A compactness theorem for Fueter sections

A compactness theorem for Fueter sections
Thomas Walpuski
Massachusetts Institute of Technology
2016-01-25
Abstract
We prove that a sequence of Fueter sections of a bundle of compact hyperkähler manifolds X over a 3–manifold M with bounded energy converges
(after passing to a subsequence) outside a 1–dimensional closed rectifiable
subset S ⊂ M . The non-compactness along S has two sources: (1) Bubblingoff of holomorphic spheres in the fibres of X transverse to a subset Γ ⊂ S,
whose tangent directions satisfy strong rigidity properties. (2) The formation
of non-removable singularities in a set of H1 –measure zero. Our analysis
is based on the ideas and techniques that Lin developed for harmonic maps
[Lin99]. These methods also apply to Fueter sections on 4–dimensional manifolds; we discuss the corresponding compactness theorem in an appendix.
We hope that the work in this paper will provide a first step towards extending the hyperkähler Floer theory developed by Hohloch–Noetzel–Salamon
[HNS09, Sal13] to general target spaces. Moreover, we expect that this work
will find applications in gauge theory in higher dimensions.
Keywords: Fueter sections, compactness, bubbling, hyperkähler manifolds
MSC2010: 58E20; 53C26, 53C43
1
Introduction
π
Let M be an orientable Riemannian 3–manifold, let X −
→ M be a bundle of
hyperkähler manifolds together with a fixed isometric identification I : ST M →
H(X) of the unit tangent bundle in M and the bundle of hyperkähler spheres1 of the
fibres of X, and fix a connection on X.
1
2
3
(ξ1 , ξ2 , ξ3 ) ∈ S ⊂ R , Iξ :=
P3 Given a hyperkähler manifold (X, g, I1 , I2 , I3 ),for each ξ 2=
is called the hyperkähler sphere
i=1 ξi Ii is a complex structure. The set H(X) := Iξ : ξ ∈ S
of X.
1
Definition 1.1. A section u ∈ Γ(X) is called a Fueter section if
(1.2)
Fu :=
3
X
I(vi )∇vi u = 0 ∈ Γ(u∗ V X)
i=1
for some local orthonormal frame (v1 , v2 , v3 ).2 Here ∇u ∈ Ω1 (M, u∗ V X) is the
covariant derivative of u, a 1–form taking values in the pull-back of the vertical
tangent bundle V X := ker(dπ : T X → T M ). The operator F is called the Fueter
operator.
The Fueter operator is a non-linear generalisation of the Dirac operator, see
Taubes [Tau99] and Haydys [Hay14, Section 3].
Remark 1.3. A construction similar to (1.2) also exists in dimension four. Since it
is more involved, we relegate its discussion to Appendix B.
/ , I is the Clifford
Example 1.4. Choose a spin structure s on M . If X = S
multiplication and ∇ denotes the induced spin connection, then the Fueter operator
is simply the Dirac operator associated with s.
Example 1.5. Let (X, g, I1 , I2 , I3 ) be a hyperkähler manifold and (v1 , v2 , v3 ) a
orthonormal frame of M . A map u : M → X satisfying
(1.6)
Fu =
3
X
Ii du(vi ) = 0
i=1
is called a Fueter map. In a local trivialisation the Fueter equation for sections of X,
takes the form (1.6) up to allowing for the Ii to depend on x ∈ M and admitting a
lower order perturbation (coming from the connection 1–form).
One of the main motivations for studying Fueter sections is the work of Hohloch–
Noetzel–Salamon [HNS09], who introduced a functional whose critical points are
precisely the solution of (1.6) and developed the corresponding Floer theory in the
case when the target X is compact and flat, and the frame on M is divergence free
and regular,3 see also Salamon [Sal13]. The requirement that X be flat is very
severe and one would like to remove it. It has been conjectured that the putative
hyperkähler Floer theory should be very rich and interesting, especially in the case
when X is a K3 surface.
2
Of course, F does not depend on the choice of (v1 , v2 , v3 ).
Every 3–manifold admits a divergence free frame by Gromov’s h–principle [Sal13, Theorem
A.1]. A frame is regular if there are no non-constant Fueter maps M → H with respect to this frame;
this is a generic condition,.
3
2
A further source of motivation is gauge theory on G2 – and Spin(7)–manifolds.
Here, Fueter sections of bundles of moduli spaces of ASD instantons naturally
appear in relation with codimension four bubbling phenomena for G2 – and Spin(7)–
instantons; see Donaldson–Segal [DS11] and the author [Wal12, Wal14] for further
details.
Remark 1.7. Sonja Hohloch brought to the author’s attention a cryptic remark in
Kontsevich–Soibelman [KS08, Section 1.5 Question 3], which indicates that their
invariants of 3D Calabi–Yau categories with stability structure can be interpreted
as “quaternionic Gromov–Witten invariants” of certain hyperkähler manifold M,
which means as a count of Fueter maps from some 4–manifold to M.
A major issue when dealing with Fueter sections is the potential failure of
compactness. This is demonstrated by the following example due to Hohloch–
Noetzel–Salamon.
Example 1.8. Consider a K3 surface X with a hyperkähler structure such that
(X, I1 ) admits a non-trivial holomorphic sphere z : S 2 → X and take M = SU(2),
the unit-sphere in the quaternions H, with a left-invariant frame (v1 , v2 , v2 ) which
at id ∈ SU(2) it is given by (i, j, k). Let ¯· : S 2 → S 2 denote complex conjugation
on S 2 = P1 . Let π : S 3 → S 2 denote the Hopf fibration whose fibres are the orbits
of v1 . It is easy to check that u = z ◦ ¯· ◦ π : S 3 → X satisfies
∂v2 u − I∂v3 u = 0,
∂v1 u = 0 and
and thus u is a Fueter map. For λ > 0 define a conformal map sλ : S 2 → S 2
by sλ (x) = λx for x ∈ R2 ⊂ S 2 and sλ (∞) = ∞. Now, the family of Fueter
maps uλ := z ◦ sλ ◦ π blows up along the Hopf circle π −1 (∞) as λ ↓ 0 and
converges to the
map on the complement of the Hopf circle. Also, note
R constant
2
that E (uλ ) = S 3 |∇uλ | is independent of λ.
The following is the main result of this article.
Theorem 1.9. Suppose X is compact. Let (ui ) be a sequence of solutions of the
(perturbed) Fueter equation
(1.10)
Fui = p ◦ ui
with p ∈ Γ(X, V X)4 and
(1.11)
E (ui ) :=
Z
M
4
|∇ui |2 ≤ cE
This sort of deformation of (1.2) is important for applications; e.g., Hohloch–Noetzel–Salamon
perturb (1.2) using a Hamiltonian function to achieve transversality.
3
for some constant cE > 0. Then (after passing to a subsequence) the following
holds:
• There exists a closed subset S with H1 (S) < ∞ and a Fueter section u ∈
∞.
Γ(M \ S, X) such that ui |M \S converges to u in Cloc
• There exist a constant ε0 > 0 and an upper semi-continuous function
Θ : S → [ε0 , ∞) such that the sequence of measures µi := |∇ui |2 H3
converges weakly to µ = |∇u|2 H3 + Θ H1 bS.
• S decomposes as
S = Γ ∪ sing(u)
with
Γ := supp(Θ H1 bS)
(
sing(u) :=
and
)
Z
1
2
x ∈ M : lim sup
|∇u| > 0 .
r Br (x)
r↓0
Γ is H1 –rectifiable, and sing(u) is closed and H1 (sing(u)) = 0.
• For each smooth point5 x ∈ Γ, there exists a non-trivial holomorphic sphere
zx : S 2 → (Xx := π −1 (x), −I(v)) with v a unit tangent vector in Tx Γ.
Moreover,
Z
Θ(x) ≥ E (zx ) :=
|dzx |2 .
S2
• If X is a bundle of simple hyperkähler manifolds with b2 ≥ 6, then there is a
subbundle d ⊂ PT M , depending only on sup Θ, whose fibres are finite sets
such that Tx Γ ∈ d for all smooth points x ∈ Γ.
Remark 1.12. The analysis of (1.2) is similar to Lin’s work on the compactness
problem for harmonic maps [Lin99]. We follow his strategy quite closely; however,
there are a number of simplifications in our case, many of the arguments have to be
approached from a different angle and our result is stronger.
Remark 1.13. In the situation of Example 1.5 if X is flat and (v1 , v2 , v3 ) is regular,
then the uniform energy bound (1.11) is automatically satisfied; see Salamon [Sal13,
Lemma 3.2 and Remark 3.5].
5
We call a point x ∈ Γ smooth if the tangent space Tx Γ exists and x ∈
/ sing(u). Since Γ is
rectifiable, Tx Γ exists almost everywhere.
4
Remark 1.14. If I is parallel (which is very rarely the case, but holds, e.g., in
the situation of Example 1.5 if M = T 3 equipped with a flat metric and the vi
are parallel), then there are topological energy bounds; see Remark 2.10. In this
case Fueter sections are stationary harmonic sections and one can derive most of
Theorem 1.9 from [Lin99]; cf. Li–Tian [LT98, Section 4] and Chen–Li [CL00],
who study triholomorphic/quaternionic maps between hyperkähler manifolds. More
recently, very important progress in the study of triholomorphic maps was made by
Bellettini–Tian [BT15].
Remark 1.15. In the situation of Example 1.5 if X is flat, then S = ∅; see Hohloch–
Noetzel–Salamon [HNS09, Section 3] and Remark 3.5. This does not immediately
follow from Theorem 1.9; however, since π2 (T n ) = 0, flat hyperkähler manifolds
admit no non-trivial holomorphic spheres and we can rule out bubbling a priori, i.e.,
Γ = ∅.
Remark 1.16. By Bogomolov’s decomposition theorem (after passing to a finite
cover) any hyperkähler manifold is a product a flat torus and a simple hyperkähler
manifold. Hohloch–Noetzel–Salamon’s compactness result says that nothing interesting happens in the torus-factors. Thus the assumption of X being a bundle
of simple hyperkähler manifolds is not restrictive. The requirement b2 ≥ 6 is an
artefact of a result of Amerik–Verbitsky that we use in Section 8.
As stated, Theorem 1.9 is very likely far from optimal. Here are some conjectural
improvements:
• We believe that the limiting section u ∈ Γ(M \ S, X) extends to M \ sing(u)
and, moreover, that sing(u) is finite (possibly countable).
• We believe that Γ enjoys much better regularity than just being H1 –rectifiable.
It seems reasonable to expect that Γ is a graph (possibly with countably many
vertices) embedded in M and Θ is constant along the edges of Γ; moreover,
we expect that the vertices (Γ, Θ) are balanced. We present some evidence
for this in Section 11.
Remark 1.17. In the situation of Remark 1.14, Bethuel’s removable singularities
theorem for stationary harmonic maps [Bet93, Theorem I.4] shows that u extends
to M \ sing(u) and a result of Allard–Almgren [AA76] affirms the conjecture in
the third bullet.
The holomorphic sphere zx can be replaced by a bubble-tree, cf. Parker–Wolfson
[PW93], such that the energy of the entire bubble tree equals Θ(x). In an earlier
version of this article it was conjectured that there can be no energy stuck on
the necks; in, particular Θ(x) is the sum of energies of holomorphic spheres in
(Xx , −I(v)). Shortly after the first version of this article was posted on the arXiv,
5
Bellettini–Tian [BT15] proved the analogue of this conjecture for triholomorphic
maps, and after a brief discussion with the author, in an updated version also the
author’s earlier conjecture. We refer the reader to [BT15, Section 7] for details.
It is an interesting and important question to ask: what happens for a generic
choice of I : ST M → H(X) and perturbation p? One would hope (perhaps too
optimistically) that generically the situation is much better and possibly good enough
to count solutions of (1.10) and thus define the Euler characteristic of the conjectural
hyperkähler Floer theory.
Assumptions and conventions Throughout the rest of the article we assume the
hypotheses of Theorem 1.9. We use c to denote a generic constant. We fix a constant
r0 > 0 which is much smaller than the injectivity radius of M and we take all radii
to be at most r0 . {·, . . . , ·} denotes a generic (multi-)linear expression which is
bounded by c.
2
Mononicity formula
The foundation of the analysis of (1.2) is the monotonicity formula which asserts
that the renormalised energy
Z
1
|∇u|2 .
r Br (x)
is almost monotone in r > 0:
Proposition 2.1. If u ∈ Γ(M, X) satisfies (1.10), then for all x ∈ M and 0 < s < r
Z
Z
Z
ecr
ecs
1
2
2
|∇u| −
|∇u| ≥
|∇r u|2 − cr2 .
r Br (x)
s Bs (x)
Br (x)\Bs (r) ρ
Here ρ := d(x, ·).
It is instructive to first prove the following which contains the essence of
Proposition 2.1.
∂
Proposition 2.2. If u : R3 → X is a Fueter map with vi = ∂x
i , then for all x ∈ M
and 0 < s < r
Z
Z
Z
1
1
1
2
2
(2.3)
|du| −
|du| = 2
|∂r u|2 .
r Br (x)
s Bs (x)
Br (x)\Bs (r) ρ
6
Proof. The derivative of
1
f (ρ) :=
ρ
is
1
f (ρ) = − 2
ρ
0
Z
|du|2
Bρ (x)
Z
1
|du| +
ρ
Bρ (x)
2
Z
|du|2 .
∂Bρ (x)
By a direct computation
|du|2 vol = |Fu|2 vol + 2
(2.4)
3
X
dxi ∧ u∗ ωi ,
i=1
see [HNS09, Lemma 2.2]. Here ωi = g(Ii ·, ·) denotes the Kähler form on X
associated with Ii . Hence,
Z
2
3
X
Z
|du| = 2
(2.5)
Bρ (x)
∗
i
dx ∧ u ωi = 2
Bρ (x) i=1
Z
Z
3
X
d(xi u∗ ωi )
Bρ (x) i=1
u∗ ω∂r
= 2ρ
∂Bρ (x)
P
xi ∂
with ∂r = 3i=1 |x|
denoting the radial vector field. On ∂Bρ (x), we can take
∂xi
the local orthonormal frame (v1 , v2 , v3 ) to a be of the form (∂r , ∂1 , ∂2 ) with (∂1 , ∂2 )
a local positive orthonormal frame for ∂Bρ (x). Now, minus twice the integrand in
the last term is
−2hI(∂r )∂1 u, ∂2 ui = 2hI1 ∂1 u, I2 ∂2 ui
(2.6)
= |I1 ∂1 u + I2 ∂2 u|2 − |I1 ∂1 u|2 − |I2 ∂2 u|2
= 2|∂r u|2 − |du|2 .
Putting everything together yields
f 0 (ρ) = 2ρ−1
Z
|∂r u|2 .
∂Br
Upon integration this yields (2.3).
Proof of Proposition 2.1. The map I yields section of π ∗ T M ⊗Λ2 V X which, using
the connection on X, can be viewed as a 3–form Λ ∈ Ω3 (X). For sections of X the
identity (2.4) is replaced by
(2.7)
|∇u|2 vol = |Fu|2 vol + 2u∗ Λ.
7
If we define f (ρ) as before, then using (2.7) its derivative can be written as
Z
Z
Z
∗
−1
2
−2
0
−2
|∇u|2 .
u Λ+ρ
|p ◦ u| − 2ρ
f (ρ) = −ρ
∂Bρ (x)
Bρ (x)
Bρ (x)
Let ∂r denote the radial vector field emanating from x and set Ω := i(v)Λ with
v := π ∗ (r∂r ). We can write Λ as
Λ = dΩ + e
where e is the sum of a form of type (1, 2) and a form of type (2, 1) satisfying
|e| = O(δr) with
(2.8)
δ := |∇I| + |FX | + |R|.
Here we use the bi-degree decomposition of Ω∗ (X) arising from T X = π ∗ T M ⊕
V X, r := d(x, π(·)), FX is the curvature of the connection on X and R is the
Riemannian curvature of M . Hence,
Z
Z
−2
u∗ Λ = −2
u∗ Ω + O(ρ2 )f (ρ) + O ρ4
Bρ (x)
∂B (x)
Z ρ
(2.9)
= −2ρ
i(∂r )u∗ Λ + O(ρ2 )f (ρ) + O ρ4 .
∂Bρ (x)
Arguing as before,
Z
Z
−2
i(∂r )u∗ Λ =
∂Bρ (x)
|I∂r ∇r u − p ◦ u| + |∇r u|2 − |∇u|2 .
∂Bρ (x)
Putting everything together one obtains
Z
0
−1
f (ρ) ≥ ρ
|∇r u|2 − cf (ρ) − cρ.
Bρ (x)
This integrates to prove the assertion.
Remark 2.10. If Λ is closed (which is rarely the case), then
Z
Z
2
∗
E (u) =
|∇u| = h[M ], [u Λ]i +
|Fu|2 .
M
M
Since the first term on the right-hand side only depends on the homotopy class of u,
this yields a priori energy bounds for Fueter sections.
Corollary 2.11. In the situation of Proposition 2.1,
Z
Z
1
1
2
|∇u| .
|∇u|2
s Bs (x)
r Br (x)
and if Bs (y) ⊂ Br/2 (x), then
Z
Z
1
1
2
|∇u| .
|∇u|2 .
s Bs (x)
r Br (x)
8
3
ε–regularity
The following is the key result for proving Theorem 1.9. It allows to obtain local
L∞ –bounds on ∇u provided the renormalised energy is not too large.
Proposition 3.1. There is a constant ε0 > 0 such that if u ∈ Γ(M, X) satisfies
(1.10) and
Z
1
|∇u|2 ≤ ε0 ,
ε :=
r Br (x)
then
(3.2)
sup
|∇u|2 (y) . r−2 ε + 1.
y∈Br/4 (x)
Remark 3.3. Given (3.2), higher derivative bounds over slightly smaller balls can
be obtained using interior elliptic estimates.
Proposition 3.1 follows from the following differential inequality and Corollary 2.11 using the Heinz trick; see Appendix A.
Proposition 3.4. If u ∈ Γ(M, X) satisfies (1.10), then
∆|∇u|2 . |∇u|4 + 1.
Proof. This is proved in [HNS09, Lemma 3.3 and Remark 3.4]. We recall the proof
¯ the induced connection on
which is a simple direct computation. Denote by ∇
∗
0
∗
0
∗
u V X and define F : Ω (M, u V X) → Ω (M, u V X) by
Fû :=
3
X
¯ v û
I(vi )∇
i
i=1
for some local orthonormal frame (v1 , v2 , v3 ). A simple computation yields
¯ ∗ ∇u + {∇u}
FFu = ∇
where {·} makes the dependence on I etc. implicit. Further
¯
¯∇
¯ ∗ ∇u + {∇u} + {∇∇u}.
¯
∇FFu
=∇
Using
¯v ∇
¯ v ∇v u = ∇
¯v ∇
¯ v ∇v u + {∇u, ∇u, ∇u}
∇
i
i
i
i
k
k
¯
¯
¯
= ∇v ∇v ∇v u + {∇u, ∇u, ∇u} + {∇∇u}
i
i
k
9
and Fu = p ◦ u we derive
¯
¯
¯
¯ ∗ ∇∇u
= ∇FFu
+ {∇u, ∇u, ∇u} + {∇∇u}
∇
¯
= {∇u, ∇u, ∇u} + {∇∇u}
+ O(1).
From this it follows that
∗
2
¯
¯
¯ ∇∇u,
∇u − 2|∇∇u|
∆|∇u|2 = 2 ∇
2
2
¯
¯
≤ c(|∇u|4 + |∇u| + |∇∇u||∇u|
) − 2|∇∇u|
. |∇u|4 + 1.
Remark 3.5. If X = M × X and X is flat, then one can prove that
∆|∇u|2 . |∇u|3 + 1
and the Heinz trick for subcritical exponents shows that k∇ukL∞ (M ) is bounded in
terms of the energy E (u); see Remark A.2 and [HNS09, Appendix B].
4
Convergence away from the blow-up locus
Proposition 4.1. There exists a subsequence (ui )i∈I ⊂ (ui )i∈N0 and a subset
S ⊂ M , called the blow-up locus, with the following properties:
• S is closed and H1 (S) < ∞.
∞.
• The sequence ui |M \S i∈I converges to a section u ∈ Γ(M \ S, X) in Cloc
• If there is a subset S 0 ⊂ M such that a subsequence ui |M \S 0 i∈I 0 ⊂I con∞ , then S 0 ⊃ S.
verges in Cloc
Proof. We proceed in four steps.
Step 1. Construction of S.
With ε0 as in Proposition 3.1, for r ∈ (0, r0 ] and i ∈ N0 , define
)
(
Z
ecr
ε0
2
Si,r := x ∈ M :
|∇ui | ≥
.
r Br (x)
2
Note that, by Proposition 2.1, Si,s ⊂ Si,r whenever s ≤ r.
Since the Si,r are compact, for each r, we can pick Jr ⊂ N0 such that the
subsequence (Si,r )i∈Jr converges to a closed subset Sr in the Hausdorff metric.
10
By a diagonal sequence argument, we can find J ⊂ N0 such that Si,2−k r0
converges to a closed subset S2−k r0 for each k ∈ N0 . Set
S :=
\
i∈J
S2−k r0 .
k∈N0
By construction S is closed.
Step 2. H1 (S) < ∞.
Given 0 < δ ≤ r0 , cover S by a collection of balls B4rj (xj ) : j = 1, . . . , m
with xj ∈ S, rj ≤ δ and B2rj (xj ) pairwise disjoint. Pick k 1 such that
2−k r0 < min {rj }. For i 1, we can find x0j ∈ Si,2−k r0 with d(x0j , xj ) < δ. Then
the balls B5rj (x0j ) still cover T and while the smaller balls Brj (x0j ) are pairwise
disjoint. By Proposition 2.1,
m
X
j=1
Z
m Z
1
cE
1 X
2
|∇ui | ≤
|∇ui |2 ≤
rj ≤
< ∞.
ε0
ε0 M
ε0
Brj (x0j )
j=1
Since this bound is uniform in δ, the assertion follows.
Step 3. Selection of (ui )i∈I and construction of u ∈ Γ(M \ S, X).
If x ∈ M \ S, then there exists r > 0 such that for all i ∈ J sufficiently large
Z
1
|∇ui |2 ≤ ε0 .
r Br (x)
By Proposition 3.1, for all i ∈ J, |∇ui | is uniformly bounded on Br/4 (x). It follows
using standard elliptic techniques and Arzelà–Ascoli that we can chose J ⊂ I such
∞ on M \ S.
that the subsequence of (ui )i∈I converges in Cloc
Step 4. M \ S is the maximal open subset on which a subsequence (ui )i∈I 0 ⊂I can
∞.
converge in Cloc
Suppose (ui )i∈I 0 ⊂I converges in C 1 a neighbourhood of x ∈ M . Then |∇ui |
is uniformly bounded in this neighbourhood. Hence, there is a slightly smaller
neighbourhood of x ∈ M which is contained in M \ Si,r for each sufficiently small
r > 0 and each i ∈ I 0 . Since limi∈I 0 Si,r = Sr ⊂ S, it follows that x ∈ M \ S.
11
5
Decomposition of the blow-up locus
We assume that we have already passed to a subsequence so that the convergence
statement in Proposition 4.1 holds. Consider the sequence of measures (µi ) defined
by
µi := |∇ui |2 H3 .
Here H3 is the 3–dimensional Hausdorff measure on M , which is simply the
standard measure on M . By (1.11) the sequence of Radon measures (µi ) is of
bounded mass; hence, it converges weakly to a Radon measure µ. By Fatou’s
lemma we can write
µ = |∇u|2 H3 + ν
for some non-negative Radon measure ν.
Definition 5.1. We call ν the defect measure and
Γ := supp ν
the bubbling locus.6 We call
(
sing(u) :=
1
x ∈ M : Θ∗u (x) := lim sup
r
r↓0
Z
)
|∇u|2 > 0
Br (x)
the singular set of u.
If we denote by Θ∗µ (x) the upper
density of µ at the
point x ∈ M , then it follows
∗
from Proposition 3.1 that S = x ∈ M : Θµ (x) > 0 ⊂ Γ ∪ sing(u). The reverse
inclusion also holds; hence, we have the following.
Proposition 5.2. The blow-up locus S decomposes as
S = Γ ∪ sing(u).
This means that there are two sources of non-compactness: one involving a loss
of energy and another one without any loss of energy.
6
Regularity of the bubbling locus
As a first step towards understanding the non-compactness phenomenon involving
energy loss, we show that the set Γ at which this phenomenon occurs is relatively
tame.
6
The justification for this terminology will be provided in Section 7.
12
Proposition 6.1. Γ is H1 –rectifiable and ν can be written as
ν = Θ H1 bΓ
with Θ : M → [0, ∞) upper semi-continuous. Moreover, H1 (sing(u)) = 0.
The interested reader can find a detailed discussion of the concept of rectifiablity
in DeLellis’ lecture notes [DL08]. For our purposes it shall suffice to recall the
definition.
Definition 6.2. A subset Γ ⊂ M is called Hk –rectifiable if there exists a countable
collection {Γi } of k–dimensional Lipschitz submanifolds such that
[ Hk Γ \
Γi = 0.
i
A measure µ on M is called Hk –rectifiable if there exist a non-negative Borel
measurable function Θ and Hk –rectifiable set Γ such that for any Borel set A
Z
µ(A) =
Θ(x) Hk .
A∩Γ
Since Γ is H1 –rectifiable, at H1 –a.e. point x ∈ Γ, it has a well-defined tangent
space Tx Γ and ν has a tangent measure, i.e., the limit
1
Tx ν := lim (exp ◦sε )∗ ν
ε→0 ε
exists and
Tx ν = Θ(x) H1 bTx Γ.
Here sε (x) := εx.
To prove Proposition 6.1 we will make use of the following deep theorem,
whose proof is carefully explained in [DL08].
Theorem 6.3 (D. Preiss [Pre87]). If µ is a locally finite measure on M and m ∈ N0
is such that for µ–a.e. x ∈ M the density
Θm
µ (x) := lim
r↓0
µ(Br (x))
.
rm
exists and is finite, then µ is Hm –rectifiable.
Proof of Proposition 6.1. The proof has five steps.
13
Step 1. With the same constant as in Proposition 2.1 and for all x ∈ M and
0<s≤r
ecs s−1 µ(Bs (x)) ≤ ecr r−1 µ(Br (x)) + cr2 .
This is not quite a trivial consequence of Proposition 2.1 because (µi ) only
weakly converges to µ; hence, we only know that µ(B̄r (x)) ≥ lim supi→∞ µi (B̄r (x))
and lim inf i→∞ µi (Br (x)) ≥ µ(Br (x)).
For x ∈ M set
Rx := {r ∈ (0, r0 ] : µ(∂Br (x)) > 0} .
If r ∈
/ Rx , then it follows from Proposition 2.1 that
ecs s−1 µ(Bs (x)) ≤ ecr r−1 µ(Br (x)) + cr2 .
The general case follows by an approximation argument. Note that Rx is at most
countable. Thus, given r ∈ Rx , we can find a sequence (ri ) such that s < ri < r,
ri ∈
/ Rx , and r := limi→∞ ri . By dominated convergence
µ(Br (x)) = lim µ(Bri (x)).
i→∞
Step 2. The limit
Θ(x) := lim r4−n µ(Br (x))
r↓0
exists for all x ∈ M . The function Θ : M → [0, ∞) is upper semi-continuous, it
vanishes outside S, is bounded and Θ(x) ≥ ε0 for all x ∈ S.
The existence of the limit is a direct consequence of Step 1.
To see that Θ is upper semi-continuous, let (xi ) be a sequence of points in M
converging to a limit point x = limi→∞ xi . Let r ∈
/ Rx and ε > 0. For i 1
Θ(xi ) ≤ ecr r−1 µ(Br (xi )) + cr2 ≤ ecr r−1 µ(Br+ε (x)) + cr2 .
Therefore, lim supi→∞ Θ(xi ) ≤ ecr r−1 µ(Br (x)) + cr2 . Taking the limit as r → 0
shows that Θ is upper semi-continuous.
The last part is clear.
Step 3. Θ∗u vanishes H1 –a.e. in M , i.e., H1 (sing(u)) = 0.
Given ε > 0, set
Eε := {x ∈ M : Θ∗u (x) > ε} .
14
Given 0 < δ, choose {x1 , . . . , xm } ⊂ Eε and {r1 , . . . , rm } ⊂ (0, δ] such that the
balls B2rj (xj ) cover Eε , but the balls Brj (xj ) are pairwise disjoint. Moreover, we
can arrange that
Z
1
|∇u|2 > ε.
rj Brj (xj )
Since u is smooth on M \ S, we must have Eε ⊂ S. Hence,
m
X
j=1
m
rj ≤
1X
ε
j=1
m
Z
|∇u|2 ≤
Brj (xj )
1X
ε
j=1
Z
|∇u|2
Nδ (S)
where Nδ (S) = {x ∈ M : d(x, S) < δ}. The right-hand side goes to zero as δ
goes to zero. Thus H1 (Eε ) = 0 for all ε > 0. This concludes the proof.
Step 4. ν is H1 –rectifiable.
By Step 2 for any x ∈ M \ sing(u) the density
Θν (x) = lim
r↓0
ν(Br (x))
r
exists and agrees with Θ(x). By Step 3, H1 (sing(u)) = 0 and, hence, ν(sing(u)) =
0. Applying Theorem 6.3 yields the assertion.
Step 5. We prove the proposition.
We have already proved the assertion about sing(u). Since ν is H1 –rectifiable
and Γ = supp(ν), it follows that Γ is H1 –rectifiable and ν can be written as
ν = Θν H1 bΓ
for some Θ̃. By Step 3, Θν (x) = Θ(x) for H1 –a.e. x ∈ Γ.
7
Bubbling analysis
We will now show that the “lost energy” goes into the formation of bubbles transverse to Γ. To state the main result recall that an orientation on Nx Γ induces a
canonical complex structure and an orientation of Nx Γ is canonically determined
by the choice of a unit tangent vector v ∈ Tx Γ ⊂ Tx M since M is oriented.
15
Proposition 7.1. If x ∈ Γ is smooth, i.e., Tx Γ exists and x ∈
/ sing(u), then there
exists a (−I(v))–holomorphic sphere zx : Nx Γ ∪ {∞} → X := Xx with
Z
(7.2)
E (zx ) :=
|dzx |2 ≤ Θ(x).
S2
Here we have picked some unit vector v ∈ Tx Γ.
Remark 7.3. It is immaterial whether we choose v or its opposite −v since this
results in changing the complex structures on both Nx Γ and X. In particular, the
above cannot be used to fix an orientation of Γ; however, the existence of zx does
restrict the possible tangent directions, see Section 8.
Remark 7.4. The reason that (7.2) may be strict is that we only extract one bubble of
what is an entire bubbling-tree, cf. Parker–Wolfson [PW93] for the general notion
of a bubbling tree and Belletini–Tian [BT15, Section 7] for a discussion on how to
extract a bubbling tree in the our situation.
The holomorphic sphere zx is obtained by blowing-up (ui ) around the point
x ∈ Γ. We assume a trivialisation of X in a neighbourhood U of x has been fixed;
see Example 1.5. We use the following notation: given any map u : U → X and a
scale factor 0 < λ, we define a rescaled map uλ : Br30 /λ (0) → X by
uλ := u(exp ◦sλ ).
(7.5)
with sλ (y) := λy. We write (z, w) to denote points in Tx Γ × Nx Γ = Tx M and
work with generalised cubes of the form
Qr,s (z0 , w0 ) := Br (z0 ) × Bs (w0 ) ⊂ Tx Γ × Nx Γ = Tx M.
Proof of Proposition 7.1. We proceed in three steps.
Step 1 (Preliminary scale fixing). There exists a null-sequence (εi ) ⊂ (0, 1) such
that
|dui;εi |2 H3 * Tx ν = Θ(x) H1 bTx Γ.
By definition, Tx ν is the weak limit of (exp ◦sε )∗ ν as ε tends to zero. Since
x∈
/ sing(u), we have
lim (exp ◦sε )∗ ν = lim (exp ◦sε )∗ µ.
ε→0
ε→0
Thus
Tx ν = lim lim (exp ◦sε )∗ µi = lim (exp ◦sεi )∗ µi
ε→0 i→∞
i→∞
for some null-sequence (εi ). This implies the assertion since
(exp ◦sεi )∗ µi = |dui;εi |2 H3 .
16
Step 2 (Asymptotic translation invariance). For any sequence (δi ) ⊂ (0, 1] and
H1 –a.e. z ∈ Tx Γ
Z
1
|∂v ui;δi εi |2 = 0.
(7.6)
lim sup
i→∞ s≤1/δ s Q
i
s,1/δ (z,0)
i
It suffices to prove this for δi = 1 and z ∈ B1 (0), because the assertion
is invariant under rescaling and unaffected by translating everything along the
direction of Tx Γ.
Step 2.1. We have
Z
lim
i→∞ Q2,1 (0)
|∂v ui,εi |2 = 0.
Denote by ∂ρ the radial vector field emanating from 4v. By Proposition 2.1, for
for 0 < s ≤ r
Z
(7.7)
ecεi τ τ −1 |∂ρ ui;εi |2
Z
Z
≤ ecεi r r−1
|dui;εi |2 − ecεi s s−1
Br (4v)\Bs (4v)
Br (4v)
|dui;εi |2 + cε2i r2 .
Bs (4v)
As i tends to infinity the first two terms on the right-hand side both converge to
Θ(x), since Tx ν = Θ(x) H1 bTx Γ and the last term tends to zero.
Since Q2,1 (0) ⊂ B8 (4v) \ B1 (4v), it follows that
Z
lim
|∂ρ ui,εi |2 = 0.
i→∞ Q2,1 (0)
This completes the proof, because along Tx Γ ∩ B2 (0) the vector fields ∂ρ and v are
colinear and |∂v ui,εi |2 H3 converges to zero outside Tx Γ.
Step 2.2. We prove (7.6).
Define fi : B2 (0) ⊂ Tx Γ → [0, ∞) by
Z
fi (z) :=
|∂v ui;εi |2 (z, ·)
B1 (0)⊂Nx Γ
and denote by M fi : B1 (0) ⊂ Tx Γ → [0, ∞) the Hardy–Littlewood maximal
function associated with fi :
Z
1
M fi (z) := sup
fi .
s≤1 s Bs (z)⊂Tx Γ
17
By the weak-type L1 estimate for the maximal operator for each δ > 0
H1 ({z : M fi (z) ≥ δ}) .
kfi kL1
.
δ
Since kfi kL1 → 0 by the previous step, the assertion follows.
Step 3. We prove Proposition 7.1.
By Step 1,
1
lim inf
max
i→∞ w∈B1 (0)⊂Nx Γ δ
Z
|dui;εi |2 = Θ(x) ≥ ε0
Qδ,δ (zi ,w)
for all δ > 0, while for fixed i ∈ N and w ∈ B1 (0) ⊂ Nx Γ
Z
1
lim
|dui;εi |2 = 0.
δ↓0 δ Qδ,δ (zi ,w)
Hence, we can find a null sequence (δi ) such that
Z
1
|dui,εi |2 = ε0 /8.
max
w∈B1/δi (0)⊂Nx Γ δi Bδ (zi ,w)
i
Define
ũi := ũi (·) := ui;δi εi δi (zi , wi ) + · .
By construction
Z
max
w∈B1 (0)⊂Nx Γ B1 (0,w)
|dũi |2 = ε0 /8
with the maximum achieved at w = 0.
Let φ : Tx M → [0, ∞) denote a smooth function, which is compactly supported
in B1 (0) and equal to one on B1/2 (0). Define ei : B(r0 /δi εi )−1 (0) → [0, ∞) by
Z
ei (ζ, ω) := φ(·) · |dũi |2 (ζ, ω) + · .
By integration by parts, for v ∈ Tx Γ,
Z
|∂v ei (ζ, ω)| . kφkC 1
|∇ũi |(|∇v ũi | + o(1)).
B1 (ζ,ω)
The o(1)–term accounts for the connection on X being non-flat. By Step 2,
|∂v ei (ζ, ω)| goes to zero uniformly on compact subsets of Tx M . This means
that, for each z ∈ Tx Γ, w ∈ B1/δi (0) ⊂ Nx Γ and i z 1,
Z
|dũi |2 ≤ ε0 /4.
B1/2 (z,w)
18
∞ –bounds on ũ which allow
From Proposition 3.1 and Remark 3.3 we obtain Cloc
i
us to pass to a limit u : Tx M → X, which solves the Fueter equation. By Step 2,
u is invariant under translations along Tx Γ and, hence, the pullback of a map
z : Nx Γ → X. We can choose the orthonormal frame (v1 , v2 , v3 ) on Tx M constant
and with v1 = v ∈ Tx Γ and v2 , v3 ∈ Nx Γ. With respect to this frame the Fueter
operator takes the form
F = I(v1 )∂v + I(v2 )∂¯
with ∂¯ = ∂v + (−I(v))∂v . Thus z is (−I(v))–holomorphic.
2
3
Question 7.8. What happens near non-smooth points of Γ?
8
Constraints on tangent directions
By Proposition 7.1, if v ∈ STx Γ, then Xx must admit a non-trivial (−I(v))–
holomorphic sphere zx of area at most Θ(x). Since Θ is upper semi-continuous, it
achieves a maximum Amax on Γ. Thus, the area of zx is bounded by Amax and the
following shows that the possible tangent directions of Γ are strongly constrained.
Proposition 8.1. Let X be a simple hyperkähler manifold with b2 (X) ≥ 6. Given
Amax > 0, there exists only finitely many Iξ ∈ H(X) for which there exists a
rational curve C in (X, Iξ ) with
area(C) = h[C], ωξ i ≤ Amax .
Here ωξ = g(Iξ , ·, ·).
If X is a K3 surface, then this is essentially contained in Bryan–Leung [BL00,
Proposition 3.1]. Its proof mainly uses some facts about the K3–lattice (H 2 (K3, Z), ∪).
The appropriate replacement of the cup-product for general simple hyperkähler
manifold is the Beauville–Bogomolov–Fujiki (BBF) form q : S 2 H 2 (X, Z) → Z.
We refer the reader to [Bea83, Bog78, Fuj87] for details about the BBF form. For
our purposes it suffices to recall that:
• q is non-degenerate, i.e., the induced map H 2 (X, Q) → H 2 (X, Q)∗ is an
isomorphism. In particular, for each C ∈ H2 (X, Z) there exists a unique
γ ∈ H 2 (X, Q) such that
(8.2)
q(γ, ·) = hC, ·i ∈ H 2 (X, Q)∗ .
• q has signature (3, b2 (X) − 3) with span [ωξ ] : ξ ∈ S 2 forming a maximal
positive definite subspace. We denote the perpendicular maximal negative
definite subspace by N .
19
Theorem 8.3 (Amerik–Verbitsky). If X is a simple hyperkähler manifold with
b2 (M ) ≥ 6, then there exists an positive integer σ ∈ N such that
q(γ, γ) ≥ −σ
for all γ ∈ H 2 (X, Q) with (8.2) for some C represented by a Iξ –holomorphic
sphere for some Iξ ∈ H(X).
Proof. This follows by observing that γ is a MBM class in the sense of [AV14,
Definition 2.14] and then appealing to [AV14, Theorem 5.3].
Remark 8.4. Theorem 8.3 generalises the fact that any class representing a holomorphic sphere in K3 has square −2.
Proposition 8.5. There exists a constant c0 > 0 such that if C is represented by a
Iξ –holomorphic sphere of area A, then γ as in (8.2) is of the form
γ = β + c0 Aωξ
(8.6)
with β ∈ N and
q(β, β) ≥ −σ − c0 A2 .
Proof. It follows from (8.2) that
q(γ, ωη ) = 0
(8.7)
for all η ⊥ ξ; hence, γ = β + c0 Aωξ with c0 = 1/q(ωξ , ωξ ), which does not depend
on ξ ∈ S 2 , and β ∈ N . Since q(γ, γ) ≥ −σ, we have
q(β, β) ≥ −σ − c0 A2max .
Proof of Proposition 8.1. There are only finitely many γ as in Proposition 8.5 with
A ≤ Amax and γ determines ξ ∈ S 2 uniquely.
9
The singular set of u
We now begin to analyse sing(u).
Proposition 9.1. The monotonicity formula, i.e., Proposition 2.1, holds for u.
Repeating the arguments used earlier immediately establishes the following.
Corollary 9.2. Θu := Θ∗u : M → [0, ∞) is upper semi-continuous and sing(u) is
closed.
20
Proof of Proposition 9.1. The proof of Proposition 2.1 uses the fact that u is smooth
on all of M only in the application of Stokes’ theorem in (2.9), i.e., to show that
Z
Z
∗
u dΩ =
u∗ Ω.
Bρ (x)
∂Bρ (x)
To see that this still holds in the current situation, fix a cut-off function χ which
is zero on [0, 1] and one on [2, ∞], and set
χr (x) := χ(d(x, S)/r).
By the dominated convergence theorem
Z
Z
∗
u dΩ = lim
χr u∗ dΩ
r↓0 Bρ (x)
Bρ (x)
Z
Z
∗
=
u Ω − lim
r↓0
∂Bρ (x)
dχr ∧ u∗ Ω.
Bρ (x)
To see that the last term vanishes, recall that Ω is of type (0, 2) and |Ω| = O(r).
Therefore, |u∗ Ω| . r|∇u|2 . Since |dχr | . 1/r, the integral in the limit is bounded
by a constant times
Z
|∇u|2 ,
supp dχr
which converges to zero as r goes to zero.
In the following section we will repeatedly use integration by parts as above.
Each application can be justified in the same way as above.
10
Tangent cones
To further analyse sing(u) it is customary to study tangent cones.
Definition 10.1. Let x ∈ M and let (ri ) be a null-sequence. Let uri be the rescaling
of u centered at x; see (7.5). After passing to a subsequence this converges to a limit
(u∗ ; Γ∗ , Θ∗ ) with Γ∗ ⊂ R3 a 1–dimensional rectifiable set, u∗ : R3 \ Γ∗ → X a
Fueter map and Θ∗ : R3 → [0, ∞) an upper semi-continuous map supported on Γ∗ .
Such a limit is called a tangent cone to u at x.
Remark 10.2. If x ∈
/ sing(u), then u∗ is the constant map with value u(x) and
Γ∗ = ∅; hence, it is only interesting to consider tangent cones to x ∈ sing(u).
Question 10.3 (Uniqueness of tangent cones). Is (u∗ ; Γ∗ , Θ∗ ) independent of the
sequence (ri )?
21
Proposition 10.4. Every tangent cone (u∗ ; Γ∗ , Θ∗ ) is conical: for all r > 0,
s∗r (u∗ ; Γ∗ , Θ∗ ) = (u∗ ; Γ∗ , Θ∗ ).
Combing this with H1 (sing(u∗ )) = 0 yields the following.
Corollary 10.5. For every tangent cone (u∗ ; Γ∗ , Θ∗ ), sing(u∗ ) ⊂ {0}.
Remark 10.6. Proposition 10.4 can be viewed as an affirmative answer to the weaker
version of Question 10.3: is (u∗ ; Γ∗ , Θ∗ ) invariant under rescaling (ri ) to (r · ri )?
The proof relies on the following estimate.
Proposition 10.7. Let U ⊂ R3 be an open subset and u ∈ Γ(U, X) be a solution
of (1.10). Let φ ∈ Cc1 (R3 ) and R ≥ 1 such that [1, R] · supp(φ) ⊂ U . Then
Z
Z
2
2
φ(x)|∇u| (Rx) − φ(x)|∇u| (x)
Z R Z
(10.8)
.
(diam supp φ)kφkC 1 |∇r u||∇u|(rx)
r=1
supp φ
+ φ(x)|p ◦ u|2 (Rx) + |Rx|2 φ(x)δ|∇u|2 (rx)
with δ = |∇I| + |FX | + |R| as in (2.8).
Proof. Let us first assume that U is flat, I is parallel and u solves (1.6). The
derivative of
Z
f (R) := φ(x)|du|2 (Rx)
is
Z
φ(x)|x|∂r |du|2 (Rx) = − ∂r (|x|φ|)|du|2 (Rx)
Z
= − φ(x)|du|2 (Rx) + |x|(∂r φ)|du|2 (Rx).
f 0 (R) =
Z
Using the identities (2.4) and (2.6) we obtain
Z
Z X
3
2
− φ|du| = −2 φ
dxi ∧ u∗ ωi
i=1
Z
(10.9)
= −2 φ d(|x|u∗ ω∂r )
Z
= 2 |x|dφ ∧ u∗ ω∂r
Z
= |x|(∂r φ)(|du|2 − 2|∂r u|2 )
+ 2|x|(∂1 φ)ω∂r (∂2 u, ∂r u) + |x|(∂2 φ)ω∂r (∂r u, ∂1 u).
22
In the expression for f 0 (R) the contributions from φ(x)|du|2 (Rx) cancel and thus
Z
0
(10.10)
|f (R)| ≤ (diam supp φ)kφkC 1
|∂r u||du|(Rx).
supp φ
This yields the desired estimate by integration.
In general, since u only solves the perturbed Fueter equation, when using (2.7)
and (2.6) in the first and the last step of (10.9) we acquire the additional terms
Z
φ(x)|p ◦ u|2 (Rx) + ∂r (|x|φ)|p ◦ u|2 (Rx)
Moreover, in the second step of (10.9) the additional error term
Z
φ(x)O(|Rx|2 δ).
appears.
Proof of Proposition 10.4. Applying Proposition 10.7 along uri shows that the
measure
µ∗ = |du∗ |2 H3 + Θ∗ H1 bΓ∗
is conical, since terms on the right-hand side of (10.8) tend to zero as ri → 0. This
implies that Γ∗ and Θ∗ are conical. Moreover, the monotonicity formula shows that
∇r u∗ = 0; hence, u∗ is conical.
10.1
Filtration by tangent cones
Following Simon [Sim96, Section 3.4] we define a filtration S0 ⊂ S1 = sing(u)
according to the maximal number of lines a non-trivial tangent cone at x splits off:
Sk := {x ∈ sing(u) : all tangent cones to u at x split off at most k lines}
Here we say that a tangent cones splits of k lines if for some orthogonal splitting
Rk ⊕ R3−k it is the pullback from R3−k .
Remark 10.11. In our situation a non-trivial tangent cone can split off at most one
line. This is why the filtration has only two steps.
The significance of this filtration is that dim Sk ≤ k; see [Sim96, Section 3.4,
Lemma 1].
Proposition 10.12. If (u∗ ; Γ∗ , Θ∗ ) is non-trivial and splits of a line R ∈ R3 , then
Γ∗ = R, Θ∗ is a positive constant and u∗ is constant.
23
Proof. Certainly we must have Γ∗ ⊂ R. The Fueter equation and R–translationinvariance forces u∗ |R2 \{0} → X to be holomorphic. Since it is also conical, it
must be constant. Clearly, if Γ∗ 6= ∅, then Γ∗ = R and Θ∗ is constant.
Question 10.13. Can the situation in Proposition 10.12 occur?
Question 10.14. Are there tangent cones with Γ∗ 6= ∅?
If the answer to Question 10.13 is no, then it would follow that sing(u) is at
most 0–dimensional.
Remark 10.15. In a closely related situation Chen–Li [CL00, Step 1 in the proof
of Theorem 4.3] essentially assert that the answer to Question 10.14 is negative.
However, the author finds himself unable to follow the details of their argument.
10.2
Holomorphic sections of the conjugate twistor fibration
Tangent maps can be interpreted as follows. Let Tw(X) := H(X) × X ∼
= S2 × X
denote the conjugate twistor space of X equipped with the tautological complex
structure on Tw(X), given at (ξ, x) by (IS 2 ⊕ −Iξ ). Note the minus in front of Iξ
compared to the usual twistor space defined in Hitchin–Karlhede–Lindström–Roček
π
[HKLR87, Section 3.F]. The projection Tw(X) −
→ S 2 is called the conjugate
twistor fibration.
Proposition 10.16. Let u : S 2 → X. The section id × u ∈ Γ(Tw(X)) is holomorphic if and only if x 7→ u(x/|x|) is a Fueter map.
An example of such a section in case X is the Atiyah–Hitchin manifold was
discovered by Chen–Li [CL05]. It is an important open question whether such
sections do exist for compact hyperkähler manifolds.
10.3
Balancing of tangent cones
An interesting property of tangent cones is that they are balanced.
Proposition 10.17. If (u∗ ; Γ∗ , Θ∗ ) is a tangent cone, then
Z
X
x · |du∗ |2 = 0 and
x · Θ∗ (x) = 0.
S2
x∈S 2 ∩Γ∗
Remark 10.18. The analogous statement for harmonic maps was proved by Lin–
Rivière as [LR02, Theorem B(iii)].
The proof hinges on the following.
24
Proposition 10.19. If u ∈ Γ(M, X) satisfies (1.10), then for all x ∈ M and
v ∈ Γ(Br (x), T M )
Z
2
2
hv, ∂r i 2|∇u| + |∇r u| − 2h∇r u, ∇v ui
∂Br (x)
Z
Z
.
rk∇vkL∞ |∇u|2 + |p ◦ u|2 +
δ|∇u|2 + |∇(p ◦ u)|2
∂Br (x)
Br (x)
with δ = |∇I| + |FX | + |R| as in (2.8).
Proof. Again, we first asume that M is flat, X = M × X, I is parallel and u
satisfies (1.6). We can assume that v is constant, because of the first term on the
right-hand side. Consider
Z
|du|2 .
f (x) =
Br (x)
By integration by parts
Z
2
Z
hv, ∂r i|du|2 .
∂v |du| =
∂v f (x) =
Br (x)
∂Br (x)
On the other hand
3
X
Z
f (x) = 2
dxi ∧ u∗ ωi ;
Br (x) i=1
hence,
Z
∂v f (x) = 2
Br (x)
Lv
3
X
dxi ∧ u∗ ωi = 2
Z
i(v)
∂Br (x)
i=1
3
X
dxi ∧ u∗ ωi .
i=1
Choose (v1 , v2 , v3 ) = (∂r , ∂1 , ∂2 ) as in the proof of Proposition 2.2. Then
!
3
X
i
∗
i(v)
dx ∧ u ωi (e1 , e2 ) = hIv ∂1 u, ∂2 ui + hI1 ∂2 u − I2 ∂1 u, ∂v ui
i=1
= hIv ∂1 u, ∂2 ui + h∂r u, ∂v ui.
A slightly length computation shows that
1
hIv ∂1 u, ∂2 ui = − hv, ∂r i|du|2 − hv, ∂r ihIr ∂1 u, ∂2 i
2
= −hv, ∂r i|du|2 − hv, ∂r i|∂r u|2 .
25
Putting everything together yields
Z
Z
2
2
hv, ∂r i 2|du| + |∂r u| = 2
h∂r u, ∂v ui.
∂Br
∂Br
In general, one has to adapt the above argument as in the proof of Proposition 2.1
and obtains further error terms.
Proof of Proposition 10.17. Apply Proposition 10.19 along the sequence uri . The
terms on the right-hand side converge to zero and, by Proposition 9.1, so do the
terms involving ∇r uri . For all v ∈ Γ(Br (x), T M )
Z
Z
X
2
lim
hv, ∂r i|∇uri | =
hv, xi · |du∗ |2 +
hv, xi · Θ∗ (x);
i→∞ S 2
hence,
S2
Z
x∈S 2 ∩Γ∗
X
x · |du∗ |2 +
S2
x · Θ∗ (x) = 0.
x∈S 2 ∩Γ∗
The same reasoning applied to u∗ directly shows that
Z
x · |du∗ |2 = 0.
S2
This concludes the proof.
11
Tangent cones to (Γ, Θ)
We conclude the this article with a slight refinement of the regularity statement of Γ.
Definition 11.1. Let x ∈ Γ and let (ri ) be a null-sequence. Set
Γi := (exp ◦sri )−1 Γ and
Θi := (exp ◦sri )∗ Θ.
After passing to a subsequence Θi H1 bΓi weakly converges to a measure of the form
Θ∗ H1 bΓ∗ , with Γ∗ ⊂ R3 a 1–dimensional rectifiable set and Θ∗ : R3 → [0, ∞)
an upper semi-continuous map supported on Γ∗ . Such a pair (Γ∗ , Θ∗ ) is called a
tangent cone to (Γ, Θ) at x.
Again, it is important to ask Question 10.3.
Using Proposition 10.7 and Proposition 10.19 one can show the following.
26
Proposition 11.2. If (Γ∗ , Θ∗ ) is a tangent cone to (Γ, Θ), then Γ∗ is the cone on
the finite set Γ∗ ∩ S 2 :
[
Γ∗ =
[0, ∞) · x.
x∈Γ∗ ∩S 2
Θ∗ is constant on each (0, ∞) · x. Moreover,
X
Θ∗ (x) · x = 0.
x∈Γ∗ ∩S 2
This means, in particular, that Γ cannot suddenly end at a point and does not
contain any kinks.
A
The Heinz trick
Throughout we consider a bounded open subset U ⊂ Rn endowed with a smooth
metric g which extends smoothly to Ū . Implicit constants are allowed to depend on
the geometry of U .
Lemma A.1 (E. Heinz [Hei55]). Fix d > 0 and set
q :=
2
+ 1.
d
Suppose f : U → [0, ∞) and p, δ ∈ {0, 1} are such that the following hold:
1. We have
∆f . f q + f p .
2. If Bs (y) ⊂ Br/2 (x) ⊂ U , then
Z
Z
d−n
d−n
s
f .r
Br (y)
f + δr2 .
Br (x)
Then there exists a constant ε0 > 0 such that for all Br (x) ⊂ U with
Z
ε = rd−n
f ≤ ε0
Br (x)
we have
sup
f (y) . r−d ε + ((1 − p) + δ)r2 .
y∈Br/4 (x)
27
Remark A.2 (Heinz trick in the subcritical case). If n < d,
1
ε0 d−n
ε ≤ ε0 whenever r ≤ R
.
Uf
In particular,
for all compact K ⊂ U , kf kL∞ (K) is bounded a priori depending only
R
on U f and d(K, ∂U ).
We use the following standard result; see [GT01, Theorem 9.20] or [HNS09,
Proof of Theorem B.1].
Proposition A.3. For all Br (x) ⊂ U and every smooth function f : Br (x) →
[0, ∞)
Z
f (x) . r−n
f dvol + r2 k∆f kL∞ .
Br (x)
Proof of Lemma A.1. Define a function θ : Br/2 (x) → [0, ∞) by
θ(y) :=
r
2
− d(x, y)
d
f (y).
Since θ is non-negative and vanishes on the boundary of B r2 (x), it achieves its
maximum
M := max θ(y)
y∈B r (x)
2
in the interior of B r2 (x). We will derive a bound for M , from which the assertion
follows at once.
Let y0 be a point with θ(y0 ) = M , set
F := f (y0 )
and denote by
1 r
− d(x, y0 )
2 2
half the distance from y0 to the boundary of B r2 (x). Each y ∈ Bs0 (y0 ) has distance
from the boundary of B r2 (x) at least s0 ; hence,
s0 :=
−d
f (y) ≤ s−d
0 θ(y) ≤ s0 θ(y0 ) . F.
Proposition A.3 applied to Bs (y0 ) together with (1) and the above bound yields
Z
−n
F .s
f + s2 (F q + F p )
Bs (y0 )
28
for all 0 ≤ s ≤ s0 . Combined with (2) this becomes
F . s−d ε + s2 (F q + F p + δ) ,
which can be rewritten as
(A.4)
sd F . ε + sd+2 (F q + F p + δ) .
This inequality will yield the desired bound on M . It is useful to make a case
distinction.
Case 1. F ≤ 1.
In this case a bound on M follows from simple algebraic manipulations. If
p = 0 or δ = 1, then (A.4) with s = s0 yields
M = θ(y0 ) . sd0 F . ε + sd+2
≤ ε + rd+2 .
0
If p = 1 and δ = 0, this bound can be sharpened. (A.4) becomes
sd F ≤
cε
.
1 − cs2
If cs20 ≤ 12 , then we obtain
M . sd0 F . ε;
1
otherwise, setting s := (2c)− 2 ≤ s0 yields
F . ε,
and thus M . ε.
Case 2. F > 1.
From (A.4) we derive
sd F . ε + sd+2 F q
for all 0 ≤ s ≤ s0 . Set t := t(s) = sF 1/d . Then the above inequality can be
expressed as
td (1 − ct2 ) ≤ cε.
For sufficiently small ε > 0, the corresponding equation td (1 − ct2 ) = cε has d
1
small roots t1 , . . . , td , which are approximately ±(cε) d , and two large roots. Since
t(0) = 0 and by continuity, for each s ∈ [0, s0 ], t(s) must be less than the smallest
1
positive root; hence, t(s) . ε d for all s ∈ [0, s0 ]. This finishes the proof.
29
B
Compactness for Fueter maps with four dimensional
source manifold
Proposition B.1. Let V be a 4–dimensional Euclidean vector space, H a quaternionic vector space, I : SΛ+ V ∗ → SH an isometric identification of the unit
length self-dual forms on V with the unit quaternions and ι : Λ+ V ∗ → so(V ). The
endomorphism Ψ ∈ End(Hom(V, H)) defined by
ΨT :=
3
X
I(ωi ) ◦ T ◦ ι(ωi )
i=1
has eigenvalues 1 and −3. Here we sum over an orthonormal basis (ω1 , ω2 , ω3 ) of
Λ+ V ∗ . We denote the (−3)–eigenspace by HomI (V, H).
π
Let M be an orientable Riemannian 4–manifold, let X −
→ M be a bundle of
hyperkähler manifolds together with a fixed identification I : SΛ+ T ∗ M → H(X)
of the unit sphere bundle of self-dual forms on M and the bundle of hyperkähler
spheres of the fibres of X and fix a connection on X.
Definition B.2. A section u ∈ Γ(X) is called a Fueter section if
(B.3)
Fu := ∇u − Ψ∇u = 0 ∈ Γ(u∗ HomI (π ∗ T M, V X)).
Remark B.4. If M = R × N for some 3–manifold N , X is the pullback of a
bundle Y of hyperkähler manifolds on N , I is obtained from an identification
J : ST M ∼
= H(X) and the connection on X is the pullback of a connection on Y,
then (B.3) can be written as
∂t u − Fu = 0
with F denoting the 3–dimensional Fueter operator. This is the form in which the
4–dimensional Fueter operator appears in [HNS09].
Remark B.5. Unlike in the 3–dimensional case, Λ+ T ∗ M need not be trivial.7 Thus
the analogue of the setup in Example 1.5 rarely makes sense globally, and one is
almost forced to work with bundles of hyperkähler manifolds.
The analogue of Theorem 1.9 in the 4–dimensional case is the following result.
Theorem B.6. Suppose X is compact. Let (ui ) be a sequence of solutions of the
(perturbed) Fueter equation
Fui = p ◦ ui
7
Λ+ T ∗ M being trivial is equivalent to 3σ(M ) + 2χ(M ) = 0 and w2 (M ) = 0.
30
with p ∈ Γ(X, HomI (π ∗ T M, V X)) and
Z
|∇ui |2 ≤ cE
E (ui ) :=
M
for some constant cE > 0. Then (after passing to a subsequence) the following
holds:
• There exists a closed subset S with H2 (S) < ∞ and a Fueter section u ∈
∞.
Γ(M \ S, X) such that ui |M \S converges to u in Cloc
• There exist a constant ε0 > 0 and an upper semi-continuous function
Θ : S → [ε0 , ∞) such that the sequence of measures µi := |∇ui |2 H4
converges weakly to µ = |∇u|2 H4 + Θ H2 bS.
• S decomposes as
S = Γ ∪ sing(u)
with
Γ := supp(Θ H1 bS)
(
sing(u) :=
and
1
x ∈ M : lim sup 2
r
r↓0
Z
)
|∇u|2 > 0 .
Br (x)
Γ is H2 –rectifiable, and sing(u) is closed and H2 (sing(u)) = 0.
• For each smooth point of Γ there exists a non-trivial holomorphic sphere
in zx : S 2 → (Xx , −I(ξ)) with ξ a unit self-dual 2–form on Tx M , whose
associated complex structure preserves the splitting Tx M = Tx Γ ⊕ Nx Γ.
Moreover,
Z
Θ(x) ≥ E (zx ) :=
|dzx |2 .
S2
• If X is a bundle ofsimple hyperkähler manifolds
with b2 ≥ 6, then there is
2
a subbundle i ⊂ I ∈ End(T M ) : I = −id , depending only on sup Θ,
whose fibres are finite sets such that Tx Γ is complex with respect to a complex
structure I ∈ ix for all smooth points x ∈ Γ.
Sketch of the proof. The proof is analogous to that of Theorem 1.9 with a few minor
modifications:
• The renormalised energy is now
1
r2
Z
Br (x)
31
|∇u|2 .
• In the proof of the monotonicity formula one now uses the 4–form Λ ∈ Ω4 (X)
obtained from the section of Λ+ π ∗ T M ⊗ Λ2 V X induced by I. Direct computation shows that (2.7) still holds. Similarly, one can verify the analogue of
(2.6).
• The proof of the ε–regularity and convergence outside S carry over mutatis
mutandis.
• In the bubbling analysis, ui;λi will be asymptotically translation invariant
in the direction of Tx Γ. Fix a unit vector v0 ∈ Tx Γ. Since, asymptotically,
everything is invariant in the direction of v0 , we arrive back at the situation in
Section 7.
Acknowledgements The author is grateful to Misha Verbitsky for pointing out
his work with Amerik [AV14] and Costante Bellettini for a discussion of [BT15].
Moreover, the author thanks Gregor Noetzel for insightful comments.
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