Servicios de Aseguramiento

We describe how to cast some of Joyce’s examples into a form to which theorem 7.1.3 applies by resolving only some orbifold singularities in a first step. This idea is similar to an outline argument for how to construct irreducible quasi-asymptotically locally Euclidean G2-manifolds in Joyce [27, p. 277].

Assume that the flat torus from which the construction starts has the form S¹ ×T⁶, where T⁶ is a torus with a flat Calabi-Yau structure. Let Γ be a finite group of G2 auto-morphisms of S¹×T⁶. Choose R-data and form the resolutionM of (S¹×T⁶)/Γ. Theorem 7.2.1 provides a family of closed G₂-structures ˜ϕ with small torsion on M, which can be perturbed to torsion-free G₂-structures. We would like to claim that these torsion-free G2-structures can also be obtained by perturbing a G2-structure with small torsion that satisfies the hypotheses of theorem 7.1.3.

Suppose that the elements of Γ act by products of isometries ofS¹ andT⁶ and let Γ^′ be the subgroup acting trivially on the S¹ factor. We assume that some elements of Γ act by reflection on S¹ (as has to be the case if b¹(M) = 0). If θ ∈S¹ is not a fixed point of any of the reflections then {θ} ×T⁶ does not meet the fixed point set of Γ\Γ^′, and its image divides (S¹×T⁶)/Γ into exactly two connected components. In order to make things work we assume furthermore that the fixed point set of Γ\Γ^′ does not meet that of Γ^′.

The R-data for (S¹×T⁶)/Γ can be restricted to give R-data for S¹ ×T⁶/Γ^′. Let M^′ be the corresponding resolution. Theorem 7.2.1 gives closed G2-structures ˜ϕ^′ with small torsion and by theorem 7.2.2 they can be perturbed to torsion-free G2-structures ϕ^′. M^′ is homeomorphic to S¹×X⁶ for a compact manifoldX.

The lemma below can be thought of as a simple version of the Cheeger-Gromoll line splitting theorem (cf. lemma 4.1.9) and ensures that there is a Calabi-Yau structure on X such that M^′ is isomorphic to S¹×X as a G2-manifold (cf. Chan [11, p. 15]).

Lemma 7.3.2. Let T^m be a torus and X a compact manifold with b¹(X) = 0. If g is a Ricci-flat metric on Tⁿ×X that is invariant under translations of the torus factor then there is a function f :X→Rⁿ such that the graph diffeomorphism

Tⁿ×X →Tⁿ×X, (t, x)7→(t+f(x), x) pulls g back to a product metric.

Sketch proof. Let _∂θ^∂1, . . . ,_∂θ^∂n be the unit coordinate vector fields onTⁿ, and setαi = _∂θ^∂i

♭.

∂

∂θⁱ is a Killing vector field, so αi is harmonic by proposition 5.2.4. Since b¹(X) = 0 the

closed forms αi|_X are exact. Define f :X →Rⁿ by picking fi such thatαi =dfi.

The compatibility conditions for the R-data ensure that the quotient group Ψ = Γ/Γ^′ acts in a well-defined way on M^′. Moreover ˜ϕ^′ is invariant under this action and hence ϕ^′ is too. We can use the R-data to resolve the singularities of M^′/Ψ and topologically this gives M. If I ⊂ S¹ is an interval not containing any fixed points of the reflections then I×X maps homeomorphically onto its imageN in M and is a candidate for a cylindrical neck. We wish to to define a closedG2-structure onM with small torsion, whose restriction to N isϕ^′.

The issue is that theorem 7.2.2 relies on the orbifold singularities that are to be resolved with small torsion being modelled on a quotient of the flat G2-structure. But ϕ^′ need not be flat near the fixed point set F of Ψ.

LetS be a tubular neighbourhood ofF. Recall thatϕ^′ = ˜ϕ^′+dη^′for somedη^′satisfying the estimates (7.3). The assumption that the fixed point sets of Γ^′ and Γ\Γ^′ are disjoint ensures that ˜ϕ^′|_S is flat. In order to ‘restore’ the flatness nearF we wish to define an exact form that is supported on S, equal to dη^′ near F, and small in the same sense that dη^′ is small. To do this we use a version of the classical Poincar´e lemma.

Lemma 7.3.3. Let F be a compact Riemannian manifold and I a bounded open interval.

For any n ≥ 0, k ≥ 0 and p ≥ 1 there is a constant C > 0 such that for any exact L^p_k m-formdηon the Riemannian product X =F×Iⁿ there is an(m−1)-formχwithdχ=dη and

kχk_L^p

k+1 < Ckdηk_L^p

k. (7.14)

Proof. The proof is by induction on n. The result holds for n= 0 by usual Hodge theory.

For the inductive step, we show that if a manifoldXsatisfies the conclusion of the theorem, then so does X×I (with the product metric).

For s ∈ I let X_s denote the hypersurface X× {s}. Let t denote the coordinate on I, and write

dη =α+dt∧β,

with α and β sections of the pull-back of Λ^∗T^∗X to X × I. Write α(s), β(s) for the corresponding forms on Xs. Fix s0 ∈I and let

χ1(s) = Z s

s0

β(t)dt.

Let ∇denote the covariant derivative on X×I, and consider χ1 as a form on X×I. For any 0≤i≤k and s∈I

k(∇ⁱχ1)(s)k^p_Lp(Xs) = Z

X

Z s

s0

(∇ⁱβ)(t)dt

p

volX

≤V^p−1 Z

X

Z s

s0

k(∇ⁱβ)(t)dtk^pdt vol_X ≤V^p−1k∇ⁱβk^p_Lp(X×I),

where V is the length of I. Hence k∇ⁱχ1k^p_Lp(X×I)≤

Z

I

k(∇ⁱχ1)(s)k^p_Lp(Xs)ds ≤V^pk∇ⁱβk^p_Lp(X×I), and

kχ₁k_L^p

k(X×I) ≤Vkdηk_L^p

k(X×I).

d(η−χ₁) has nodt-component, so thedt-component ofd²(η−χ₁) is _∂t^∂d(η−χ₁) = 0. Hence d(η−χ1) is the pull-back to X×I of an exact form on X. By the inductive hypothesis there is a form χ2 such that dχ2 =d(η−χ1) and χ=χ1+χ2 satisfies (7.14) for some C independent of dη.

Therefore there is a 2-formχ on S such that dχ=dη^′|_S

and χ satisfies estimates proportional to (7.3) (the uniform estimate comes from theorem 7.2.3). If ρis a cut-off function which is 1 near F and 0 outsideS then

kd(ρχ)k_L² < K^′t⁴, kd(ρχ)k_C⁰ < K^′t^1/2, k∇d(ρχ)k_L¹⁴ < K^′, (7.15) with K^′ independent of t. ˜ϕ^′+d(η^′−ρχ) is a family of closedG2-structures which are flat near F. It is clear from the chain rule that torsion is small, but we need to take care to choose the small formψ^′ such thatd∗ψ^′ =dΘ( ˜ϕ^′+d(η^′−ρχ)) in such a way that it vanishes not only on the cylindrical neck region, but also near F. Because F has dimension 3 any closed 4-form on the tubular neighbourhood S is exact. By lemma 7.3.3 we can write

(Θ( ˜ϕ^′ +dη^′)−Θ( ˜ϕ^′))|_S =dχ^′

for some 3-formχ^′ onS, so that d(ρχ^′) satisfies estimates of the form (7.15). We can then

take

ψ^′ =∗(Θ( ˜ϕ^′+d(η^′−ρχ))−Θ( ˜ϕ^′+dη^′) +d(ρχ^′)), which is supported in S but 0 near F.

We can ensure that all forms are invariant under Ψ, sod(η^′−ρχ) descends to an exact 3-form β on the orbifold M^′/Ψ. As it is supported away from the singular set, β is also defined on the resolution M. In the same way ψ^′ descends to a small 3-formψ1 onM.

Recall that we denoted by ˜ϕ the G2-structure on M with small torsion obtained by resolving (S¹×T⁶)/Γ. ˜ϕ^′+d(η^′−ρχ) descends to an orbifold G2-structure on M^′/Ψ with small torsion. Its orbifold singularities are modelled on quotients of the flat G2-structure, so the singularities can be resolved like in theorem 7.2.1 to define a closed G2-structure on M. This is precisely ˜ϕ+β. The torsion introduced by the resolution is small, in the sense that there is a smooth 3-form ψ2 on M, supported near the pre-image F^′ of the singular set, such that d^∗ψ2 = d^∗ϕ˜ near F^′ and ψ2 satisfies the estimate (i) in theorem 7.2.1. Now ( ˜ϕ +β, ψ₁ +ψ₂) satisfies the hypotheses of theorem 7.1.3 (for t sufficiently small). Moreover, proposition 7.2.4 shows that the torsion-free G2-structures obtained by perturbing the ‘one-step’ resolution ˜ϕand the ‘two-step’ resolution ˜ϕ+βare diffeomorphic.