Annales de l’institut Fourier (AIF)

Garofalo, Nicola

Lanconelli, Ermanno

Frequency functions on the Heisenberg group, the uncertainty principle and unique continuation

Hardt, R.

Rosenberg, Harold

ANNALES DE L’INSTITUT FOURIER
R. HARDT
HAROLD ROSENBERG
Open book structures and unicity of
minimal submanifolds
Annales de l’institut Fourier, tome 40, no 3 (1990), p. 701-708
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Ann. Inst. Fourier, Grenoble
40, 1 (1990), 701-708
OPEN BOOK STRUCTURES
AND UNICITY OF MINIMAL SUBMANIFOLDS
by R. HARDT^*) and H. ROSENBERG^**)
0. Introduction.
Suppose r is a compact codimension 2 submanifold of a compact
orientable smooth Riemannian manifold N. In this note, we consider some
ways that a foliation of N — F by minimal hypersurfaces controls the
uniqueness of minimal hypersurfaces in M—F. Our discussion is motivated
by the examples of certain disks and annuli in S3. These are described below
in §1.2, 1.3, 1.4 and 2.2. We are also reminded of the beautiful theorem
of Shiftman [Sh] concerning minimal annuli in R3 bounded by two convex
curves (respectively, circles) that lie in parallel planes. His conclusion is that
all intermediate parallel planar sections are also convex curves (respectively,
circles). \
Our argument in §1 combines the maximum principle and the Hopf
boundary point lemma with various topological conditions. Somewhat
analogous discussions occur (without the topological conditions) in [HS],
pp. 478-479, and (without boundary considerations) in [So], Lemma 1. In
§2 we examine, in an analytic 3—manifold, the intersection between a fixed
immersed minimal surface and an open book structure whose leaves are
minimal surfaces and whose binding consists of analytic curves bounding
(*) Research partially supported by the National Science Foundation.
(**) Research undertaken during a visit to the Institute for Mathematics and its
Applications.
Key-words : Minimal submanifolds - Open book structures.
A.M.S. Classification : 53A10 - 30C15.
702 R. HARDT & H. ROSENBERG
the fixed minimal surface. The behavior away from the binding has been
studied by J. Hass [H] §2. F. Morgan [M] gives conditions under which area-
minimizing embeddings form open book structures. Some open problems
suggested by our work are given in §3.
The authors would like to thank the referee for helpful suggestions
and considerably simplifying our proof of 2.1.
1. Embedded minimal hypersurfaces and open books.
1.1. Open books. — Suppose TV is a compact connected n + 1
dimensional orientable smooth Riemannian manifold, and r is a compact
n — 1 dimensional smooth submanifold of N. A fibration C of N — r into
smooth hypersurfaces is called an open book structure for N with binding
r if r is covered by open sets U for which there is a smooth diffeomorphism
mapping U onto (T H U) x {z € C : \z\ < 1} that sends [L D U : L e C}
onto the product foliation
{(r n u) x {\v: o < A < 1 } : v e s1}.
It follows that each leaf L of C is the interior of a compact manifold with
boundary. Moreover, the orientability of N implies that the foliation £ is
transversally orientable.
1.2. THEOREM. — Suppose that the closure of each leaf of C is
an embedded minimal hypersurface with boundary F and (p : M —^ N
is a minimal immersion of a compact connected orientable n dimensional
manifold with boundary such that ^^(T) = 9M. If either
(1) M is simply connected, or
(2) Hn(N,R) == 0 and ^ is an embedding,
then M must be the closure of a leaf of C.
Proof. — For each L € £, ClosL = LUT. Moreover, each point a of
N —r has a neighborhood in N whose intersection with'the leaf through a
is a single embedded disk. Thus by the connectedness of7V—r, the quotient
space (N — r)/C has naturally the structure of a smooth circle with the
quotient map being a smooth submersion. Let LJ be the pull-back under the
quotient map of an orienting 1 form of this circle. Then duj = 0 in N — F,
and hence d((p^(jj) = 0 in M — F. Moreover, by the maximality of the rank
OPEN BOOK STRUCTURES AND UNICITY OF MINIMAL SUBMANIFOLDS 703
of (p along 9M, <p*uj extends up to the boundary (as in the argument of
[HS] p. 478) to a smooth 1 form on all of M.
In case M is simply connected, the closed form ^uj is exact, that
is, ^uj = d0 for some real-valued smooth function 0 on M. Using 0, we
can choose a point a € M where 0 attains a maximum. Near a, one leaf
of C will lie entirely on one side of M, with tangential contact at a. Then,
(as in [HS] p. 479), the maximum principle, in case a C M - 9M, or the
Hopf boundary point lemma [GT] 3.2, in case a € <9M, implies that 0 is
identically equal to 0(a). Thus, M - T is completely contained in a single
leaf L of C. Being open, as well as closed, relative to L, M — T actually
equals L.
To complete the proof we will show that ^uj is also exact for non-
simply connected M provided that (p is embedded and that Hn{N, R) = 0.
For this we assume, for notational convenience, that y? is the inclusion map
and observe that it now suffices to verify that
/ uj = 0 for any smooth closed curve f3 in M — T.
h
Since M is orientable and embedded, f uj = f cc for some smooth closed
curve 7 in N — M; here, 7 may be found by lifting f3 a small positive
distance off of M in a normal direction, obtained from the orientability of
M. We may also choose 7 to be transverse to all but finitely many leaves
of £. Noting that 7 does not intersect M, we find, after orienting M, that
the total intersection numbers of the corresponding oriented chains satisfy
f3^M = 7JJM == 0.
Choose a leaf L which intersects 7 transversely and orient it so that
9L = 9M. Since Hn(N,R) = 0, L - M == 9E for some n + 1 chain E
in N , and so
7»L = (7(M) + (^9E) =0+0.
Now observe that
f uj= [ (^=^L=O. D
h A
1.3. Embedded Minimal Surfaces in S3. — In
s^K^ec^bp+H2^!},
d = S3 n [y = 0} and £2 == S3 H {z = 0}
704 R. HARDT & H. ROSENBERG
are great circles that are linked and constant spherical distance Tr/2 apart.
For 0 € [0,27r)
De = S3 H {arg(2/) = 0}
is a (totally geodesic) embedded minimal disk with boundary C\. Applying
Theorem 1.2 with N = S3, r = Ci, and C = {De : 0 € [0,27r)}, we find
(as in [HS] p. 441) that :
Any connected embedded orientable minimal surface in S3 that has
boundary C\ must be one of the disks De for some 0 € [0,27r).
Also, for 0 € [0,27r),
Ac = S3 H {arg(^) = 0}
is an embedded minimal annulus with boundary C\ U C^. Reflection (as in
[La]) of Ac about either boundary component gives the Clifford torus
AQ U AQ^ = AQ U Ae^ U Ci U (°2 for each 0 e [0, TI-).
Applying Theorem 1.2 with N ==. S3, r = C\ U £2, and £ = {A^ : ̂  €
[0,27r)}, we similarly find that :
Any connected embedded orientable minimal surface in S3 that has
boundary C\ U Cs must be one of the annuli AQ for some 0 € [0,27r).
Recalling that the proof of Theorem 1.1 required only that the closed
one form y?*cj on M be exact we may replace the assumption on the ambient
manifold TV by an assumption on M.
2. Immersed minimal surfaces and open books.
In this section we obtain, for n = 2, some results allowing minimal
surfaces that are a priori only immersed and open book structures whose
leaves may have higher boundary multiplicity. We assume that N is a
compact orientable real analytic Riemannian 3—manifold, that r is a finite
disjoint union of analytic Jordan curves in N , and that £ is a transversally
orientable open book structure on N with binding r and with leaves that
are embedded minimal surfaces.
As in §1, for any point a € N — r, the leaf through a intersects some
neighborhood of a in a single disk. It also implies that for any point b 6 T
each leaf intersects some neighborhood of b in a finite number of disjoint
half disks.
OPEN BOOK STRUCTURES AND UNICITY OF MINIMAL SUBMANIFOLDS 705
We will now consider a smooth immersion ip from a connected
compact orientable bordered surface M into N for which (/^(F) = 9M.
Such a (p induces a 2 chain [up] (= y^M) in -/V.
Suppose U is a tubular neighborhood ofFin TV. For almost all positive
numbers e < dist(F,c)U), the slice
[^=9([^L.{x:dist(x,T)<e})
is a finite sum of oriented smooth curves. Here the 2 chain [p\ L A denotes
the restriction of [(?] to A in the sense that
([(^] L A)(a) = M(a • char./cn(A))
for any differential 2 form a on TV. For almost all 0 < 6 < e, the 1 chains
[(p}s and [<^]e are homologous in U — F because
bL - [̂  = <9(M L ̂  : 6 < dist(:r,F) < e}).
More generally, we say any two such 2 chains [up] and ['0] are iinic homologous
at T if [(^]g and [^\e are homologous in U — F for some positive number
e < dist(F,<9E/). Note that, by orienting N and F, the link homology
class at F may be described by associating, to each component of F,
a pair of integers, one being a boundary multiplicity and the other a
linking multiplicity (see e.g. §2.2 below) corresponding to generators of
the homology of a torus link of the component.
2.1. THEOREM. — Suppose M is an annulus and TV,F,£ and
(p : M —> N are as above with (p a minimal immersion that maps distinct
components of9M onto distinct components off. If(p is link homologous
at F to an oriented leaf of C then y?(M) — F is a leaf of C.
Proof. — Let 7 be a loop on M that generates the fundamental
group and lies arbitrarily close to a component of 9M.
Then the link homology condition implies that, in TV, the correspond-
ing I—chain (p * 7 is homologous to a I—chain supported in a single leaf L
of C. Thus, in the notation of the proof of 1.4,
ip ^ ( j j = I ^==0,
J^ J(p^^
which implies as before, that (p{M — 9M) lies entirely in a single leaf L
of C. The image (p(M — 9M) is open in L because y? is an immersion and
closed in L because ^-l(^) = 9M' Thus ^(M) - F = L. D
706 R. HARDT & H. ROSENBERG
2.2. Immersed disks in S3. — Recall the notation of 1.2.
Any immersed minimal disk (p : D —>• S3 with (^"^(Ci) = <9J9 must
be De or a multiple covering of De for some 0 € [0,27r).
In fact, if \p : D —> S3 were such an immersion, then the closed form
y?*di/, (which is, as in the proof of 2.1, smooth up to 9D), would be exact
because ^(D, R) = 0. Then, as in the proof of 2.1, the image of y? would
have to lie in some leaf De.
2.3. Some minimal annuli in S3. — Next we consider, for each
0 e [0,27r) and each pair, p, q of nonzero integers with |p[, \q\ relatively
prime, the set
A^ =S3n{aIg(ypzq)=0}.
As in [L] one can easily verify that A^9 — (C\ U £2) is an (open) embedded
minimal annulus. Moreover, by suitably orienting AS'9, C\ and (72, we
obtain a 2—chain A^'9 satisfying
<9A^=gCi+pC2.
Any immersed minimal annulus ̂  : A —> S3, with (p~l(Ci UC^) == <9A
and with
<^[{a € A : 0 < dist(a, QA) < e},
being injective-for some e > 0, mus^ be embedded with image equal to A^
for some 0, p, q-
To see this, we may choose a small positive 6 < e so that each torus
T i = { x ( ^ S 3 :dist(x,Cz)=6}
intersects y?(A) transversely in a simple closed curve 7^ for i = 1,2. We
fix a usual pair of generators for the homologies of the T{ with the first of
each pair corresponding to a shortest geodesic on 7^. We find that 71 is
homotopic in T\ to a (p,q) geodesic, while 72 is homotopic in T^ to a (q,p)
geodesic, for some nonzero integers p, q with |p|, \q\ relatively prime. It also
follows that (f> is link homologous at 6^ U 62 to ±A^'9 for any 0 € [0,27r).
Thus we may apply Theorem 2.1 with
M=A, 7 v = S 3 , r=(7iUC2, and/;={A^:0€ [0,27r)}. D
OPEN BOOK STRUCTURES AND UNICITY OF MINIMAL SUBMANIFOLDS 707
3. Remarks and questions.
1. The sets A^1 U C^ are embedded minimal Mobius bands with
boundary C\ (and with waist 62). This shows the necessity of the ori-
entability assumption in Theorem 1.1. Are these, along with the disks DQ
of §1.2, the only embedded minimal surfaces in S3 with boundary C\ ?
2. The minimal annuli A^9 of §2.3 are embedded in S2 x S1. This
shows the necessity of the assumption in Theorem 1.1 on the vanishing
of either ^(M.IR) or Hn(N,R). Are these the only embedded minimal
surfaces in S3 with boundary I\ U Fa?
3. Is the near-boundary injectivity assumption necessary in §2.2 and
§2.3? In particular, is there an oriented immersed minimal annulus in S3
•^ r\ r\
whose (2—chain) boundary is 2(7i 4- 2(72 other than 2A^ ?
4. Are there analogues of 2.2 and 2.3 with F being two great circles
in S3 that are a constant distance apart, other than Tr/2?
5. What complex polynomials give rise to open book structures of
the sphere?
BIBLIOGRAPHY
[GT] D. GILBARG & N. TRUDINGER, Elliptic Partial Differential Equations of Second
Order, Springer-Verlag, Berlin-Heidelberg-New York, 1977.
[H] J. HASS, Minimal surfaces in foliated manifolds, Comm. Math. Helv., 61 n° 1
(1986), 1-32.
[HS] R. HARDT & L. SIMON, Boundary regularity and embedded solutions for the
oriented Plateau problem, Ann. of Math., 110 (1979), 439-486.
[L] H.B. LAWSON, Complete minimal surfaces in 5^, Ann. of Math., 92 (1970), 335-
374.
[M] F. MORGAN, On nniteness of the number of stable minimal hypersurfaces with a
fixed boundary, Indiana U. Math. J., 35 (1986), 779-833.
708 R. HARDT & H. ROSENBERG
[Sh] M. SHIFFMAN, On surfaces of stationary bounded by two circles, or convex curves
in parallel planes, Ann. of Math., 3 (1956), 77-90.
[So] B. SOLOMON, On the Gauss map of an area-minimizing hypersurface J Diff
Geom., 24 (1984), 221-232. , . .
Manuscrit recu Ie 8 fevrier 1990,
revise Ie 25 juin 1990.
R. HARDT,
Mathematics Department
Rice University
Houston, Texas 77251 (USA).
&
H. ROSENBERG,
U.F.R. de Mathematiques
Universite Paris VII
Paris (France).


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