Obstruction-flat asymptotically locally Euclidean metrics Jeff - - PowerPoint PPT Presentation

Origin Linearized Equation Gauge-fixing Approach Delicate Issues

Obstruction-flat asymptotically locally Euclidean metrics

Jeff Viaclovsky (joint work with Antonio Ach´ e) January 11, 2012

Origin Linearized Equation Gauge-fixing Approach Delicate Issues

Obstruction tensor

(Mn, g) Riemannian Manifold, n ≥ 4, n even.

Question

When is g conformal to an Einstein metric? Ric(g) = Rg n g,

Origin Linearized Equation Gauge-fixing Approach Delicate Issues

Theorem (Fefferman-Graham,1985)

If n ≥ 4 is even there exists a nontrivial 2-tensor O(g) satisfying

• O(u2g) = u2−nO(g)
• If g is locally conformally Einstein ⇒ O(g) = 0.

Origin Linearized Equation Gauge-fixing Approach Delicate Issues

Other properties of O

• O(g) is a local tensor invariant of g
• Trace-free
• Divergence-free (variational structure, Q- curvature).

Origin Linearized Equation Gauge-fixing Approach Delicate Issues

Origin

The Poincar´ e Metric gPoincar´

e =

4ds2 (1 − |y|2)2 Defined on Bn+1 = Unit Ball in Rn+1

Origin Linearized Equation Gauge-fixing Approach Delicate Issues

Cauchy problem

• 1-Parameter family of metrics {gr}r, with r ∈ [0, 1] defined in

M.

• Set g+ = r−2

dr2 + gr

• ,

Problem (C. Fefferman, R. Graham ’85)

Solve for {gr} at the level of power series satisfying Ric(g+) + ng+ = 0, Subject to

• g0 = g,
• gr even in r.

Origin Linearized Equation Gauge-fixing Approach Delicate Issues

Theorem (Fefferman-Graham 1985)

• If n is odd there is a unique formal power series solution to

the Cauchy problem (up to diffeomorphims fixing M).

• If n is even, we can only prescribe g+ to solve

Ric(g+) + ng+ = O(rn−2), r → 0 the solution is unique up to diffeomorphisms fixing M and terms of order rn−2.

Origin Linearized Equation Gauge-fixing Approach Delicate Issues

Question

What obstructs the existence of Poincar´ e metrics in even dimensions?

Origin Linearized Equation Gauge-fixing Approach Delicate Issues

Theorem (Graham-Hirachi, ’05)

Given (Mn, g) with n even, take a solution of Ric(g+) + ng+ = O(rn−2). Then O(g) is given by O(g) = tf

• r2−n [Ric(g+) + ng+]
• ,

where tf = “Trace-free part”.

Origin Linearized Equation Gauge-fixing Approach Delicate Issues

Two types of results

• Decay improvement for Asymptotically Locally Euclidean

Metrics

• Singularity Removal Theorems for isolated orbifold

singularities

• In both cases: “Ellipticity” of the system O(g) = 0

Origin Linearized Equation Gauge-fixing Approach Delicate Issues

Spaces with coordinates at infinity

We are interested in spaces satisfying

• (M, g) complete Riemannian manifold,
• Non-compact
• Outside of a compact set M is “locally Euclidean”, i.e., there

exists a compact set K ⊂ M and a diffeomorphism ψ such that ψ : M/K → (Rn \ BR(0)) /Γ,

• Γ finite subgroup of SO(n),
• Γ acts freely on Rn/BR(0).

Origin Linearized Equation Gauge-fixing Approach Delicate Issues

Asymptotically locally Euclidean (ALE) metrics

A complete Riemannian manifold (M, g) is called asymptotically locally Euclidean or ALE of order τ > 0 if we can find coordinates at infinity as before such that g − gEuc = O(r−τ), r → ∞, r = “distance to some fixed base-point”.

• Decay in derivatives

∂αg = O(r−τ−|α|) as r → ∞, |α| ≥ 1.

Origin Linearized Equation Gauge-fixing Approach Delicate Issues

Special case: ALE of order 0

(M, g) is ALE of order 0 if

• There exists a coordinate system as before
• g − gEuc has pointwise decay at infinity, i.e.,

g − gEuc = o(1) as r → ∞,

• ∂mg decays like

∂mg = o(r−m) as r → ∞, m ≥ 1.

Origin Linearized Equation Gauge-fixing Approach Delicate Issues

Decay improvement theorem

Theorem (Ach´ e - V)

Let (Mn, g) (n even)

• g Obstruction-flat,
• Scalar-flat,
• ALE of order 0 then (M, g) is ALE of order 2.

Origin Linearized Equation Gauge-fixing Approach Delicate Issues

Example

• (Mn, g) even-dimensional compact Einstein manifold with

positive scalar curvature,

• Gx the Green’s function of the conformal Laplacian at the

point x.

• ˆ

M = M \ {x}

• ˆ

g = G

4 n−2

x

g The manifold ( ˆ M, ˆ g) satisfies

• Asymptotically-flat
• Obstruction-flat (locally conformally Einstein) and scalar-flat

Origin Linearized Equation Gauge-fixing Approach Delicate Issues

Explicit formulas (Bach tensor)

Rm = W + A g, where

• W is the Weyl tensor ( conformally invariant),
• A is the Schouten tensor

A = 1 n − 2

• Ric −

R 2(n − 1)g

• Then the Bach tensor is given by

Bij = ∆gAij − ∇k∇iAkj + AklWikjl,

Origin Linearized Equation Gauge-fixing Approach Delicate Issues

Properties of the Bach tensor

• B is the first variation of the functional

g →

• M

|W|2dvol,

• Conformally invariant in dimension 4,
• Schematically

B = ∆Ric + ∇2Rg + ∆gRgg + Rm ∗ Rm,

• In dimension 4, O = B.

Origin Linearized Equation Gauge-fixing Approach Delicate Issues

Explicit formulas

• In dimension n > 4, B is not conformally invariant.
• In dimension n = 6

O =∆gBij − 2WkijlBkl − 4Ak

kBij

+ 8Akl∇lC(ij)k − 4Ckl

i Cljk + 2Ckl i Cjkl

+ 4∇lAk

kCl (ij) − 4WkijlAk mAml.

C is the Cotton tensor given by C = d▽A.

Origin Linearized Equation Gauge-fixing Approach Delicate Issues

• In general there are no explicit formulas available.
• R. Graham and K. Hirachi proved the following

O = ∆

n 2 −2

g

Bg + l.o.t.

• The terms l.o.t., are quadratic and higher in Rm and its

derivatives

• More precisely

l.o.t. =

n/2

• j=2

 

• α1+...+αj=n−2j

∇α1Rm ∗ . . . ∗ ∇αjRm   .

Origin Linearized Equation Gauge-fixing Approach Delicate Issues

If g is scalar-flat, this system looks like ∆

n 2 −1

g

Ric = l.o.t. A more general theorem

Theorem (Ach´ e- V)

If (Mn, g) with n ≥ 4 (not necessarily even) is ALE of order 0, Scalar-flat and satisfies ∆kRic = l.o.t., for 1 ≤ k ≤ n

2 − 1, then (M, g) is ALE of order n − 2k. Also true

for k = 1 and n = 3.

Origin Linearized Equation Gauge-fixing Approach Delicate Issues

Previous works

Theorem (J. Cheeger, G. Tian, ’94)

(Mn, g) n ≥ 3 ALE of order 0 and Ricci-flat then (M, g) is ALE of

• rder n,
• Gauge-fixing approach by means of the implicit function

theorem (divergence-free gauge),

• Three annulus lemma (L. Simon).
• Improvement of earlier work of Bando-Kasue-Nakajima, who

proved order n − 1 using refined Kato inequalities (n = 4 they

• btained optimal order 4).

Origin Linearized Equation Gauge-fixing Approach Delicate Issues

Previous works

Theorem (G. Tian - V)

(M4, g) ALE of order 0, Scalar-flat and either

• 1. Self-dual or anti-self-dual
• 2. Harmonic curvature tensor

Then (M, g) is ALE of order τ for any τ < 2

Origin Linearized Equation Gauge-fixing Approach Delicate Issues

Previous works (higher order equations)

Theorem (J. Streets)

(M4, g) ALE of order 0, scalar-flat and Bach-flat. Then (M, g) is ALE of order 2.

• Bach-flat condition generalizes (anti)self-duality and Ricci-flat

conditions.

• Gauge-fixing approach (as in Cheeger-Tian)

Origin Linearized Equation Gauge-fixing Approach Delicate Issues

The compact case

• Bρ(0) metric ball on a flat cone C(Sn−1/Γ), Γ ⊂ SO(n) is a

finite subgroup acting freely on Sn−1.

• g defined on Bρ(0) \ {0}.
• The origin is a C0-orbifold point for g. That is, there exists a

coordinate system around the origin such that g − gEuc = o(1) r → 0, and also, for any m ≥ 1 ∂mg = o(r−m) as r → 0.

Origin Linearized Equation Gauge-fixing Approach Delicate Issues

Singularity removal

Theorem (Ach´ e-V)

If g defined on Bρ(0) \ {0} is

• Obstruction-flat,
• Constant scalar curvature
• If 0 is a C0-orbifold singularity for g, then g extends to a

smooth orbifold metric in Bρ(0). That is, after diffeomorphism, there is a smooth Γ-invariant metric ˜ g on the universal cover of Bρ(0) \ {0} such that ˜ g descends to g.

Origin Linearized Equation Gauge-fixing Approach Delicate Issues

General problem

Study the system O(g) = 0 as a Nonlinear PDE on g. Important Difficulties

• O(g) = 0 is a higher order system (at least fourth order),
• g → O(g) is not elliptic due to diffeomorphism invariance.

Origin Linearized Equation Gauge-fixing Approach Delicate Issues

If g is ALE of order 0 the difference g − gEuc has pointwise decay at infinity. Set h = g − gEuc O(gEuc + h) = 0, Linearize at the flat metric gEuc 0 = O(gEuc + h) = O′

gEuc(h) +

R(h)

Error term

.

Origin Linearized Equation Gauge-fixing Approach Delicate Issues

• From scaling properties of O, if h is small in the right norm

then the error term R(h) should be “small” O′(h) = − R(h)

small

One gets useful information by looking at the linear system O′

gEuc(h) = 0

Origin Linearized Equation Gauge-fixing Approach Delicate Issues

Overview

• Study of the linearized equation at a flat metric.
• Estimate on the nonlinear terms.
• Use gauge invariance to “eliminate” the degeneracy of the

system.

• Detailed construction of the divergence-free gauge.

Origin Linearized Equation Gauge-fixing Approach Delicate Issues

The linearized equation

Linearization of O in the direction of h at gEuc O′

gEuc(h) := ∂

∂ǫO(gEuc + ǫh)|ǫ=0,

Idea

Before studying the non-linear equation, we study first the linear system O′

gEuc(h) = 0.

Origin Linearized Equation Gauge-fixing Approach Delicate Issues

Recall O(g) = ∆

n 2 −2B + l.o.t.

At a flat metric the terms l.o.t. linearize to zero. Therefore O′

gEuc(h) = ∆

n 2 −2B′

gEuc(h).

Schematically, we can write O′

gEuc(h) = −

1 2(n − 2)∆n/2h + L(∂n−1δh, ∂n−2∆tr(h)), here L(A, B) = Linear combination of A and B.

Origin Linearized Equation Gauge-fixing Approach Delicate Issues

Idea

Prescribe h to be divergence-free and its trace to be harmonic. The linearized equation becomes ∆n/2h = 0, δh = 0, ∆tr(h) = 0. In that case, each component of h satisfies ∆

n 2 f = 0.

Origin Linearized Equation Gauge-fixing Approach Delicate Issues

Lemma

In Rn, let h satisfy ∆

n 2 h = 0,

δh = 0, h = O(r1−ǫ) as r → ∞. Then h has an expansion at infinity of the form h = A + O(r−2), where the components of A are constant.

Origin Linearized Equation Gauge-fixing Approach Delicate Issues

Key observations

From the condition δh = 0 one can rule out

• The solution r−1h1 (least decaying),
• Any logarithmic solution,

A similar argument works for the other cases, i.e., for ∆k+1h = 0, to conclude h = A + O(r−n+2k), as r → ∞.

Origin Linearized Equation Gauge-fixing Approach Delicate Issues

Equation on the trace

The linearized equation subject to δh = 0 becomes − 1 2(n − 2)∆n/2h + L(∂n−2∆tr(h)) = 0,

Idea

Trace the equation and obtain an equation on tr(h).

Issue

The tensor O′

gEuc(h) is traceless for any h.

Origin Linearized Equation Gauge-fixing Approach Delicate Issues

We use the linearized scalar-flat equation R′(h)gEuc(h) = −∆tr(h) + δδh,            O′

gEuc(h) = 0,

R′

gEuc(h) = 0,

δh = 0. Is equivalent to    ∆n/2h = 0, δh = 0 ∆tr(h) = 0.

Origin Linearized Equation Gauge-fixing Approach Delicate Issues

Gauge-Fixing Approach

Use the diffeomorphism invariance of the system O(g) = 0, Rg = 0, to fix a gauge φ such that h = φ∗g − gEuc is divergence-free with respect to gEuc. φ∗g = gEuc + h,

Origin Linearized Equation Gauge-fixing Approach Delicate Issues

The obstruction-flat equation is now 0 = O(gEuc + h) = O′

gEuc(h) + R(h),

where R(h) = Remainder, Study O′

gEuc(h) = −R(h).

Origin Linearized Equation Gauge-fixing Approach Delicate Issues

Estimate on the remainder

∇φ∗g = ∇gEuc+h, ∇gEuc+hT = ∇gEucT + (gEuc + h)−1 ∗ ∇h ∗ T, Schematically Rm(gEuc + h) = ∇gEuc+hΓ(gEuc + h), B(gEuc + h) = ∇2

gEuc+hRm(gEuc + h)

Keep track of all error terms

Origin Linearized Equation Gauge-fixing Approach Delicate Issues

|R(h)| ≤ C  |h||∇nh| +

In

• j=2
• α1+...+αj=n

|∇α1h| . . . |∇αjh|   , Under scalar-flat and divergence-free conditions ∆n/2h = −R(h), Fix a Gauge φ such that h = φ∗g − gEuc satisfies

• δh = 0
• h = O(r−β), β > 0

Then h = O(r−β) and ∆n/2h = O(r−2β−n).

Origin Linearized Equation Gauge-fixing Approach Delicate Issues

Application to the obstruction-flat equation

Recall

• h = O(r−β), β > 0 as r → ∞,
• δh = 0,
• ∆n/2h = R(h) = O(r−2β−n), r → ∞

(Better than expected!) Write h = h′ + r−1h1 + O(r−2), where

• h′ is roughly O(r−2β) at infinity (Better decaying)
• The components of h1 are spherical harmonics of order 1

Origin Linearized Equation Gauge-fixing Approach Delicate Issues

Bootstrap to obtain h = r−1h1 + O(r−1−ǫ), r → ∞. From δh = 0 we conclude that δ(r−1h1) = 0 ⇒ r−1h1 ≡ 0. h = O(r−1−ǫ), use a bootstrap argument to conclude h = O(r−2), r → ∞.

Origin Linearized Equation Gauge-fixing Approach Delicate Issues

Delicate issues

• The ALE of order 0 condition does not give us any decay

improvement

• How to construct a divergence-free gauge?
• If φ∗g is divergnce-free, can we guarantee that

h = φ∗g − gEuc decays as r−β at infinity?

Origin Linearized Equation Gauge-fixing Approach Delicate Issues

Existence of divergence-free gauges

Implicit function theorem: Consider the nonlinear map φ → δgEuc (φ∗g) , In order to apply the implicit function theorem, we need to consider the linearized equation δgEucLXgEuc = −δEuch, i.e., δgEucLXgEuc = O(r−1) as r → 0.

Origin Linearized Equation Gauge-fixing Approach Delicate Issues

• The map X → δgEucLXgEuc has kernel that makes the

implicit function theorem fail (linear kernel).

• If g is ALE of positive order, the implicit function theorem can

be applied and produce a δ-free gauge.

• One has to modify the gauge condition

δth = δh − tir−1 ∂

∂r , t = 0,

We try to prescribe δth = 0,

Origin Linearized Equation Gauge-fixing Approach Delicate Issues

• Apply the implicit function theorem to

δtX = δtLXgEuc,

• The only linear kernel of t are forms which are dual to

Killing fields,

• On Rn \ Bρ(0), the operator t is invertible given that we fix

appropriate boundary conditions.

Origin Linearized Equation Gauge-fixing Approach Delicate Issues

Theorem (Cheeger-Tian)

If g is ALE of order 0, there exists a a gauge φ such that for t = 0 small δt (φ∗g − gEuc) = 0, moreover, h = φ∗g − gEuc has pointwise decay at infinity.

Origin Linearized Equation Gauge-fixing Approach Delicate Issues

Set h = φ∗g − gEuc, then O(gEuc + h) = 0, R(g + h) = 0 δh = tir−1 ∂

∂r h,

The obstruction-flat equation takes the form Pth + R(h) = 0, where Pt is a small perturbation of ∆n/2.

Origin Linearized Equation Gauge-fixing Approach Delicate Issues

Using separation of variables, we can write solutions of Pth = 0 as a sum h = h+ + h− + h0 where h+ =

∞

• j=1

nj

• k=1

log(r)nkrα+

k Tj

(growth solution), h− =

∞

• j=1

nj

• k=1

log(r)nkrα−

k Tj

(decay solution), h0 =

∞

• j=1

nj

• k=1

log(r)nkrα0

kTj

(degenerate solution)

Origin Linearized Equation Gauge-fixing Approach Delicate Issues

Growth estimate

• Let β = inf{|Re(α±)|} > 0.
• Consider the weighted norm

|||h|||2

a,b =

b

a

• Sn−1 h2dVgSn−1r−1dr,

Fix 0 < β′ < β. Write a solution of Pth = 0 as h = h+ + h− + h0,

Origin Linearized Equation Gauge-fixing Approach Delicate Issues

Growth estimate

For L > 0 large the growth solution satisfies Lβ

′

|||h+|||1,L ≤ |||h+|||L,L2 The decay solution satisfies |||h−|||L,L2 ≤ L−β

′

|||h−|||1,L

Proof.

Turan’s Lemma.

Origin Linearized Equation Gauge-fixing Approach Delicate Issues

Three annulus lemma

Write a solution of Pth = 0 as h = h+ + h− + h0, and assume that h0 ≡ 0. If Lβ′|||h|||1,L ≤ |||h|||L,L2 then Lβ′|||h|||L,L2 ≤ |||h|||L2,L3 i.e., if h grows in A1,L(0), then it continues growing. (continued)

Origin Linearized Equation Gauge-fixing Approach Delicate Issues

If |||h|||L2,L3 ≤ L−β′|||h|||L,L2, then |||h|||L,L2 ≤ L−β′|||h|||1,L. Moreover one of the two inequalities has to hold.

• It is crucial to rule out degenerate solutions.

Origin Linearized Equation Gauge-fixing Approach Delicate Issues

Nonlinear three annulus theorem

Lemma

Let g = gEuc + h be obstruction-flat, scalar flat and δt-free on Rn \ Bρ(0). There exists χ > 0 such that if

• hCm,α < χ on Rn \ Bρ(0),
• Pth = 0 has no degenerate solutions.

Then h satisfies the Three Annulus Lemma.

Proof (Outline).

If the growth estimate in the Three Annulus Lemma is not satisfied, use a rescaling construction to produce a solution of the nonlinear system with the wrong behavior.

Origin Linearized Equation Gauge-fixing Approach Delicate Issues

Proof of main theorem

• Prove there are no degenerate solutions for t = 0 small.
• From the Three Annulus Lemma and the ALE condition one

rules out growth solutions.

• Using standard elliptic estimates one obtains h = O(r−β′),

r → ∞.

• Fix a divergence-free gauge where g is ALE of positive order.
• Bootstrap argument from above.

Origin Linearized Equation Gauge-fixing Approach Delicate Issues

Degenerate solutions

Lemma

For t = 0 there are no degenerate solutions of the linear system satisfying δth = 0. All degenerate solutions in separated variables satisfy δh = O(r−γ−1) as r → 0, δh = O(rγ−1) as r → ∞, with 0 < γ < 1 small. Consider the map X = δgEucLXgEuc,

Origin Linearized Equation Gauge-fixing Approach Delicate Issues

Using the theory of Nirenberg-Walker one solves for X satisfying X = O(r1+γ) r → ∞, X = O(r1−γ) r → 0, X = δh. It follows that h0 = h − LXgEuc satisfies δh0 = 0. We now prove that h has to be a Lie derivative

Origin Linearized Equation Gauge-fixing Approach Delicate Issues

• LXgEuc is a solution of the linearized system
• h0 is also a solution and hence ∆n/2h0 = 0
• h0 is O(r−γ) near zero and O(rγ) at infinity,
• The condition δgEuch0 = 0 rules out this behavior unless h0 is

constant

Origin Linearized Equation Gauge-fixing Approach Delicate Issues

All degenerate solutions are Lie derivatives LXgEuc that satisfy δtLXgEuc = 0 X is “essentially linear” i.e., X =O(r1−δ) r → 0, X =O(r1+δ) r → ∞. The only essentially linear solutions of δtLXgEuc = 0, for t = 0 small are dual to Killing fields.