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) December 12, 2011

Origin Linearized Equation Gauge-fixing Approach Delicate Issues Obstruction tensor ( M n , g ) Riemannian Manifold, n ≥ 4 , n even. Question When is g conformal to an Einstein metric? Ric ( g ) = R g 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 ( u 2 g ) = u 2 − n O ( 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 4 ds 2 g Poincar´ e = (1 − | y | 2 ) 2 Defined on B n +1 = Unit Ball in R n +1

Origin Linearized Equation Gauge-fixing Approach Delicate Issues Cauchy problem • 1-Parameter family of metrics { g r } r , with r ∈ [0 , 1] defined in M . • Set g + = r − 2 � dr 2 + g r � , Problem (C. Fefferman, R. Graham ’85) Solve for { g r } at the level of power series satisfying Ric ( g + ) + ng + = 0 , Subject to • g 0 = g , • g r 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 ( r n − 2 ) , r → 0 the solution is unique up to diffeomorphisms fixing M and terms of order r n − 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 ( M n , g ) with n even, take a solution of Ric ( g + ) + ng + = O ( r n − 2 ) . Then O ( g ) is given by � r 2 − n [ Ric ( g + ) + ng + ] � O ( g ) = tf , 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 → ( R n \ B R (0)) / Γ , • Γ finite subgroup of SO ( n ) , • Γ acts freely on R n /B R (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 − g Euc = O ( r − τ ) , r → ∞ , r = “distance to some fixed base-point”. • Decay in derivatives ∂ α g = 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 − g Euc has pointwise decay at infinity, i.e., g − g Euc = o (1) as r → ∞ , • ∂ m g decays like ∂ m g = o ( r − m ) as r → ∞ , m ≥ 1 .

Origin Linearized Equation Gauge-fixing Approach Delicate Issues Decay improvement theorem Theorem (Ach´ e - V) Let ( M n , 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 • ( M n , g ) even-dimensional compact Einstein manifold with positive scalar curvature, • G x the Green’s function of the conformal Laplacian at the point x . ˆ • M = M \ { x } 4 • ˆ n − 2 g = G g x 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 � � 1 R A = Ric − 2( n − 1) g n − 2 Then the Bach tensor is given by B ij = ∆ g A ij − ∇ k ∇ i A kj + A kl W ikjl ,

Origin Linearized Equation Gauge-fixing Approach Delicate Issues Properties of the Bach tensor • B is the first variation of the functional � | W | 2 dvol, g → M • Conformally invariant in dimension 4 , • Schematically B = ∆ Ric + ∇ 2 R g + ∆ g R g g + 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 =∆ g B ij − 2 W kijl B kl − 4 A k k B ij + 8 A kl ∇ l C ( ij ) k − 4 C kl i C lj k + 2 C kl i C jkl + 4 ∇ l A k k C l ( ij ) − 4 W kijl A k m A ml . 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 n 2 − 2 O = ∆ B g + l.o.t. g • The terms l.o.t. , are quadratic and higher in Rm and its derivatives • More precisely   n/ 2 � � ∇ α 1 Rm ∗ . . . ∗ ∇ α j Rm  . l.o.t. =  j =2 α 1 + ... + α j = n − 2 j

Origin Linearized Equation Gauge-fixing Approach Delicate Issues If g is scalar-flat, this system looks like n 2 − 1 ∆ Ric = l.o.t. g A more general theorem Theorem (Ach´ e- V) If ( M n , g ) with n ≥ 4 (not necessarily even) is ALE of order 0 , Scalar-flat and satisfies ∆ k Ric = l.o.t., for 1 ≤ k ≤ n 2 − 1 , then ( M, g ) is ALE of order n − 2 k . Also true for k = 1 and n = 3 .

Origin Linearized Equation Gauge-fixing Approach Delicate Issues Previous works Theorem (J. Cheeger, G. Tian, ’94) ( M n , g ) n ≥ 3 ALE of order 0 and Ricci-flat then ( M, g ) is ALE of order n , • Gauge-fixing approach by means of the implicit function theorem (divergence-free gauge), • Three annulus lemma (L. Simon).

Origin Linearized Equation Gauge-fixing Approach Delicate Issues Previous works Theorem (G. Tian - V) ( M 4 , 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) ( M 4 , 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 • B ρ (0) metric ball on a flat cone C ( S n − 1 / Γ) , Γ ⊂ SO ( n ) is a finite subgroup acting freely on S n − 1 . • g defined on B ρ (0) \ { 0 } . • The origin is a C 0 -orbifold point for g . That is, there exists a coordinate system around the origin such that g − g Euc = o (1) r → 0 , and also, for any m ≥ 1 ∂ m g = o ( r − m ) as r → 0 .

Origin Linearized Equation Gauge-fixing Approach Delicate Issues Singularity removal Theorem (Ach´ e-V) If g defined on B ρ (0) \ { 0 } is • Obstruction-flat, • Constant scalar curvature • If 0 is a C 0 -orbifold singularity for g , then g extends to a smooth orbifold metric in B ρ (0) . That is, after diffeomorphism, there is a smooth Γ -invariant metric ˜ g on the universal cover of B ρ (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 − g Euc has pointwise decay at infinity. Set h = g − g Euc O ( g Euc + h ) = 0 , Linearize at the flat metric g Euc 0 = O ( g Euc + h ) = O ′ g Euc ( 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 ′ g Euc ( 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.