SOME PROGRESS IN CONFORMAL GEOMETRY

Sun-Yung A. Chang, Jie Qing and Paul Yang

Dedicated to the memory of Thomas Branson

Abstract. In this paper we describe our current research in the theory of partial differential equations in conformal geometry. We introduce a bubble tree structure to study the degeneration of a class of Yamabe metrics on Bach flat manifolds satisfying some global conformal bounds on compact manifolds of dimension 4. As applications, we establish a gap theorem, a finiteness theorem for diffeomorphism type for this class, and diameter bound of the σ₂-metrics in a class of conformal 4-manifolds. For conformally compact Einstein metrics we introduce an eigenfunction compactification. As a consequence we obtain some topological constraints in terms of renormalized volumes.

1. Conformal Gap and finiteness Theorem for a class of closed 4-manifolds

1.1 Introduction

Suppose that (M⁴, g) is a closed 4-manifold. It follows from the positive mass theorem that, for a 4-manifold with positive Yamabe constant,

Z

σ₂dv ≤ 16π²

M

and equality holds if and only if (M⁴, g) is conformally equivalent to the standard 4-sphere, where

• 1
• 1

2

σ₂[g] =

R² − |E| ,

• 24
• 2

R is the scalar curvature of g and E is the traceless Ricci curvature of g. This is an interesting fact in conformal geometry because the above integral is a conformal invariant like the Yamabe constant.

Typeset by A S-T X

M E

1

• 2
• CONFORMAL GEOMETRY

One may ask, whether there is a constant ꢀ₀ > 0 such that a closed 4-manifold
M⁴ has to be diffeomorphic to S⁴ if it admits a metric g with positive Yamabe constant and

Z

σ₂[g]dv_g ≥ (1 − ꢀ)16π².

M

for some ꢀ < ꢀ₀? Notice that here the Yamabe invariant for such [g] is automatically close to that for the round 4-sphere. There is an analogous gap theorem of Bray and Neves for Yamabe invariant in dimension 3 [BN]. One cannot expect the Yamabe invariant alone to isolate the sphere, and it is more plausible to consider the integral of σ₂. We will answer the question affirmatively in the class of Bach flat 4-manifolds.
Recall that Riemann curvature tensor decomposes into

R_ijkl = W_ijkl + (A_ikg_jl − A_ilg_jk − A_jkg_il + A_jlg_ik),

in dimension 4, where W_ijkl is the Weyl curvature,

• 1
• 1

A_ij = (R_ij − Rg_ij)

• 2
• 6

is Weyl-Schouten curvature tensor and R_ij is the Ricci curvature tensor. Also recall that the Bach tensor is

1

B_ij = W_kijl,lk + R_klW_kijl

.

2
We say that a metric g is Bach flat if B_ij = 0. Bach flat metrics are critical metrics

2

R

for the functional _M |W| dv. Bach flatness is conformally invariant in dimension 4. It follows from Chern-Gauss-Bonnet,

Z

2

8π²χ(M⁴) =

(σ₂ + |W| )dv,

M

R

that _M σ₂dv is conformally invariant.
The gap theorem is as follows:

Theorem 1.1.1. Suppose that (M⁴, [g]) is a Bach flat closed 4-manifold with positive Yamabe constant and that

Z

2

(|W| dv)[g] ≤ Λ₀.

M

for some fixed positive number Λ₀. Then there is a positive number ꢀ₀ > 0 such that, if

Z

σ₂[g]dv_g ≥ (1 − ꢀ)16π²

M

• CHANG, QING AND YANG
• 3

holds for some constant ꢀ < ꢀ₀, then (M⁴, [g]) is conformally equivalent to the standard 4-sphere.

Our approach is based on the recent work on the compactness of Bach flat metrics on 4-manifolds of Tian and Viaclovsky [TV1][TV2], and of Anderson [An]. Indeed our work relies on a more precise understanding of the bubbling process near points of curvature concentration. For that purpose we develop the bubble tree structure in a sequence of metrics that describes precisely the concentration of curvature. Our method to develop bubble tree structure is inspired by the work of Anderson and Cheeger [AC] on the bubble tree configurations of the degenerations of metrics of bounded Ricci curvature. Our construction is modeled after this work but differs in the way that our bubble tree is built from the bubbles at points with the smallest scale of concentration to bubbles with larger scale; while the bubble tree in [AC] is constructed from bubbles of large scale to bubbles with smaller scales. The inductive method of construction of our bubble tree is modeled on earlier work of [BC], [Q] and [St] on the study of concentrations of energies in harmonic maps and the scalar curvature equations.
As a consequence of the bubble tree construction we are able to obtain the following finite diffeomorphism theorem:

Theorem 1.1.2. Suppose that A is a collection of Bach flat Riemannian manifolds (M⁴, g) with positive Yamabe constant, satisfying

Z

2

(|W| dv)[g] ≤ Λ₀,

M

for some fixed positive number Λ₀, and

Z

(σ₂dv)[g] ≥ σ₀,

M

for some fixed positive number σ₀. Then there are only finite many diffeomorphism types in A.

It is known that in each conformal class of metrics belonging to the family A, there is a metric g¯ = e^2wg such that σ₂(A_g¯) = 1, which we shall call the σ₂ metric. The bubble tree structure in the degeneration of Yamabe metrics in A is also helpful to understand the behavior of the σ₂-metrics in A. For example:

Theorem 1.1.3. For the conformal classes [g₀] ∈ A the conformal metrics g = e^2wg₀ satisfying the equation σ₂(g) = 1 has a uniform bound for the diameter.

1.2 The Neck Theorem

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• CONFORMAL GEOMETRY

The main tool we need to develop the bubble tree picture is the neck theorem which should be compared with the neck theorem in the work of Anderson and Cheeger [AC]. Due to the lack of point-wise bounds on Ricci curvature, our version of the neck theorem will have weaker conclusion. But it is sufficient to allow us to construct the bubble tree at each point of curvature concentration.
Let (M⁴, g) be a Riemannian manifold. For a point p ∈ M, denote by B_r(p) the geodesic ball with radius r centered at p, S_r(p) the geodesic sphere of radius r centered at p. Consider the geodesic annulus centered at p:

¯

A_{r ,r} (p) = {q ∈ M : r₁ ≤ dist(q, p) ≤ r₂}.

In general, A_{r ,r} (p) may have more than one connected components. We will consider any one component

• 1
• 2

¯

• 1
• 2

¯

A_{r ,r} (p) ⊂ Ar₁, r₂(p)

• 1
• 2

that meets the geodesic sphere of radius r₂:

\

A_{r ,r} (p) S_r (p) = ∅
Let H³(S_r(p)) be the 3D-Hausdorff measure of the geodesic sphere S_r(p).

• 1
• 2
• 2

Theoerm 1.2.1. Suppose (M⁴, g) is a Bach flat and simply connected 4-manifold with a Yamabe metric of positive Yamabe constant. Let p ∈ M, α ∈ (0, 1), ꢀ > 0, v₁ > 0, and a < dist(p, ∂M). Then there exist positive numbers δ₀, c₂, n depending on ꢀ, α, C_s, v₁, a such that the following holds. Let A_{r ,r} (p) be a connected compo-

• 1
• 2

nent of the geodesic annulus in M such that

• r₂ ≤ c₂a,
• r₁ ≤ δ₀r₂,

H³(S_r(p)) ≤ v₁r³, ∀r ∈ [r₁, 100r₁],

and

Z

2

|Rm| dv ≤ δ₀.

A_{r 2} (p)

,r

1

Then A_{r ,r} (p) is the only such component. In addition, for the only component

• 1
• 2

14
1

• A
• (p),

(δ₀⁻ −ꢀ)r₁,(δ ⁴ +ꢀ)r₂

0

(p), there exist some Γ ⊂ O(4), acting freely on S³,

1

which intersects with S

(δ₀⁴ +ꢀ)r₂

with |Γ| ≤ n, and an quasi isometry ψ, with

• 1
• 1

A

(p)

(δ₀^{− 4} +ꢀ)r₁,(δ ⁴ −ꢀ)r₂

0

• CHANG, QING AND YANG
• 5

(S³/Γ))

• 1
• 1

⊂ Ψ(C

δ₀^{− 4} r₁,δ₀⁴ r₂

14
1

⊂ A

(p)

(δ₀⁻ −ꢀ)r₁,(δ ⁴ +ꢀ)r₂

0

such that for all C _r,r(S³/Γ) ⊂ C has
(S³/Γ), in the cone coordinates, one

12

• 1
• 1

δ₀^{− 4} r₁,δ₀⁴ r₂

|(Ψ^∗(r⁻²g))_ij − δ_ij|_C ≤ ꢀ.

1,α

The first step in the proof is to use the Sobolev inequality to show the uniqueness of the connected annulus A_{r ,r} (p). The second step is to establish the growth of volume of geodesic spheres

• 1
• 2

H³(S_r(p) ≤ Cr³

for all r ∈ [r₁, ¹₂ r₂]. Here we rely on the work of Tian and Viaclovsky [TV1] [TV2] where they analyzed the end structure of a Bach-flat, scalar flat manifolds with finite L² total curvature. The last step is to use the Gromov and Cheeger compactness argument as in the work of Anderson and Cheeger [AC] to get the cone structure of the neck.

1.3 Bubble tree construction

In this section we attempt to give a clear picture about what happen at curvature concentration points. We will detect and extract bubbles by locating the centers and scales of curvature concentration.
We will assume here that (M_i, g_i) are Bach flat 4-manifolds with positive scalar curvature Yamabe metrics, vanishing first homology, and finite L² total curvature. Choose δ small enough according to the ꢀ-estimates and the neck Theorem in the previous section. Suppose that X_i ⊂ M_i contains a geodesic ball of a fixed radius r₀ and

Z

δ
|Rmⁱ|²dvⁱ ≤ ,

2

T_η0 (∂X_i)

where

T_η (∂X_i) = {p ∈ M_i : dist(p, ∂X_i) < η₀},

0

for some fixed positive number 4η₀ < r₀. Define, for p ∈ X_i,

Z

δ

s¹_i (p) = r such that

|Rmⁱ|²dvⁱ = .

2

B_rⁱ (p)

Let p¹_i = p such that s_i¹(p) = inf s¹_i (p).

B_tⁱ₀ (p_i)

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• CONFORMAL GEOMETRY

We may assume λ¹_i = s_i¹(p¹_i ) → 0, for otherwise there would be no curvature concentration in X_i. We then conclude that (M_i, (λ¹_i )⁻²g_i, p_i¹) converges to (M_∞¹ , g_∞¹ , p_∞¹ ), which is a Bach flat, scalar flat, complete 4-manifold satisfying the Sobolev inequality, having finite L² total curvature, and one single end.

Definition 1.3.1. We call a Bach flat, scalar flat, complete 4-manifold with the Sobolev inequality, finite L² total curvature, and a single ALE end a leaf bubble, while we will call such space with finitely many isolated irreducible orbifold points

an intermediate bubble.

Now, we define, for p ∈ X_i \ Bⁱ ₁ (p_i¹),

K¹λ_i

s²_i (p) = r such that

Z

δ
|Rmⁱ|²dvⁱ = .

2

B_rⁱ (p)\Bⁱ

₁ (p¹_i )

i

1

• K
• λ

Let

p²_i = p

such that

• s²_i (p) =
• inf

K

s²_i (p).

B_rⁱ (p)\Bⁱ

₁ (p¹_i )

i

1

λ

Again let λ²_i = s_i²(p²_i ) → 0. Otherwise there would be no more curvature concentration. Then

Lemma 1.3.2.

λ_i² λ_i¹

dist(p¹_i , p²_i )

+

→ ∞.

λ_i¹

There are two possibilities: dist(p¹_i , p²_i )
Case 1.

Case 2.

→ ∞

λ_i²

dist(p¹_i , p²_i )

λ_i²

≤ M¹.

In case 1, we certainly also have dist(p¹_i , p²_i )

→ ∞.

λ_i¹

• CHANG, QING AND YANG
• 7

Therefore, in the convergence of the sequence (M_i, (λ²_i )⁻²g_i, p_i²) the concentration which produces the bubble (M_∞¹ , g_∞¹ ) eventually escapes to infinity of M_∞² and hence is not visible to the bubble (M_∞² , g_∞² ), likewise, in the converging sequence (M_i, (λ_i¹)⁻²g_i, p¹_i ) one does not see the concentration which produces (M_∞² , g_∞² ). There are at most finite number of such leaf bubbles.

Definition 1.3.3. We say two bubbles (M_∞^j1 , g_∞^j1 ) and (M_∞^j2 , g_∞^j2 ) associated with (p^j_i¹ , λ^j_i¹ ) and (p_i^j2 , λ^j_i² ) are separable if