POSITIVE RICCI CURVATURE METRICS HAVE
INFINITELY MANY MINIMAL HYPERSURFACES
FERNANDO C. MARQUES AND ANDRÉ NEVES
Abstract. In the early 80’s, Yau conjectured that any compact Riemannian 3-manifold admits an infinite number of closed immersed minimal surfaces. We use Min-Max theory to show that every compact Riemannian (n + 1)-manifold with positive Ricci curvature and 2 ≤ n ≤ 6
admits an infinite number of closed embedded smooth minimal hypersurfaces.
In the last section we propose some open problems related with the
geometry of these minimal hypersurfaces.
1. Introduction
A foundational question in Differential Geometry, asked by Poincaré [26],
is whether every closed surface admits a closed geodesic.
If the surface is not simply connected then we can minimize length in a
homotopy class and produce a closed geodesic. Hence the question becomes
considerably more interesting on a sphere and the first breakthrough was
due to Birkhoff [6] who used min-max methods to find a closed geodesic
for any metric on a 2-sphere. Later, in another notable work, Lusternick
and Schnirelmann [22] showed that every metric on a 2-sphere admits three
simple (i.e., no self-intersections) closed geodesics (see also [3, 12, 20, 17, 21,
30]) and this suggests the question whether we can find an infinite number
of closed geodesics in any surface. If the genus of the surface is positive it
is not hard to find infinitely many closed geodesics and Ballmann showed
that if the surface not a sphere, projective plane, or Klein bottle, then one
can find infinitely many simple closed geodesics for every metric. The case
of the sphere was finally settled by Franks [11] and Bangert [5], whose work
implies that every metric on a sphere admits an infinite number of closed
geodesics. Later, Hingston [16] estimated the number of closed geodesics of
length at most L when L is very large.
Likewise, one can ask whether every manifold admits a minimal hypersurface and Pitts [25], using min-max methods, showed that every compact
Riemannian (n + 1)-manifold with n ≤ 5 admits a smooth closed embedded
minimal hypersurface and later Schoen-Simon [27] extended this result for
The first author was partly supported by CNPq-Brazil, FAPERJ, Math-Amsud and
the Stanford Department of Mathematics. The second author was partly supported by
Marie Curie IRG Grant and ERC Start Grant.
1
2
FERNANDO C. MARQUES AND ANDRÉ NEVES
every dimension and showed the existence of a closed embedded minimal
hypersurfaces with a singular set of Hausdorff codimension at least 7.
Motivated by these results, Yau conjectured in [32] (first problem in the
Minimal Surfaces section) that every compact Riemannian 3-manifold admits an infinite number of closed immersed minimal surfaces.
The purpose of this paper is to show
1.1. Theorem. Let (M, g) be a compact orientable Riemannian (n + 1)manifold with 2 ≤ n ≤ 6. Assume that one of the two conditions below is
met:
• The Ricci curvature of g is positive;
• (M, g) contains no closed embedded minimal stable hypersurfaces.
Then (M, g) admits an infinite number of distinct closed embedded minimal
hypersurfaces.
1.2. Remark. It is well known that metrics with positive Ricci curvature
admit no closed orientable embedded minimal stable hypersurface but can
have, nonetheless, closed non-orientable minimal stable hypersurfaces, the
simplest example being the minimal projective plane in RP3 with the round
metric.
The existence of the minimal hypersurfaces in Theorem 1.1 comes from
Almgren-Pitts Min-Max Theory, which we now very briefly explain. Guth
in [14] constructed, for every p ∈ N, a p-cycle in Zn (M ; Z2 ) whose homotopy classes are distinct for infinitely many p ∈ N. Almgren-Pitts Theory
applies to such homotopy classes and so we find closed embedded minimal
hypersurface (with possible multiplicities) Σp for every p ∈ N. Under the
conditions of Theorem 1.1 we show that infinitely many of those minimal
hypersurfaces Σp have to be geometrically distinct, i.e., not multiples of each
other.
In [18, 19] Kapouleas describes in detail an alternative approach to construct an infinite number of minimal embedded surfaces in a three-manifold
with a bumpy metric by either desingularizing two intersecting minimal surfaces or by doubling an existing unstable minimal surface. Note that for S 3
with a metric of positive Ricci curvature, White [31] showed the existence
of two distinct embedded minimal spheres, which must intersect by [15] and
are necessarily unstable.
The minimal hypersurfaces obtained via our construction have, conjecturally, area tending to infinity and thus should be different from the minimal surfaces proposed by Kapouleas.
2. Almgren-Pitts Min-Max Theory
Let (M, g) be an orientable compact Riemannian (n + 1)-manifold with
boundary ∂M . We assume M is isometrically embedded in RL .
POSITIVE RICCI CURVATURE METRICS HAVE INFINITELY MANY MINIMAL HYPERSURFACES
3
Let X be a cubical subcomplex of the m-dimensional cube I m = [0, 1]m .
Every polyhedron is homeomorphic to the support of some cubical subcomplex X [7, Chapter 4].
We describe the necessary obvious modifications to the Almgren-Pitts
Min-Max Theory so that the m-dimensional cube I m is replaced by X as
the parameter space.
2.1. Basic notation. The spaces we will work with in this paper are:
• the space Ik (M ; Z2 ) of k-dimensional modulo 2 currents in RL with
support contained in M (see [9, 4.2.26] for more details);
• the space Zk (M ; Z2 ) (or Zk (M, ∂M ; Z2 )) of integral currents T ∈
Ik (M ; Z2 ) with ∂T = 0 (or spt(∂T ) ⊂ ∂M ), respectively;
• the closure Vk (M ), in the weak topology, of the space of k-dimensional
rectifiable varifolds in RL with support contained in M . Integral
rectifiable k-dimensional varifolds with support contained in M are
denoted by IV k (M ).
Given T ∈ Ik (M ), we denote by |T | and ||T || the integral varifold and
Radon measure in M associated with T , respectively; given V ∈ Vk (M ),
||V || denotes the Radon measure in M associated with V . If U ⊂ M is an
open set of finite perimeter, we abuse notation and denote the associated
current in In+1 (M ; Z2 ) by U .
The above spaces come with several relevant metrics. The flat metric and
mass of T ∈ Ik (M ; Z2 ), denoted by F(T ) and M(T ), are defined in [9, page
423] and [9, page 426] respectively. The F-metric on Vk (M ) is defined in
Pitts book [25, page 66] and induces the varifold weak topology on Vk (M ).
Finally, the F-metric on Ik (M ; Z2 ) is defined by
F(S, T ) = F(S − T ) + F(|S|, |T |).
We assume that Ik (M ; Z2 ), Zk (M ; Z2 ), and, Zk (M, ∂M ; Z2 ) have the
topology induced by the flat metric. When endowed with the topology of the
mass norm, these spaces will be denoted by Ik (M ; M; Z2 ), Zk (M ; M; Z2 ),
and Zk (M, ∂M ; M; Z2 ), respectively. The space Vk (M ) is considered with
the weak topology of varifolds.
Given A, B ⊂ Vk (M ), we also define
F(A, B) = inf{F(V, W ) : V ∈ A, W ∈ B}.
For each j ∈ N, I(1, j) denotes the cube complex on I 1 whose 1-cells and
0-cells (those are sometimes called vertices) are, respectively,
[0, 3−j ], [3−j , 2 · 3−j ], . . . , [1 − 3−j , 1]
and
[0], [3−j ], . . . , [1 − 3−j ], [1].
We denote by I(m, j) the cell complex on I m :
I(m, j) = I(1, j) ⊗ . . . ⊗ I(1, j)
(m times).
α = α1 ⊗ · · · ⊗P
αm is a q-cell of I(m, j) if and only if αi is a cell of I(1, j)
for each i, and m
i=1 dim(αi ) = q. We often abuse notation by identifying a
q-cell α with its support: α1 × · · · × αm ⊂ I m .
4
FERNANDO C. MARQUES AND ANDRÉ NEVES
The cube complex X(j) is the union of all cells of I(m, j) whose support
is contained in some cell of X. We use the notation X(j)q to denote the set
of all q-cells in X(j). Two vertices x, y ∈ X(j)0 are adjacent if they belong
to a common cell in X(j)1 .
Given i, j ∈ N we define n(i, j) : X(i)0 → X(j)0 so that n(i, j)(x) is the
element in X(j)0 that is closest to x (see [25, page 141] or [23, Section 7.1]
for a precise definition).
Given a map φ : I(n, j)0 → Zn (M ; Z2 ), we define the fineness of φ to be
f (φ) = sup {M(φ(x) − φ(y)) : x, y adjacent vertices ∈ X(j)0 } .
The reader should think of the notion of fineness as being a discrete
measure of continuity with respect to the mass norm.
2.2. Homotopy notions. Let φi : X(ki )0 → Zn (M ; Z2 ), i = 1, 2. We say
that φ1 is X-homotopic to φ2 in (Zn (M ; M; Z2 ), Φ0 ) with fineness δ if we
can find k ∈ N and a map
ψ : I(1, k)0 × X(k)0 → Zn (M ; Z2 )
such that
(i) f (ψ) < δ;
(ii) if i = 1, 2 and x ∈ X(k)0 , then
ψ([i − 1], x) = φi (n(k, ki )(x));
Instead of considering continuous maps from X into Zn (M ; M; Z2 ), AlmgrenPitts consider sequences of discrete maps into Zn (M ; Z2 ) with fineness tending to zero.
2.3. Definition. A
(X, M)-homotopy sequence of mappings into Zn (M ; M; Z2 )
is a sequence of mappings S = {φi }i∈N ,
φi : X(ki )0 → Zn (M ; Z2 ),
such that φi is X-homotopic to φi+1 in Zn (M ; M; Z2 ) with fineness δi and
(i) limi→∞ δi = 0;
(ii) sup{M(φi (x)) : x ∈ X(ki )0 , i ∈ N} < +∞.
The next definition explains what it means for two distinct homotopy
sequences of mappings into Zn (M ; M; Z2 ) to be homotopic.
2.4. Definition. Let S 1 = {φ1i }i∈N and S 2 = {φ2i }i∈N be (X, M)-homotopy
sequences of mappings into Zn (M ; M; Z2 ). We say that S 1 is homotopic
with S 2 if there exists {δi }i∈N such that
• φ1i is X-homotopic to φ2i in Zn (M ; M; Z2 ) with fineness δi ;
• limi→∞ δi = 0.
POSITIVE RICCI CURVATURE METRICS HAVE INFINITELY MANY MINIMAL HYPERSURFACES
5
The relation “is homotopic with” is an equivalence relation on the set of
all (X, M)-homotopy sequences of mappings into Zn (M ; M; Z2 ). We call
the equivalence class of any such sequence an (X, M)-homotopy class of
mappings into Zn (M ; M; Z2 ). We denote by [X, Zn (M ; M; Z2 )]# the set of
all equivalence classes.
Analogous homotopic definitions hold if we consider discrete maps where
the fineness is measured with respect to the flat metric instead of the mass
norm or if we consider continuous maps in the flat topology Φ : X →
Zn (M ; Z2 ).
2.5. Width. Given Π ∈ [X, Zn (M ; M; Z2 )]# , let
L : Π → [0, +∞]
be defined by
L(S) = lim sup max{M(φi (x)) : x ∈ dmn(φi )},
where S = {φi }i∈N .
i→∞
Note that L(S) is the discrete replacement for the maximum area of a continuous map into Zn (M ; M; Z2 ).
Given S = {φi }i∈N ∈ Π we also consider the following compact subset
K(S) of Vn (M ) given by
K(S) = {V : V = lim |φij (xj )| as varifolds, for some increasing
j→∞
sequence {ij }j∈N and xj ∈ dmn(φij )},
which can be thought of as the discrete replacement for the range of a
continuous map into Zn (M ; M; Z2 ).
2.6. Definition. The width of Π is defined by
L(Π) = inf{L(S) : S ∈ Π}.
We say S ∈ Π is a critical sequence for Π if
L(S) = L(Π)
and the critical set C(S) of a critical sequence S ∈ Π is given by
C(S) = K(S) ∩ {V : ||V ||(M ) = L(S)}.
2.7. Min-Max Theorem. Consider Π ∈ [X, Zk (M ; M; Z2 )]# .
2.8. Proposition. Assume M has no boundary. There exists a critical sequence S ∗ ∈ Π. For each critical sequence S ∗ , there exists a critical sequence
S ∈ Π such that
• C(S) ⊂ C(S ∗ );
• every Σ ∈ C(S) is a stationary varifold.
The sequence S is obtained from a pull-tight procedure applied to S ∗ .
The proof is essentially the same of Theorem 4.3 of [25].
To formulate the regularity part of Almgren-Pitts Min-Max Theory, we
need to introduce the notion of almost minimizing varifold.
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FERNANDO C. MARQUES AND ANDRÉ NEVES
2.9. Definition. V ∈ Vn (M ) is Z2 almost minimizing in an open set U ⊂ M
if and only if for all ε > 0 there is δ > 0 and T ∈ Zn (M, M \ U ; Z2 ) with
F(V, |T |) < ε and satisfying the following property:
If {Ti }qi=0 is a sequence in Zn (M, M \ U ; Z2 ) with
• T0 = T and spt(T − Ti ) ⊂ U for all i = 1, . . . , q;
• M(Ti − Ti−1 ) ≤ δ for all i = 1, . . . , q;
• M(Ti ) ≤ M(T ) + δ for all i = 1, . . . , q;
then M(Tm ) ≥ M(T ) − ε.
Loosely speaking V ∈ Vn (M ) is almost minimizing in U if every continuous deformation supported in U that decreases area by more than ε must
have passed through a stage where area increased by more than δ.
Given real numbers 0 < s < r, let A(p, s, r) = {x ∈ RL : s < |x − p| < r}.
2.10. Definition. The varifold V ∈ Vn (M ) is Z2 almost minimizing in annuli if for each p ∈ M , there exists r = r(p) > 0 such that V is Z2 almost
minimizing in M ∩ A(p, s, r) for all 0 < s < r.
If V ∈ Vn (M ) is Z2 almost minimizing in annuli and stationary in M ,
then V ∈ IV n (M ) by Lemma 3.13 of [25].
The next theorem follows by simple adaptation of Theorem 4.10 of [25].
2.11. Theorem. Assume M has no boundary and let Π ∈ [X, Zn (M ; M; Z2 )]# .
Then there exists V ∈ IV n (M ) such that the following three statements are
true:
(1) ||V ||(RL ) = L(Π),
(2) V is stationary in M ,
(3) V is Z2 almost minimizing in annuli.
Moreover, if S ∗ is a critical sequence then we can choose V ∈ C(S ∗ ).
The key combinatorial result of Almgren-Pitts which implies both Theorem 4.10 of [25] and the theorem above is stated in Section 2.13.
The regularity of almost minimizing integral varifolds was first done by
Pitts in [25, Section 7] when n ≤ 5 and extended by Schoen-Simon to every
dimensionan in [27, Theorem 4]. Schoen-Simon work with integer coefficients
but, as we explain below, the arguments extend to Z2 coefficients.
2.12. Theorem. Assume M has no boundary. Let 0 6= V ∈ IV n (M ) be
Z2 almost minimizing in annuli and stationary in M , n ≤ 6. Then V is a
smooth closed embedded minimal hypersurface with possible multiplicities.
Proof. Let A be the collection of all nontrivial V ∈ IV n (M ) that are Z2
almost minimizing in annuli and stationary in M .
Pitts shows in [25, Theorem 3.11] that for any given p ∈ spt||V ||, there
exists r(p) > 0 such that for any 0 < s < t < r(p) there exists a replacement
varifold V ∗ ∈ A with
(i) ||V ∗ ||(M ) = ||V ||(M ),
(ii) V ∗ xGn (M \ A(p, s, t)) = V xGn (M \ A(p, s, t)),
POSITIVE RICCI CURVATURE METRICS HAVE INFINITELY MANY MINIMAL HYPERSURFACES
7
(iii) V ∗ xGn (M ∩ A(p, s, t)) = (limj→∞ |Tj |)xGn (M ∩ A(p, s, t)),
with {Tj } ⊂ In (M, Z2 ), {M(Tj )} bounded independently of j, spt(∂Tj ) ∩
A(p, s, t) = ∅, Tj locally area minimizing in M ∩ A(p, s, t) and |Tj | stable in
M ∩ A(p, s, t).
It follows from the regularity theory for mod 2 area minimizing flat chains
in [24, Regularity Theorem 2.4] (all conditions are satisfied by Remark 1 in
[24, page 249]) that there exists a smooth minimal hypersurface Mj properly
embedded in A(p, s, t) such that
(spt Tj ) ∩ A(p, s, t) = M j ∩ A(p, s, t).
Since A(p, s, t) is simply connected, Mj is orientable for each j. Therefore
spt ||V ∗ || ∩ A(p, s, t) = M ∩ A(p, s, t),
where M is an orientable stable smooth minimal hypersurface exactly like in
Schoen-Simon [27, page 789]. The proof that spt||V || is a smooth embedded
minimal hypersurface proceeds just like in the proof of [27, Theorem 4]. 2.13. Almost minimizing varifolds. We now describe a key combinatorial result of Pitts [25], which was based on a previous construction of
Almgren [2].
0
With m ∈ N fixed, set L0 = 3m and c = LL
0 .
Consider a finite set of stationary varifolds B ⊂ Vn (M ) such that
• there is A0 so that ||V ||(M ) = A0 for all V ∈ B;
• for all V ∈ B we can find p ∈ spt ||V||, and positive numbers
{ri }ci=1 , {si }ci=1 with
ri − 2si > 2(ri+1 + 2si+1 ),
i = 1, . . . , c − 1,
rc − 2sc > 0
so that V is not Z2 almost minimizing in ai (V ) = A(p, ri − si , ri + si )
for all i = 1, . . . , c.
Set s(V ) = mini {si } and s = min{s(V ) : V ∈ B}.
From the second property we can find ε1 > 0 so that for each V ∈ B, each
j = 1, . . . , c, each δ > 0, and each T ∈ Zn (M ; Z2 ) with F(V, |T |) < ε1 , there
is a sequence {Ti }qi=0 in Zn (M ; Z2 ) such that
• T0 = T and spt(T − Ti ) ⊂ aj (V ) for all i = 1, . . . , q;
• M(Ti − Ti−1 ) ≤ δ for all i = 1, . . . , q;
• M(Ti ) ≤ M(T ) + δ for all i = 1, . . . , q;
but M(Tq ) < M(T ) − ε1 .
The theorem we now state corresponds to Part 5 of Theorem 4.10 in [25,
page 165]. The idea is that if we have a discrete map φ so that its elements
of higher mass are close enough to B, then we can find another map ψ
homotopic to φ so that the maximum of the masses is decreased by a fixed
value. The proof is an elegant (but long) combinatorial argument given from
Part 6 to Part 20 in [25, page 165–page 174]. Pitts’ parameter spaces are
cell complexes I(m, k) but everything holds when the parameter spaces are
cubical subcomplexes of I(m, k).
8
FERNANDO C. MARQUES AND ANDRÉ NEVES
2.14. Theorem. (Almgren-Pitts Combinatorial Construction)Assume M
has no boundary and consider a set B as described above.
Suppose we have 0 < ε2 < ε1 /2 and, for some X a cubical subcomplex of
I(m, k), a map φ : X0 → Zn (M ; Z2 ) such that
(a)
M(φ(x)) ≥ A0 − 2ε2 =⇒
F(|φ(x)|, V ) < ε1 /2 for some V ∈ B;
(b) M(φ(x) − φ(y)) ≤ ε2 whenever σ is a cell of X and x, y ∈ σ0 ;
(c) whenever σ is a cell of X and x, y ∈ σ0 , there is Q ∈ In+1 (M ; Z2 )
such that
∂Q = φ(x) − φ(y)
and
M(Q) ≤ M(φ(x) − φ(y));
(d) mf (φ) < ε1 /2.
Set δ = 2mf (φ)(1 + 4(L0 − 1)s−1 ).
There is
ψ : X(l)0 → Zn (M ; Z2 )
X-homotopic to φ in Zn (M ; M; Z2 )
with finesses smaller than δ and such that, with n = n(0, l),
• ψ(x) = φ(n(x)) if M(φ(n(x))) < A0 − 2ε2 ;
• M(ψ(x)) ≤ M(φ(n(x))) + cδ if
A0 − 2ε2 ≤ M(φ(n(x))) < A0 − ε2 ;
• M(ψ(x)) ≤ M(φ(n(x))) + cδ − ε2 if M(φ(n(x))) ≥ A0 − ε2 ;
In particular
max {M(φ(x))} ≤ A0 +
x∈X0
ε2
=⇒
2
max {M(ψ(x))} ≤ A0 + cδ −
x∈X(l)0
ε2
.
2
2.15. Remark. From Corollary 1.14 of [1] there is a constant νM depending
only on M so that condition (c) in the theorem is implied by condition (b)
provided ε2 ≤ νM .
3. Almgren’s homomorphism and Interpolation results
We describe some of the maps defined by Almgren in [1, Section 3]1
1Almgren uses the group Z instead of Z and the domain [0, 1] instead of S 1 but
2
everything extends to this setting.
POSITIVE RICCI CURVATURE METRICS HAVE INFINITELY MANY MINIMAL HYPERSURFACES
9
3.1. Discrete setting. Suppose we have a map φ : I(1, k)0 → Zn (M, ∂M ; Z2 )
so that φ([0]) = φ([1]) and
F(φ(aj ), φ(aj+1 )) ≤ νM,∂M
for all j = 0, . . . , 3k − 1,
where aj = [j3−k ] and νM,∂M , defined in [1, Theorem 2.4], is a small constant
that depends only on M . This condition ensures us the existence of ρ =
ρ(M ) and an isoperimetric choice Aj ∈ In+1 (M ; Z2 ) such that
∂Aj − (φ(aj+1 ) − φ(aj )) ∈ In (∂M ; Z2 ) and M(Aj ) < ρF(φ(aj ), φ(aj+1 ))
P k −1
Aj ∈ Zn+1 (M, ∂M ; Z2 ) and so defines
for all j = 0, . . . , 3k − 1. Hence 3j=0
an element in integral homology class (see [9, Section 4.4])


k −1
3X
#
(φ) = 
FM,∂M
Aj  ∈ Hn+1 (M, ∂M ; Z2 ).
j=0
The following simple lemma shows that the isoperimetric choice is unique.
3.2. Lemma. νM,∂M can be chosen so that if Cj ∈ In+1 (M ; Z2 ) has
M(Cj ) ≤ νM,∂M
and
∂Cj − (φ(aj+1 ) − φ(aj )) ∈ In (∂M ; Z2 ),
then Aj = Cj .
Proof. We have spt(∂(Aj − Cj )) ⊂ ∂M and so, by the [28, Constancy Theorem 26.27], Aj − Cj = kM , where k = 0, 1. Furthermore
M(Aj ) ≤ ρF(φ(aj ), φ(aj+1 )) ≤ ρνM,∂M .
Thus M(Aj − Cj ) ≤ (ρ + 1)νM,∂M and so the result follows if (ρ + 1)νM,∂M
is strictly smaller than M(M ).
Almgren [1] shows that if φ0 is (S 1 , F)-homotopic to φ with fineness
#
#
(φ0 ).
(φ) = FM,∂M
smaller than νM,∂M , then FM,∂M
3.3. Continuous setting. Assume ∂M = ∅ for simplicity.
Given a continuous map in the flat topology
Φ : S 1 → Zn (M ; Z2 ),
consider k large so that,
(1)
F(Φ(e2πix ), Φ(e2πiy )) ≤ νM
for all x, y in common cell of I(1, k).
Set φ : I(1, k)0 → Zn (M ; Z2 ) so that φ([x]) = Φ(e2πix ) and we define
#
FM (Φ) = FM
(φ).
We have that FM (Φ) does not depend on k chosen provided (1) is satisfied
and that if Φ is homotopic to another continuous map Φ0 : S 1 → Zn (M ; Z2 ),
then FM (Φ) = FM (Φ0 ).
Continuous map in the flat topology Φ : S 1 → Zn (M ; Z2 ) with FM (Φ) 6= 0
are called sweepouts.
10
FERNANDO C. MARQUES AND ANDRÉ NEVES
3.4. Area bounds for sweepouts. Given p ∈ M , let Br (p) denote the
geodesic ball of radius r centered at p. .
3.5. Proposition. Assume M has no boundary. There is α0 = α0 (M )
and r0 = r0 (M ) so that for all x ∈ M , r ≤ r0 , and sweepout Φ : S 1 →
Zn (M ; Z2 ), we have
sup M(Φ(θ)x(Br (x)) ≥ α0 rn .
θ∈S 1
Proof. Scaling considerations show that we can find α1 = α1 (M ) and r1 =
r1 (M ) so that, recalling the constant νBr (x),∂Br (x) given by Lemma 3.2, we
have
νBr (x),∂Br (x) ≥ α1 rn+1 for all x ∈ M, r ≤ r1 .
With α0 and r0 to be chosen later, pick k large so that
F(Φ(e2πix ), Φ(e2πiy )) ≤ α0 rn+2
for all x, y in common cell of I(1, k)
and set
φ : I(1, k)0 → Zn (M ; Z2 ) φ([x]) = Φ(e2πix ).
With α0 , r0 suitably small we can find, for all j = 0, . . . , 3k − 1, Qj in
In+1 (M ; Z2 ) such that with aj = [j3−k ] we have
∂Qj = φ(aj+1 ) − φ(aj )
and M(Qj ) ≤ ρF(φ(aj+1 ) − φ(aj )),
where ρ = ρ(M ) is defined in Section 3.1. The fact that Φ is a sweepout
implies that we can also assume that
(2)
k −1
3X
Qj = M
in In+1 (M ; Z2 ).
j=0
We can find r/2 ≤ s ≤ r (by [28, Lemma 28.5]) so that
φ(aj )xBs (x) ∈ Zn (Bs (x), ∂Bs (x); Z2 ),
Lj = ∂ (Qj xBs (x)) − ∂Qj xBs (x) ∈ In (∂Br (x); Z2 ),
and
2
M(Lj ) ≤ M(Qj ) ≤ 2ρα0 rn+1 ,
r
k
for all j = 0, . . . , 3 − 1.
Let
φ̄ : I(1, k)0 → Zn (Bs (x), ∂Bs (x); Z2 ),
φ̄(x) = φ(x)xBs (x).
We have, assuming α0 (rρ + 2) < α1 ,
F(φ̄(aj+1 ) − φ̄(aj )) ≤ M(Qj xBs (x)) + M(Lj ) ≤ α0 (rρ + 2)rn+1
≤ α1 rn+1 ≤ νBr (x),∂Br (x)
POSITIVE RICCI CURVATURE METRICS HAVE INFINITELY MANY MINIMAL HYPERSURFACES
11
and so Lemma 3.2 implies that


k −1
3X
FB#s (x),∂Bs (x) (φ̄) = 
Qj xBs (x) = [M xBs (x)] = [Bs (x)].
j=0
Claim: If M(φ̄(x)) ≤ α0 rn for all x ∈ I(1, k)0 then
FB#s (x),∂Bs (x) (φ̄) = 0,
which is a contradiction.
From [1, Proposition 1.22] we see that if α0 and r0 are chosen small enough
then we can find Sj ∈ In+1 (Bs (x)) for all j = 0, . . . , 3k so that
∂Sj − φ̄(aj ) ∈ In (∂Bs (x))
and M(Sj ) ≤ ν2 M(φ̄(aj ))(n+1)/n ,
(n+1)/n
for some constant ν2 = ν2 (n). Thus, assuming 2ν2 α0
< α1 , we obtain
from Lemma 3.2 that Aj = Sj+1 − Sj is an isoperimetric choice and so
FB#r ,∂Br (φ̄) = S3k − S0 = 0.
3.6. Interpolation results. Let X be a cubical subcomplex of some I m =
[0, 1]m .
Given a continuous map Φ : X → Z2 (M ; Z2 ), with respect to the flat
topology, we say that Φ has no mass concentration if
lim sup{||Φ(x)||(Br (p)) : x ∈ X, p ∈ M } = 0.
r→0
In Lemma 15.2 of [23] we showed
3.7. Lemma. Every continuous map Φ : X → Z2 (M ; M; Z2 ) in the mass
topology with
sup{M(Φ(x)) : x ∈ X} < +∞
has no mass concentration.
We now state the two main interpolation theorems we will use. The
first one passes from a continuous function in the flat metric to a sequence
of discrete maps with finesses tending to zero and the second one from a
discrete map with small finesses to a continuous map in the mass topology.
Theorem 3.1. Let Φ : X → Zn (M ; Z2 ) be a continuous map in the flat
topology with no mass concentration and such that
sup{M(Φ(x)) : x ∈ X} < +∞.
There is a sequence of positive numbers {δi }i∈N tending to zero and
φi : X(ki )0 → Zn (M ; Z2 )
so that
(i) S = {φi }i∈N is a (X, M)-homotopy sequence of mappings into Zn (M ; M; Z2 )
and f (φi ) < δi ;
12
FERNANDO C. MARQUES AND ANDRÉ NEVES
(ii)
sup{F(φi (x) − Φ(x)) : x ∈ X(ki )0 } ≤ δi ;
(iii)
sup{M(φi (x)) : x ∈ X(ki )0 } ≤ sup{M(Φ(x)) : x ∈ X} + δi .
This theorem follows from Theorem 13.1 in [23].
Theorem 3.2. There are constants C0 = C0 (M, X) and δ0 = δ0 (M, X) so
that if
φ : X(k)0 → Zn (M ; Z2 )
has f (φ) < δ, then there is a continuous map in the mass norm
Φ : X → Zn (M ; M; Z2 )
with
(i) Φ(x) = φ(x) for all x ∈ X(k)0 ;
(ii) If α is some j-cell in X(k)j , then Φ restricted to α depends only on
the values of φ assumed in α0 ;
(iii)
sup{M(Φ(x) − Φ(y)) : x, y lie in a common cell of X(k)} ≤ C0 f (φ).
This theorem follows from Theorem 14.1 in [23].
4. Min-Max Volumes
From now on we assume that the (n + 1)-dimensional Riemannian manifold (M, g) is compact and closed
4.1. Definition. We say that a continuous map Φ : X → Zn (M ; Z2 ) detects
a p-class if
• X is a cubical subcomplex of some I m = [0, 1]m ;
• there is λ ∈ H 1 (X; Z2 ) so that for every closed curve γ ⊂ X with
λ(γ) 6= 0 then Φ ◦ γ is a sweepout;
• the cup product λp = λ ^ . . . ^ λ is non zero in H p (X; Z2 ).
If γ1 , γ2 are two curves homotopic to each other in X then Φ ◦ γ1 is a
sweepout if and only if Φ ◦ γ2 is a sweepout. One could also show that the
same property holds under the weaker condition that γ1 and γ2 represent
the same element in H1 (X) but we will not need that property.
The notion of detecting a p-class is invariant under homotopies and small
perturbations.
4.2. Lemma. Given m ∈ N there is µ = µ(m, M ) so that given Φi : X →
Zn (M ; Z2 ), i = 1, 2 continuous maps such that
sup{F(Φ1 (x) − Φ2 (x)) : x ∈ X} ≤ µ,
where X is a cubical subcomplex of I m = [0, 1]m , then Φ1 detects a p-class
if and only if Φ2 detects a p-class.
POSITIVE RICCI CURVATURE METRICS HAVE INFINITELY MANY MINIMAL HYPERSURFACES
13
Proof. In [1, Theorem 8.2] it is shown the existence of µ so that given Φ1 , Φ2
as above then Φ1 , Φ2 are homotopic maps in the flat topology. Almgren’s
result is stated when the parameter space is I m but everything extends to
the case when X is a cubical subcomplex of I m . As a result we have that
for every closed curve γ ⊂ X that
FM (Φ1 ◦ γ) = FM (Φ2 ◦ γ)
and so the result follows.
We define Pp to be the set of all continuous maps Φ : X → Zn (M ; Z2 )
that detect a p-class and that have no mass concentration.
Following Guth [14, Appendix 2] we consider
4.3. Definition. The p-width of M is defined as
ωp (M ) = inf sup M(Φ(x)).
Φ∈Pp x∈X
There is no upper bound for the dimension of the parameter spaces of all
Φ ∈ Pp and so one cannot conclude that ωp (M ) can be represented by the
area of a stationary varifold whose support is a smooth embedded minimal
hypersurface.
4.4. Proposition. Assume 2 ≤ n ≤ 6 and suppose there is only a finite
number of smooth embedded minimal hypersurfaces in (M, g).
For every p ∈ N there is δ = δ(M, p) so that if Φ ∈ Pp has
sup{M(Φ(x)) : x ∈ X} ≤ ωp + δ,
then with
• S the (X, M)-homotopy sequence of mappings into Zn (M ; M; Z2 )
given by Theorem 3.1 (i) applied to Φ;
• Π the (X, M)- homotopy class of mappings into Zn (M ; M; Z2 ) generated by S;
we have L(Π) = ωp (M ).
Proof. By hypothesis, the set of stationary integral varifolds with area smaller
than wp (M ) + 1 and whose support is a smooth compact embedded minimal hypersurface is finite. If DM denotes the set consisting of the masses
of these varifolds, choose δ small so that DM has at most one element in
[ωp (M ), ωp (M ) + 2δ], which we call l0 .
Consider Φ ∈ Pp with
l1 = sup{M(Φ(x)) : x ∈ X} ≤ ωp (M ) + δ,
denote by S = {φi }i∈N the (X, M)-homotopy sequence of mappings into
Zn (M ; M; Z2 ) given by Theorem 3.1 (i) applied to Φ, and denote by Π the
(X, M)- homotopy class of mappings into Zn (M ; M; Z2 ) generated by S. It
follows from Theorem 3.1 (iii) that L(S) ≤ l1 and thus
L(Π) ≤ L(S) ≤ l1 .
14
FERNANDO C. MARQUES AND ANDRÉ NEVES
4.5. Lemma. For every S ∗ = {φ∗i } ∈ Π we have L(S ∗ ) ≥ ωp (M ) and so
ωp (M ) ≤ L(Π) ≤ l1 .
Proof. From Theorem 3.2 we obtain for all i large the existence of Φ∗i , Φi
continuous maps in the mass norm from X into Zn (M ; M; Z2 ) that are
homotopic to each other and such that Φi = φi , Φ∗i = φ∗i on dmn(φi ),
dmn(φ∗i ) respectively.
Note that by Lemma 4.2 and Lemma 3.7 we have that Φi ∈ Pp for all
i sufficiently large because, by Theorem 3.2 (iii) and Theorem 3.1 (ii), we
have
lim sup{F(Φi (x) − Φ(x)) : x ∈ X} = 0.
i→∞
Thus Φ∗i ∈ Pp for all i sufficiently large as well and so we obtain from
Theorem 3.2 that
L(S ∗ ) = lim sup = sup{M(Φi (x)) : x ∈ X} ≥ ωp (M ).
i→∞
Applying Theorem 2.11 and Theorem 2.12 to Π we have that L(Π) ∈ DM
and so, in virtue of Lemma 4.5, we obtain that
L(Π) ∈ DM ∩ [ωp (M ), l1 ] = {l0 }.
If l0 > ωp (M ) we could have chosen l1 < l0 and this would give us a contradiction. Thus L(Π) = l0 = ωp (M ).
5. Upper bounds
5.1. Theorem (Guth). The set Pp is nonempty for all p ∈ N and we can
find C = C(M ) so that
1
ωp (M ) ≤ Cp n+1
for all p ∈ N.
Guth proved this theorem in [14, Section 5] when the ambient space is a
unit ball but his arguments carry over to the case when the ambient space is
a compact manifold M . We present them here for the reader’s convenience
From [7, Chapter 4] we have the existence of a (n+1)-dimensional cubical
subcomplex K of some I m and a Lipschitz homeomorphism G : K → M
with G−1 a Lipschitz homeomorphism. For each k ∈ N denote by c(k) ⊂ M
the set consisting of the image under G of all centers of cubes σ ∈ K(k)n+1 .
In what follows we abuse notation and identify cells in K(k) with their
support.
POSITIVE RICCI CURVATURE METRICS HAVE INFINITELY MANY MINIMAL HYPERSURFACES
15
5.2. Proposition. There is C1 = C1 (M ) and ε0 so that for all k ∈ N and
ε ≤ ε0 we can find a Lipschitz map F : M → M homotopic to the identity
and such that
• F (M \ Bε3−k (c(k))) ⊂ G(K(k)n );
• |DF | ≤ C1 ε−1 .
Proof. Denote the center of the cube I n+1 by x0 . Using the map ψ0 described
in [28, Lemma 29.4] we obtain fε : I n+1 → I n+1 Lipschitz such that
fε (x) = x all x ∈ ∂I n+1 ,
fε (I n+1 \ Bε (x0 )) ⊂ ∂I n+1 ,
and |Dfε | ≤ c/ε,
where c = c(n). Moreover, H(t, x) = (1 − t)f (x) + tx shows f is homotopic
to the identity through maps that are the identity on ∂I n+1 .
For each σ ∈ K(k)n+1 we have an affine linear homomorphism Lσ :
I n+1 → σ and so we define
Fσ : G(σ) → G(σ),
−1
Fσ = G ◦ Lσ ◦ fε ◦ L−1
σ ◦G .
The map Fσ is the identity on ∂G(σ) and homotopic to the identity
through maps that are the identity on ∂G(σ). Thus we can define F to
be Fσ on G(σ) and obtain a map defined on M that is homotopic to the
identity
We can find c1 = c1 (M ) so that for all ε sufficiently small we have
F M \ Bc1 ε3−k (c(k)) ⊂ G(K(k)n )
and so, after relabelling ε, F is the desired map.
Proof of Theorem 5.1. Consider an isometric embedding of M into
1
p ∈ N and choose k ∈ N so that 3k ≤ p n+1 ≤ 3k+1 .
RL ,
fix
5.3. Lemma. There is v ∈ S L−1 = {x ∈ RL : |x| = 1} so that
(i) the height function f (x) = x.v defined on M is Morse;
(ii) f −1 (t) ∩ c(k) contains at most one point for all t ∈ R;
(iii) no critical point of f belongs to c(k).
Proof. Consider the following open subset of S L−1 with full measure:
A = {v ∈ S L−1 : v.(u − w) 6= 0 for all u, w ∈ c(k) with u 6= w}.
Because the function f (x) = x.v is Morse for all v in a subset of S N −1 with
full measure, we can choose such v lying in A as well, which means that the
first two conditions are satisfied. Because A is open and the condition of
being Morse is open as well, we can choose some ṽ ∈ A close to v so that all
three conditions are satisfied.
Choose the function f given by Lemma 5.3 and scale it so that its range
is strictly contained in [−1, 1]. Because all the (finite) critical points of f
are non degenerate, it is straightforward to check the existence of a constant
D0 , depending on f and M , so that
Z
(3)
|∇f |−1 ≤ D0 rn−1 for all x ∈ M, r > 0, and |t| ≤ 1.
f −1 (t)∩Br (x)
16
FERNANDO C. MARQUES AND ANDRÉ NEVES
The above estimate implies that {x ∈ M : f (x) < t} has finite perimeter
for all t. Hence, by [28, Theorem 30.3], we have that
f −1 (t) = ∂{x ∈ M : f (x) < t} ∈ Zn (M ; Z2 )
and, more generally, that the function Ψ below is well defined
Ψ : {a ∈ Rp+1 : |a| = 1} → Zn (M ; Z2 )
(
)
p
X
Ψ(a0 , . . . , ap ) = ∂ x ∈ M :
ai (f (x))i < 0 .
i=0
The fact that we are using Z2 coefficients implies that Ψ(a) = Ψ(−a) and
thus induces a map Ψ : RPp → Zn (M ; Z2 ).
5.4. Lemma. Ψ is continuous in the flat topology and belongs to Pp .
Proof. Define Ψ0 : RPp → I0 ([−1, 1]; Z2 ) by
(
t ∈ [−1, 1] :
Ψ0 ([a0 , . . . , ap ]) = ∂
p
X
)
i
ai t < 0 .
i=0
p
Pick {θj }j∈N a sequence in RP tending to θ ∈ RPp . In [14, Lemma 6.2]
Guth shows the existence of Qj ∈ I1 ([−1, 1]; Z2 ) so that
spt (∂Qj − ψ0 (θj ) + ψ0 (θ)) ⊂ {−1, 1}
and
lim M(Qj ) = 0.
j→∞
Because M(∂Qj ) < +∞, we obtain from [28, Constancy Theorem 26.27]
that Qj is a finite disjoint union of closed intervals in [−1, 1]. Thus, it
follows from the fact that the range of f is strictly contained in [−1, 1] that
Ψ(θ) − Ψ(θj ) = ∂ f −1 (Qj )
for all j ∈ N.
The co-area formula and (3) imply that
Z Z
−1
volume (f (Qj )) =
|∇f |−1 dt ≤ D0 (diam M )n−1 M(Qj )
Qj
f −1 (t)
and so the volume of f −1 (Qj ) tends to zero, which means Ψ(θj ) tends to
Ψ(θ) in the flat norm.
Consider
γ : S 1 → RPp ,
eiθ 7→ [(cos(θ/2), sin(θ/2), 0, . . . , 0)],
which is a generator of π1 (RPp ). Then
Ψ ◦ γ : S 1 → Zn (M ; Z2 ),
eiθ 7→ ∂{x ∈ M : f (x) < − tan θ}
is a sweepout of M . The generator λ ∈ H 1 (RPp ; Z2 ) satisfies λ(γ) = 1 and
λp 6= 0 and so Ψ detects a p-class. Finally we see from (3) that Ψ has no
mass concentration and thus Ψ ∈ Pp .
POSITIVE RICCI CURVATURE METRICS HAVE INFINITELY MANY MINIMAL HYPERSURFACES
17
Lemma 5.3 (iii) implies we can choose ε small so that
M(f −1 (t)xBε3−k (x)) ≤ 2ωn εn 3−kn
for all x ∈ c(k) and |t| ≤ 1,
where ωn is the volume of the unit n-ball. For that choice of ε, consider the
map F given by Proposition 5.2 and set
Φ : RPp → Zn (M ; Z2 ),
Φ(θ) = F# (Ψ(θ)).
This map belongs to Pp because F is homotopic to the identity and Lipschitz.
For each |t| ≤ 1 we have from Lemma 5.3 (ii) that
M F# f −1 (t)xBε3−k (c(k)) ≤ sup |DF |n M f −1 (t)xBε3−k (c(k))
M
≤ 2 sup |DF |n ωn εn 3−nk ≤ 2C1n ωn 3−kn .
M
Hence, because each Ψ(θ) consists of at most p level surfaces of f we obtain
(4)
M (F# (Ψ(θ)xBε3−kn (c(k)))) ≤ 2pC1n ωn 3−kn for all θ ∈ RPp .
Set B = M \ Bε3−kn (c(k)). From the first property of Proposition 5.2 we
have that the support of F# (Ψ(θ)xB) is contained in G(K(k)n ). Thus, the
fact that we are using Z2 coefficients implies that
M F# (Ψ(θ)xB) ≤ M(G(K(k)n )) ≤ C2 sup |DG|n 3k(n+1) 3−kn = C3 3k ,
K
where C2 is the number of (n + 1)-cells in the cell complex K and C3 a
constant that depends only on M . Combining this inequality with (4) and
recalling how p was chosen we have, for some constant C = C(M ),
1
M(Φ(θ)) ≤ 2pC1n ωn 3−kn + C3 3k ≤ Cp n+1 for all θ ∈ RPp .
6. Lower Bounds
The following result was proven by Gromov [13, Section 8].
6.1. Theorem (Gromov). There is C = C(M ) so that
1
ωp (M ) ≥ Cp n+1
for all p ∈ N.
For the readers convenience we present a proof of this theorem following
closely the proof given by Guth in [14, Section 3].
Proof of Theorem 6.1. Choose Φ ∈ Pp with parameter space some cubical
complex X. Denote by λ ∈ H 1 (X; Z2 ) the element responsible for Φ detecting a p-class.
There is a constant ν = ν(M ) such that for all p ∈ N we can find p disjoint
1
balls {Bi }pi=1 with radius r = νp− n+1 . With α0 being the constant given by
Proposition 3.5, set
n
α0 n o
Si = x ∈ X : M(Φ(x)xBi ) ≤
r , i = 1, . . . , p.
2
18
FERNANDO C. MARQUES AND ANDRÉ NEVES
If we can find x ∈ X \ (S1 ∪ . . . ∪ Sp ), the theorem follows because in this
case
p
X
1
1
α0 n n+1
M(Φ(x)) ≥
M(Φ(x)xBi ) ≥
ν p
= Cp n+1 ,
2
i=1
where C is a constant that depends only on M .
If X = S1 ∪ . . . ∪ Sp , we are going to find a curve γ ⊂ X contained in some
Si so that λ(γ) 6= 0. If true, then Φ ◦ γ : S 1 → Zn (M ; Z2 ) is a sweepout
that contradicts Proposition 3.5 applied to the ball Bi .
Consider the inclusions ι : Si ,→ X, i = 1, . . . , p.
6.2. Lemma. For some i = 1, . . . , p we have ι∗ (λ) 6= 0 in H 1 (Si , Z2 ).
Proof. Suppose ι∗ (λ) = 0 for all i = 1, . . . , p and consider the exact sequence
q∗
ι∗
i
H 1 (X, Si ; Z2 ) →
H 1 (X; Z2 ) → H 1 (Si ; Z2 ).
Then we can find λi ∈ H 1 (X, Si ; Z2 ) so that qi∗ λi = λ. Thus doing the cup
product we obtain
q1∗ λ1 ^ . . . ^ qk∗ λk = λ ^ . . . ^ λ 6= 0 in H p (X; Z2 ).
On the other hand, from the definition of relative homology, each qi∗ λi has
a representative which vanishes in Si which means that q1∗ λ1 ^ . . . ^ qk∗ λk
has a representative that vanishes on S1 ∪ . . . ∪ Sp = X (a proof of this fact
can be seen in [8, page 3]) and so it must be trivial on H p (X; Z2 ). This is a
contradiction.
Choose Si given by the lemma above. By the Universal Coefficient Theorem we have that Hom (H1 (Si ); Z2 ) = H 1 (Si ; Z2 ) and thus we can find
closed curve γ ⊂ Si where (ι∗ γ)(λ) = γ(ι∗ λ) 6= 0. Therefore ι∗ γ is nontrivial
in X, which is exactly what we wanted to prove.
7. Equality case
The idea to prove the next theorem is due to Lusternick and Schnirelmann
[22] when they found three geodesics on a 2-sphere with any metric.
7.1. Theorem. Assume that 2 ≤ n ≤ 6 and that (M, g) has only finitely
many closed embedded minimal hypersurfaces. Then ωp+1 (M ) > ωp (M ) for
all p ∈ N.
Proof. Apply Proposition 4.4 and obtain Π ∈ [X, Zn (M ; M; Z2 )]# so that
L(Π) = ωp+1 (M ). From Proposition 2.8 choose S ∗ = {φ∗i }i∈N ∈ Π so that
every Σ ∈ C(S ∗ ) is a stationary varifold with mass ωp+1 (M ). During the
proof of Proposition 4.4 we saw that if Φ∗i are the continuous maps in the
mass norm from X into Zn (M ; M; Z2 ) obtained from φ∗i via Theorem 3.2,
then Φ∗i ∈ Pp+1 for all i sufficiently large.
Let S be the finite set of all stationary integral varifolds with area not
bigger than wp+1 (M ) and support a smooth closed embedded hypersurface.
Because S is a finite set we can find x ∈ M and r small enough so that
POSITIVE RICCI CURVATURE METRICS HAVE INFINITELY MANY MINIMAL HYPERSURFACES
19
||V ||(Br (x)) = 0 for all V ∈ S. Choose η1 so that for all V ∈ Vn (M ) we
have
α0 n
(5)
F(V, S) ≤ 2η1 =⇒ ||V ||(Br (x)) ≤
r ,
2
where α0 is the constant given by Proposition 3.5.
Set
Ωi = {x ∈ X : F(|Φ∗i (x)|, S) ≥ 2η1 }.
7.2. Lemma. Assume i is large enough so that Φ∗i ∈ Pp+1 . If for some
k ∈ N, Y is a subcomplex of X(k) such that Ωi ⊂ Y , then Φ∗i : Y →
Zn (M ; M; Z2 ) belongs to Pp .
Proof. Let λ ∈ H 1 (X; Z2 ) be the element responsible for Φ∗i detecting the
(p + 1)-class (see Definition 4.1).
Let Z be the closure of the complement of Y in X (thus a subcomplex of
X(k) as well) and consider the injections i : Y, Z ⊂ X. The result follows if
we show that (i∗ λ)p 6= 0 in H p (Y ; Z2 ).
If γ is a closed curve in Z then γ lies in the complement of Ωi and
so Proposition 3.5 and (5) imply that Φ∗i ◦ γ is not a sweepout. As a
result, i∗ λ(γ) = 0. By the Universal Coefficient Theorem we have that
Hom (H1 (Z); Z2 ) = H 1 (Z; Z2 ) and thus i∗ λ = 0 in H 1 (Z). The exact sequence
j∗
i∗
1
H 1 (X; Z2 ) → H 1 (Z; Z2 )
H 1 (X, Z; Z2 ) →
implies that λ = j1∗ λ1 for some λ1 ∈ H 1 (X, Z; Z2 ).
If ι∗ (λp ) = 0 then the exact sequence
j∗
ι∗
2
H p (X, Y ; Z2 ) →
H p (X; Z2 ) → H 1 (Y ; Z2 )
implies that j2∗ λ2 = λp for some λ2 ∈ H p (X, Z; Z2 ). Thus
j2∗ λ2 ∪ j1∗ λ1 = λp+1 6= 0 in H p+1 (X; Z2 ).
On the other hand, from the definition of relative cohomology, j1∗ λ1 , j2∗ λ2
can be represented by cochains which vanish on Z, Y respectively, which
means that j1∗ λ1 ^ j2∗ λ2 has a representative that vanishes on Z ∪ Y = X
(a proof of this fact can be seen in [8, page 3]) and so it must be trivial on
H p (X; Z2 ). This is a contradiction.
With X(ki )0 = dmn(φ∗i ), consider Y i to be the cubical subcomplex of
X(ki ) consisting of all cells α of X(ki ) so that for every vertex x in α0 we
have
F(|φ∗i (x)|, S) ≥ η1 .
20
FERNANDO C. MARQUES AND ANDRÉ NEVES
7.3. Lemma. There is ε3 so that for all i sufficiently large we can find ψi
Y i -homotopic to φ∗i in Zn (M ; M; Z2 ) with fineness tending to zero and such
that
limi→∞ sup{M(ψi (x)) : x ∈ dmn(ψi )} ≤ ωp+1 (M ) − ε3 .
Proof. Let C = {V ∈ C(S ∗ ) : F(V, S) ≥ η1 /2}. Recall that X is a cubical
subcomplex of I m .
No element in C is Z2 almost minimizing in annuli because otherwise we
know from Theorem 2.12 that its support would be a smooth closed embedded hypersurface and thus belong to S. The idea is to use the combinatorial
construction of Almgren-Pitts Theorem 2.14 applied to φ∗i : Yi → Zn (M ; Z2 )
in order to produce ψi : Yi → Zn (M ; Z2 ) satisfying the conditions of the
lemma.
Therefore for all V ∈ C we can find p ∈ spt ||V||, and positive numbers
{ri }ci=1 , {si }ci=1 with
ri − 2si > 2(ri+1 + 2si+1 ),
i = 1, . . . , c − 1,
rc − 2sc > 0,
so that V is not Z2 almost minimizing in ai (V ) = A(p, ri − si , ri + si ) for all
i = 1, . . . , c.
Thus we can find ε(V ) > 0 so that for each V ∈ C, each j = 1, . . . , c, each
δ > 0, and each T ∈ Zn (M ; Z2 ) with F(V, |T |) < ε(V ), there is a sequence
{Ti }qi=0 in Zn (M ; Z2 ) such that
• T0 = T and spt(T − Ti ) ⊂ aj (V ) for all i = 1, . . . , q;
• M(Ti − Ti−1 ) ≤ δ for all i = 1, . . . , q;
• M(Ti ) ≤ M(T ) + δ for all i = 1, . . . , q;
but M(Tq ) < M(T ) − ε(V ).
Because C is compact we can find {Vi }N
i=1 ⊂ C so that
C ⊂ ∪N
i=1 {S ∈ Vn (M ) : F(S, vi ) < ε(Vi )/4}.
Set ε1 = min{ε(Vi ) : i = 1, . . . , N }.
By definition of C(S ∗ ) there is 0 < ε2 < min{η1 /2, ε1 /2} and n0 ∈ N so
that for all i ≥ n0 and x ∈ X(ki )0 we have
ε1
M(φ∗i (x)) ≥ ωp+1 (M ) − 2ε2
=⇒
F(|φ∗i (x)|, C(S ∗ )) < .
4
i
In particular, if x ∈ Y0 we obtain that
ε1
M(φ∗i (x)) ≥ ωp+1 (M ) − 2ε2
=⇒
F(|φ∗i (x)|, C) < .
4
Thus, for all i sufficiently large we can apply Theorem 2.14 with B =
i
∗
i
{Vi }N
i=1 , A0 = ωp+1 (M ), X being Y , and φ being φi , to obtain ψi Y ∗
homotopic to φi in Zn (M ; M; Z2 ) with with fineness tending to zero and
such that
limi→∞ sup{M(ψi (x)) : x ∈ dmn(ψi )} ≤ ωp+1 (M ) − ε2 /2.
POSITIVE RICCI CURVATURE METRICS HAVE INFINITELY MANY MINIMAL HYPERSURFACES
21
Applying Theorem 3.2 to the homotopy between ψi and φ∗i given by the
previous lemma we have maps Ψi , Φ̄∗i from Y i into Zn (M ; M; Z2 ) which are
homotopic in the mass norm, continuous in the mass norm, and such that
(6) limi→∞ sup{M(Ψi (x)) : x ∈ Yi }
≤ limi→∞ sup{M(ψi (x)) : x ∈ dmn(ψi )} ≤ ωp+1 (M ) − ε3
and
lim sup{M(Φ∗i (x) − Φ̄∗i (x)) : x ∈ Yi } = 0.
i→∞
For all i sufficiently large we have Ωi ⊂ Y i and so we obtain from Lemma
3.7, Lemma 4.2, and Lemma 7.2 that Φ̄∗i ∈ Pp for all i sufficiently large.
Therefore Ψi ∈ Pp for all i sufficiently large as well.
If ωp (M ) = ωp+1 (M ) then (6) implies that Ψi does not belong to Pp for
all i sufficiently large and this is a contradiction.
8. Proof of Main Theorem
We start with the following lemma.
8.1. Lemma. Under the conditions of Theorem 1.1, every two closed embedded smooth minimal hypersurfaces must intersect.
Proof. If (M, g) has positive Ricci curvature, then the result was proven by
Frankel [15]. Assume now that (M, g) has no stable minimal hypersurfaces.
The first thing we show is that every closed embedded hypersurface Σ
in (M, g) must be orientable. Indeed if Σ is non-orientable then M \ Σ
must be connected (because M is orientable) and so we can find a curve γ
which intersects Σ only once. Thus [Σ] 6= 0 in Hn (M, Z2 ) and so we can
minimize area in the homology class of Σ to obtain, by [24], a smooth closed
embedded minimal hypersurface which is area-minimizing and thus stable.
Contradiction.
Pick Σ1 , Σ2 closed embedded orientable minimal hypersurfaces in M . The
previous argument implies that M \ Σi must have two distinct connected
components for i = 1, 2 and thus, if Σ1 ∩ Σ2 = ∅, we find an open set Ω of
M whose boundary is Σ1 ∪ Σ2 . Because Σ1 , Σ2 are unstable minimal hypersurfaces, we can perturb them slightly using the lowest eigenfunction for the
Jacobi operator and obtain a new region Ω1 (which is a small perturbation
of Ω) whose boundary consists of two strictly mean convex hypersurfaces S1
and S2 .
We have that [S1 ] 6= 0 in Hn (Ω1 ; Z2 ) and so we can minimize area in
the homology class of [S1 ]. The fact the boundary of Ω1 is strictly mean
convex implies we can find an area-minimizing T ∈ Zn (Ω1 ; Z2 ) with support
strictly contained in the interior of Ω1 . Thus, by [24], the support of T is
a smooth closed embedded minimal stable hypersurface and this gives us a
contradiction.
22
FERNANDO C. MARQUES AND ANDRÉ NEVES
Under the conditions of Theorem 1.1, suppose that the set L of all closed
connected embedded minimal hypersurfaces of M is finite. We want to
obtain a contradiction.
From Proposition 4.4 we find homotopy classes Πp so that ωp (M ) =
L(Πp ). Theorem 2.11 and Theorem 2.12 give us closed embedded minimal
hypersurfaces with possible multiplicities Vp and such that M(Vp ) = ωp (M ).
From Theorem 7.1 we know that M(Vp+1 ) > M(Vp ) for all p ∈ N and
from Lemma 8.1 we have that, for all p ∈ N, Vp = np Σp for some Σp ∈ L.
The fact L is finite implies we can find δ0 so that
ωp (M ) = M(Vp ) > δ0 p for all p ∈ N.
On the other hand we know from Theorem 5.1 and Theorem 6.1 that we
can find some constant C so that
1
1
C −1 p n+1 ≤ ωp (M ) ≤ Cp n+1
for all p ∈ N
and this is a contradiction.
9. Some open problems
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Instituto de Matemática Pura e Aplicada (IMPA), Estrada Dona Castorina
110, 22460-320 Rio de Janeiro, Brazil
E-mail address: coda@impa.br
Imperial College London, Huxley Building, 180 Queen’s Gate, London SW7
2RH, United Kingdom
E-mail address: a.neves@imperial.ac.uk