MINIMAL SURFACES WITHOUT AREA BOUNDS 49

Proof of Theorem 1 for g ¼ 2. Given any three-manifold M, we can embed O2 in

M. Choose a metric g on M so that O2 is strictly mean convex (by Proposition 1, the

set of such g’s is open and nonempty). Let fn:S2 ! O2 be a map such that the

induced map fn# : p1ðS2Þ ! p1ðO2Þ is the following:

fn# ðx1Þ ¼ ðbaÞnc1ðbaÞ�na�1ba; fn# ðy1Þ ¼ d1;

fn# ðx2Þ ¼ c2; fn# ðy2Þ ¼ d2

It is easy to see that there exist such maps fn which are embeddings.

One can check that fn# ðx1Þ minimizes the word metric for its conjugacy class: by

conjugating fn# ðx1Þ by any element of p1ðO2Þ and using the relations of p1ðO2Þ, one

can not decrease the length of fn# ðx1Þ in the word metric (for the definition of word

metric, see [CM1]).

So, for each n, we have an embedded incompressible genus two surface

S2;n ¼ fnðS2Þ. By [ScY], there are immersed least-area (minimal) genus two surfaces

G2;n O2 with G2;n \ @O2 ¼ ; so that G2;n and S2;n induce the same mapping from

p1ðS2Þ to p1ðO2Þ for each n. Since the S2;n are embedded, [FHS] implies that the

G2;n are embedded.

We claim that the areas of the G2;n’s are unbounded. Assume not. Then, there

exists a constant C1 > 0 such that AreaðG2;nÞ4C1 for all n. The G2;n are stable since

they are area-minimizing. So, by [Sc], we get a uniform curvature estimate: there

exists a constant C2 > 0 such that, for small enough r and all s 2 ð0; r�,

supBr�s

jAnj2 4

C2

s2

for all n and all balls Br�s O2, where An is the second fundamental form of G2;n.

Since the G2;n are all without boundary, we get a uniform curvature estimate on

all of G2;n, instead of just on balls. Therefore, by Lemma 1, a subsequence of

fG2;ng converges to a compact, connected, embedded minimal surface without

boundary G2 O2 of finite multiplicity.

For large n, the G2;n are coverings of G2 by the maximum principle, and the degree

of the covering is proportional to n. Let n ! 1. Then, G2 has infinite multiplicity, a

contradiction. Therefore, the areas of the G2;n’s are unbounded. This completes the

proof of Theorem 1 for the case g ¼ 2.

2. The General Case: g5 2

We now move to the general case. The arguments for fixed genus g5 2 are essen-

tially the same as in the genus 2 case.

Let Sg denote the standard genus g surface, g5 2. This has fundamental group

p1ðSgÞ ¼ hx1; y1; . . . ; xg; yg j ½x1y1� � � � ½xgyg�i

where the xi are freely homotopic to meridians of the handles, all with the same

orientation, the yi are freely homotopic to lines of latitude of the handles, all with

50 BRIAN DEAN

the same orientation, and ½xiyi� ¼ xiyix�1i y�1

i for i ¼ 1; . . . ; g. Let Og be a solid genus

g surface with a solid genus g surface and two solid tori removed, where the solid tori

both lie in one of the end handles of the ambient genus g surface (the case g ¼ 3 is

shown in Figure 2, where the top and bottom of the picture are identified).

The fundamental group of Og is

p1ðOgÞ ¼ ha; b; c1; d1; . . . ; cg; dg j ½ad1�; ½bd1�; ½c1d1� � � � ½cgdg�i

where the generators are defined as in the case g ¼ 2 (so, a, b, and all ci are freely

homotopic to meridians with the same orientation, and all di are freely homotopic

to lines of latitude with the same orientation).

Proof of Theorem 1. Given any three-manifold M, we can embed Og in M.

Choose a metric g on M so that Og is strictly mean convex (by Proposition 1, the set

of such g’s is open and nonempty). Let fn:Sg ! Og be a map such that the induced

map fn#: p1ðSgÞ ! p1ðOgÞ is the following:

fn#ðx1Þ ¼ ðbaÞnc1ðbaÞ�na�1ba

fn#ðxiÞ ¼ ci for i ¼ 2; . . . ; g

fn#ðyiÞ ¼ di for i ¼ 1; . . . ; g

It is easy to see that there exist such maps fn which are embeddings.

The proof then proceeds exactly as in the genus 2 case. The results of [ScY, FHS],

and [Sc] again apply.

References

[CM1] Colding, T. H. and Minicozzi II, W. P.: Examples of embedded minimal tori withoutarea bounds, Internat. Math. Res. Notes No. 20 (1999), 1097–1100.

[CM2] Colding, T. H. and Minicozzi II, W. P.: The space of embedded minimal surfaces of

fixed genus in a 3-Manifold IV; locally simply connected, Preprint.

Figure 2. O3. The top and bottom of the picture are identified.

COMPACT MINIMAL SURFACES WITHOUT AREA BOUNDS 51

[CW] Choi, H. I. and Wang, A. N.: A first eigenvalue estimate for minimal hypersurfaces,

J. Differential Geom. 18 (1983), 559–562.[Es] Escobar, J. F.: Conformal deformation of a Riemannian metric to a scalar flat metric

with constant mean curvature on the boundary, Ann. of Math. 136 (1992), 1–50.

[FHS] Freedman, M. H., Hass, J. and Scott, P.: Least area incompressible surfaces in3-manifolds, Invent. Math. 71 (1983), 609–642.

[Sc] Schoen, R.: Estimates for stable minimal surfaces in three dimensional manifolds, In:

Seminar on Minimal Submanifolds, Ann. of Math. Stud. 103, Princeton Univ. Press,1983.

[ScY] Schoen, R. and Yau, S. T.: Existence of incompressible minimal surfaces and thetopology of three-dimensional manifolds with non-negative scalar curvature, Ann.

of Math. ð2 Þ 110 (1979), 127–142.

52 BRIAN DEAN