Minimal surfaces with low genus in lens spaces
-
Xingzhe Li
and
Xuan Yao
Abstract
Given a Riemannian
1 Introduction
In 1917, Birkhoff [5] proposed a min-max method and showed the existence of a closed geodesic in any Riemannian 2-sphere
In one higher dimension, investigating minimal surfaces in
Conjecture 1.1 (S. T. Yau [52])
There are four distinct embedded minimal spheres in any manifold diffeomorphic to
Towards this conjecture, White [51] first used degree methods to show the existence of at least two embedded minimal spheres in any Riemannian
Inspired by the work of Wang–Zhou [48], we set out this paper to investigate minimal surfaces with low genus in lens spaces.
1.1 Minimal projective planes in
RP
3
The motivation of Yau’s conjecture is closely connected to the topological structure of the space of embedded spheres in
In this paper, we use equivariant min-max theory and a variant multiplicity one result to investigate the above question. Our first main result is the following dichotomy theorem.
Given a Riemannian
there exist at least four distinct embedded minimal projective planes, or
there exist one embedded minimal projective plane and two embedded minimal spheres.
In [20], Haslhofer–Ketover proved in
In recent years, we have witnessed the tremendous progress of Almgren–Pitts min-max theory. One peak of the development is the resolution of Yau’s conjecture [52, Problem 88] on the existence of infinitely many closed minimal surfaces due to Marques–Neves [34] and Song [43]. Meanwhile, several significant advancements [22, 36, 44] in the spatial distribution of closed minimal hypersurfaces were made by the multi-parameter min-max theory coupled with the Weyl law for the volume spectrum [29]. Moreover, after the establishment of a min-max theory for prescribed mean curvature hypersurfaces [57, 56], Zhou [54] solved the multiplicity one conjecture in the Almgren–Pitts setting. Combining with the Morse index estimates, Marques–Neves [33, 35] complete their celebrated program on the Morse theory for the area functional (see also [32]). We refer to [55] for a more detailed history.
1.2 Strategy of the proof
Although our proof shares the same spirit as Wang–Zhou’s resolution [48], one major difference is that the sweepouts are formed by 1-sided projective planes.
However, Wang–Zhou’s multiplicity one theorem [48, Theorem B] relies heavily on the min-max theory [48, Theorem 2.4] for prescribing mean curvature functionals
Let
Heuristically, note that a projective plane
As one key novelty of this paper, we show the following multiplicity one theorem in a certain equivariant setting analogy to the above, which provides an alternative way to handle the multiplicities in the 1-sided Simon–Smith min-max theory.
Let
if
if
(1) By re-merging the components
(2) For a general compact Lie group 𝐺 acting by isometries on
the 𝐺-equivariant min-max theory for closed minimal 𝐺-hypersurfaces has been established by Ketover [23], Liu [30], and the second author [46, 47] in different settings.
However, our scenarios cannot be covered directly by these works since we also need an equivariant min-max for the
After applying the above theorem to
Meanwhile, another difficulty arises as
However, in general, we can neither eliminate minimal spheres in
Indeed, we would conjecture that, under a certain metric on
1.3 Minimal surfaces with Euler characteristic 0 in lens spaces
L
(
p
,
q
)
The above idea can also be applied to investigate minimal Klein bottles and minimal tori in lens spaces
In particular, we have a very similar dichotomy theorem for minimal Klein bottles in
Let
there exist
there exist one embedded minimal Klein bottle and three embedded minimal spheres.
One notices that an embedded Klein bottle 𝐾 in
Hence, using the idea in Remark 1.4, we can lift the sweepout in
contains exactly one embedded minimal Klein bottle
Moreover, we also establish a new result regarding minimal tori in lens spaces.
Let
Using the weighted genus bound [25, (2.6)] and the catenoid estimates [27], Ketover concluded under the positive Ricci assumption that there are three minimal tori in
Ketover–Liokumovich [26, Theorem 1.4] showed as an equivalent formulation of the Smale conjecture in lens spaces that the space of Heegaard tori in
It is also noteworthy that the space of Clifford tori in
The main challenges in finding minimal tori lie in two aspects.
Firstly, the moduli space 𝒳 of embedded tori in
while such an 𝛼 is crucial for applying the Lusternik–Schnirelmann theory.
In Section 8.1, we address this by considering part of the boundary
2 Preliminaries
In this section, we collect necessary preliminaries and notation in the equivariant setting.
Let
To signify the 𝐺-invariance, we will add a superscript or a subscript “𝐺” to the usual terminologies.
We also use “
𝜋: the projection
- (2.1)
For any
For any
Let
Note that a map
Since the group actions in this paper are free and effective, we can make the following definition.
A 𝐺-invariant open set
Namely, an appropriately small open 𝐺-set 𝑈 shall have
2.1
A
h
-functional and
V
C
G
±
-space
For any pairs
and
Note that
A pair
In addition,
for any
In the following lemma, we shall prove that
Given any 𝐺-invariant open set
is
Proof
It is sufficient to show that, for any
In particular, if the open 𝐺-set 𝑈 is appropriately small (Definition 2.2), then we also have the equivalence between the 𝐺-stability and stability in 𝑈.
Indeed, since the stability of
where
Motivated by Almgren’s VZ-space [2], we make the following definition for the weak topological completion of the diagonal pairs
The
For any
where 𝐅 is the varifolds 𝐅-metric [39, Section 2.1] and ℱ is the flat metric [42, §31].
Note that
Moreover, combining Lemma 2.4 with [48, Lemmas 1.4–1.7], we have the following results.
Set
if
2.2
C
1
,
1
almost embedded
(
G
±
,
h
)
-surfaces
In this subsection, we consider the
Let
for each 𝑖, any other component
For simplicity, we will denote
Definition 2.9 (
C
1
,
1
G
±
-boundary)
Let
as 2-currents with
Let
if Σ decomposes into ordered 𝐺-connected sheets
For any
It then follows from (2.3) and [48, Lemma 1.12] that, for any
where
Definition 2.11 (
C
1
,
1
(
G
±
,
h
)
-boundary)
A
Similar to Lemma 2.4, we have the following result.
Given any 𝐺-invariant open set
Proof
For any
By the above lemma,
with respect to the unit normal
2.3 Strong
A
h
-stationarity
In [48], Wang–Zhou introduced the strong
Definition 2.13 (Strong
A
h
-stationarity)
A
For any
One can also say
In particular, the results in [48, Section 1.3] remain valid for strongly
Moreover, for the generalized mean curvature
2.4 𝐺-stability and compactness
Since we only consider 𝐺-equivariant deformations of
Definition 2.15 (𝐺-stable
C
1
,
1
(
G
±
,
h
)
-boundary)
Let
If in addition
where
Note that if the touching set
Let
Proof
The “only if” part is direct.
For the “if” part, we only need to show
Combining Lemma 2.16 with [48, Proposition 1.24], we conclude the following compactness result.
Suppose that
if Σ is smooth, then
if
Proof
By Lemma 2.16, we can apply the compactness result [48, Proposition 1.24] and the
2.5 𝐺-isotopy minimizing problem
Let
be a sufficiently small constant, where
For any open 𝐺-subset
Similar to Lemma 2.16, we also have the following equivalence between isotopy minimizers and 𝐺-isotopy minimizers in appropriately small open 𝐺-sets by the definitions of
Let
Proof
Denote by
Therefore, for any 𝐺-isotopy minimizing sequence
Since any isotopy
for
By the choice of
Let
3 Min-max theory for
C
1
,
1
(
G
±
,
h
)
-boundaries
In this section, we follow the approach in [48] to set up the relative 𝐺-equivariant min-max problem for the
3.1 𝐺-equivariant min-max problem
Fix a 𝐺-connected closed surface
where
For convenience, write
endowed with the oriented smooth topology in the usual sense.
Let 𝑋 be a finite-dimensional cubical complex, and let
Given
A sequence
A subsequence
The critical set of a minimizing sequence
We have the following min-max theorem, and the proof will be given later.
Theorem 3.2 (PMC min-max theorem)
With all notions as above, suppose
Then there exist a minimizing sequence
3.2 Tightening
We follow the pull-tight process in [48, Section 2.2] with some alterations.
Theorem 3.3 (Pull-tight)
Let Π be an
Proof
Given
which satisfies (Lemma 2.4)
Then, combining with the constructions in [48, Section 2.2, Steps 1, 2], we obtain a continuous map
is continuous in the ℱ-metric satisfying
Without loss of generality, we may assume that
where
Given a min-max sequence
Hence
Since
3.3 Almost minimizing
We now adapt the almost minimizing property to the
Given
Definition 3.5 (
(
G
,
A
h
)
-almost minimizing pairs)
Given a 𝐺-invariant open subset
Moreover, we say that
The lemma below emerges as a consequence of being
Let
We also need the following 𝐺-equivariant notions generalized from [13, 48].
Given
be a collection of 𝐺-annuli centered at
In addition, a pair
for each 𝑗,
Comparing with [48, Definition 3.9], our 𝐿-admissible collection
Now, our goal is to find a
Theorem 3.8 (Existence of
(
G
,
A
h
)
-almost minimizing pairs)
Let Π be an
In addition, up to a subsequence of
Proof
The proof is essentially the same as that of [13, Appendix] (for the area functional) and [48, Section 3.3] (for the
An immediate consequence of Lemma 3.6 and Theorem 3.8 is the following.
The limit
3.4 Regularity of min-max pairs: Part I
In this subsection, we will introduce the notions concerning
with a certain
Given an open 𝐺-subset
As above,
Proposition 3.12 (Classification of tangent cones)
Let
Definition 3.13 (
(
G
±
,
A
h
)
-replacement chain property)
Let
satisfying that
for another sequence
of open 𝐺-subsets, the sequence of
Note that the
Theorem 3.14 (First regularity)
Let
3.5 Regularity of min-max pairs: Part II
In this subsection, we prove the regularity of the
with connected components
To begin with, consider a constrained
Then a sequence
Similarly, if
then we say that
Using the arguments in Lemma 2.18, we also have the following result indicating the equivalence between minimizing in problem
Given
define
Then
Proof
By (2.6) and the above definitions, we have
where
Thus
Moreover, the above formulae also imply that
if and only if
In the following lemma, we show that any isotopy in a small enough open 𝐺-set which does not increase
Let
then there is
Moreover,
Proof
Let
Indeed, take any
where
After recovering
Combining Lemma 3.16, 2.18 and Proposition 2.17, we have the following regularity result for the minimizers in problem
Suppose
is minimizing in problem
Proof
Statement (i) follows directly from Definition 3.4.
Similar to the proof of Lemma 3.6 (cf. [48, Lemma 3.3]), one can show that
Next, for any
Now, we can show the regularity for
Theorem 3.18 (Regularity of
(
G
±
,
A
h
)
-almost minimizing pairs)
Given any appropriately small open 𝐺-set
Proof
Since 𝑈 is appropriately small, it follows from Lemma 3.6 that
Fix any open 𝐺-subsets
Note that
(up to a subsequence) so that
and
Next, for another open 𝐺-set
of
3.6 Proof of Theorem 3.2
By Theorem 3.8, there exists an
that is
where
4 Compactness for min-max
(
G
±
,
h
)
-boundaries
4.1 Strong convergence and weighted genus bound
Given an
and an associated min-max sequence
In this part, our goal is to show the smooth regularity of a subsequential varifold limit
To begin with, by Corollary 3.9, we note that
Arguing similarly to the proof of [48, Proposition 5.1], we have the following proposition, which is crucial for the removable singularity step.
There exists a subsequence of
Proof
The theorem is readily verified by combining Proposition 2.17, property (S), and the standard removable singularity theorem (see [41]). ∎
Although
Theorem 4.3 (Genus bound)
Let
such that
where
Proof
Let
4.2 Existence of supersolution
For a sequence of strongly
Let
where
Proof
As we explained before, [48, Proposition 6.4] can be applied directly to get a nonnegative function
5 Multiplicity one for generalized Simon–Smith min-max theory
In this section, we will generalize two multiplicity one theorems in [48] to the equivariant setting, namely the relative min-max in the space of oriented 𝐺-equivariant
5.1 Multiplicity one for relative equivariant min-max
To start with, we have the following compactness theorem, which admits a proof similar to [48].
Let
Recall that the space
Let
Then there exists a closed embedded 𝐺-invariant minimal surface Γ with connected components
if
if
Proof
It is sufficient to verify the theorem when
if
if
Choose
By the choice of ℎ, we have
5.2 Multiplicity one for classical equivariant min-max
In this part, we show an equivariant version of a multiplicity one type result in analogy to [48, Theorem 7.3].
This relies on a version of Simon–Smith min-max theory for unoriented 𝐺-equivariant
Consider
with the unoriented smooth topology.
Let 𝑋 be a finite-dimensional cubical complex.
Given a fixed continuous map
In our setting, the space
Consider the above setup and let Π be a homotopy class of sweepouts by
and
if
if
Proof
We follow the proof of [48, Theorem 7.3] (also see [54, Theorem 5.2]) with some alterations.
Similar to before, we only need to check the theorem for 𝐺-bumpy metric
Let 𝒮 be the collection of all stationary 2-varifolds with mass lying in
whose support is a closed embedded 𝐺-invariant minimal surface satisfying (R′).
Note that 𝒮 is a finite set by bumpiness.
Given a small
Since each
Denote
Then the proof is completed by applying Theorem 5.2 to
In the above theorem, there exists a subset
Proof
Extracting a diagonal subsequence, one easily obtains a sequence
6 Minimal
RP
2
in
RP
3
In this section, let
endowed with the unoriented smooth topology, where
6.1 Sweepouts formed by real projective planes
We now describe three classes of sweepouts that detect three nontrivial cohomology classes in
To begin with, use
where
One notices that, for any
given by
in the double cover
Next, denote by
satisfies
Finally, define
Clearly,
6.2 Proof of Theorem 1.2
We have the following direct corollary by applying Theorem 5.3 and Proposition 5.4 to any homotopy class in
Suppose
there is one 𝐺-component,
every other 𝐺-component
Proof
Firstly,
Fix any Riemannian metric
Let
Using the above notation, suppose
where
Proof
The fact that
Suppose by contradiction that
is a closed set.
Fix a
For any
Proof of Claim 1
Suppose by contradiction that, for some
For each 𝑖, we may pick some
we have
which contradicts the choice of
Let
At the end of this section, we prove Theorem 1.2.
Proof of Theorem 1.2
Firstly, we have a minimizing
Case I:
i.e. there is no minimal 2-sphere in
Next, without loss of generality, we assume
whose quotients in
Additionally, although each
Since manifolds with positive Ricci curvature contain no 2-sided stable minimal hypersurface, we see
Case II:
i.e.
For each
Also, by Corollary 6.1, we can assume
Sub-case II (a):
If
If
Sub-case II (b):
7 Minimal Klein bottles in
L
(
4
m
,
2
m
±
1
)
Consider the unit 3-sphere
Then the lens space
By [8, Corollary 6.4], the Klein bottle embeds into
where
endowed with the smooth unoriented topology.
One can verify that, for any 𝐺-surface
7.1 Sweepouts formed by Klein bottles
We now describe the sweepouts that detect the nontrivial cohomology classes of
Proposition 7.1 ([25])
Let
For the sake of completeness, we provide a relatively detailed explanation following the constructions in [25, Section 4] (see also [45]).
Firstly, one can identify
Note that any orientation preserving isometry
Note that
Next, for
This is known to be a Clifford torus if
an oriented equator.
Since
where
Case A:
We can now define three maps into
One notices that
we conclude
Furthermore, since
satisfies
Clearly, the above
Case B:
is
We can define the map
which is a sweepout in the sense of Almgren–Pitts since
and we can similarly define
with
7.2 Proof of Theorem 1.5
Using Theorem 5.3 and Proposition 5.4, we have the following corollary, which gives the existence of min-max and minimizing minimal Klein bottles.
Suppose
there is exactly one 𝐺-component, say
any other 𝐺-component
then, up to a subsequence,
Proof
Firstly,
a
a connected 𝐺-invariant minimal torus not in
a 𝐺-invariant disjoint union of minimal spheres, i.e.
Suppose
Next, we can use the Lusternik–Schnirelmann theory to show the following result.
Given any Riemannian metric
where
Proof
Using Corollary 7.2 in place of Corollary 6.1 and [6, Proposition 2.3], the proof in Theorem 6.2 can be taken almost verbatim to show the desired results. ∎
At the end of this subsection, we prove Theorem 1.5.
Proof of Theorem 1.5
Firstly, given any Riemannian metric
Case I:
Without loss of generality, we assume that
Sub-case I (A):
For each
Therefore, we have found four distinct embedded minimal Klein bottles in
Sub-case I (B):
Case II:
Sub-case II (A):
where
Suppose Theorem 1.5 (i) fails, i.e. the number of distinct embedded minimal Klein bottles in
By cutting
at least one more minimal sphere in 𝑁 that is different from 𝑆 if
at least two more minimal spheres in 𝑁 if
Sub-case II (B):
8 Minimal torus in lens spaces
Note that Theorem 1.6 (ii) has been proven by Ketover in [25, Theorem 4.8].
Hence, in this section, we only consider minimal tori in
where
the natural inclusion, and by
8.1 Sweepouts formed by tori
To begin with, we have a family of
Proposition 8.1 (Ketover [25])
Let
For the sake of completeness, we adopt the notation as in Section 7 and explain the constructions of sweepouts formed by tori in
Firstly, for any
to be an oriented circle parallel to
Case A:
Note that the oriented torus
with the opposite orientation for
where
and
In addition, one notices that the parameter space
the parameter space of Clifford tori. Then, by Poincaré–Lefschetz duality,
has a generator
is a sweepout in the sense of Almgren–Pitts, which implies
The proof of Lemma 8.2 is left to Appendix A. Now, we know that
is a 4-sweepout in the sense of Almgren–Pitts as
satisfies
where
Case B:
Note that
and we can define the collection of continuous maps
where
8.2 Proof of Theorem 1.6
Since
Let
for a smooth connected embedded minimal torus Γ in
Proof
Firstly, we know that
and (by the genus bound) Γ is either a minimal
torus with multiplicity one, or
2-sphere, or
real projective plane with a stable connected double cover (only if
Klein bottle with a stable connected double cover (only if
Recall that
Let
when
Proof
This result follows from the Lusternik–Schnirelmann theory. One can easily use Corollary 8.3 and [20, Lemma 5.3] to extend the proof of [34, Theorem 6.1] and [20, Theorem 5.2]. ∎
We mention that, for a general Riemannian metric
Finally, we prove Theorem 1.6.
Proof of Theorem 1.6
By combining Corollary 8.3 and Theorem 8.4, we find four embedded minimal tori in
To begin with, we recall the setup of White’s manifold structure theorem [51].
Let
is smooth Fredholm with Fredholm index 0.
Now, let 𝑁 be an unoriented torus,
which implies that the number of elements in
Funding source: Natural Science Foundation of Shanghai Municipality
Award Identifier / Grant number: 25ZR1402252
Funding source: National Natural Science Foundation of China
Award Identifier / Grant number: 12501076
Funding source: National Science Foundation
Award Identifier / Grant number: DMS-2243149
Funding statement: Tongrui Wang is supported by the Natural Science Foundation of Shanghai 25ZR1402252 and the National Natural Science Foundation of China 12501076. Xingzhe Li and Xuan Yao are supported by NSF grant DMS-2243149.
A Proof of Lemma 8.2
Proof of Lemma 8.2
Using the notation in Section 8.1, Case A, for any 𝑘-dimensional closed submanifold 𝑁 in
Firstly, after regarding
where
Next, set
Then
Since
Hence it follows that
Moreover, define
Similarly, one notices that
Finally, we notice that
Hence
Acknowledgements
The authors would like to thank Professor Gang Tian for his interest in this work, and Professor Xin Zhou for the guidance and discussions. They also thank Professor Zhichao Wang, Professor Jintian Zhu, Alex Xu and Nick Sun for helpful discussions. Xingzhe Li and Xuan Yao are grateful to Tongrui Wang and Westlake University for hosting them for a visit during which many of these ideas were conceived. Part of this work was done when Xingzhe Li and Xuan Yao visited Princeton University and we would also like to thank Princeton University for its hospitality. We also thank the anonymous reviewers for helpful comments.
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