Multiple solutions for a higher order variational problem in conformal geometry

Transcrição

1 Multiple solutions for a higher order variational problem in conformal geometry Departamento de Matemática Instituto de Matemática e Estatística Universidade de São Paulo August 17th, 2017

2 Outiline of this talk Q-curvature and the Paneitz operator Intro: best constant in Sobolev embeddings Conformal invariance Paneitz operator and Q-curvature in Riemannian manifolds Multiplicity results Yamabe type invariants Aubin inequality A topological argument Bifurcation Examples (explicit computations) homogeneous fibrations Hopf bundles, Berger spheres

3 Acknowledgements My co-authors Renato Bettiol (UPenn) Yannick Sire (Johns Hopkins) My sponsors

4 Intro: Sobolev embeddings (M. Gursky) Sobolev embedding: H 2 (R n ) L 2n n 4 (R n ) (n 5) u 2 L 2 (R n ) λ n u 2 L 2n n 4 Best λ n known (Lions, Edmunds, Fortunato, Jannelli) it is attained at some radial function u satisfying: 2 u = u n+4 n 4 minimizers are positive equation is conformally invariant: solutions u invariant by: translations u(x) u(x + v) dilations u(x) t n 4 2 u(t x) ( inversions u(x) x 4 n u Question ) x x 2 Does there exist an analogous operator in Riemannian manifolds? (conformally invariant, positive minimizers,...)

5 The Paneitz operator Theorem (Paneitz 1983) Given (M n, g), n 5, there exists a differential operator P g such that: P g = 2 g + lower order terms of order 2 if ĝ = u 4 n 4 g, then Pĝ(φ) = u n+4 n 4 Pg (u φ). Examples. P R n = 2 P S n = S n( S n c).

6 Definition of Q-curvature and Paneitz operator (M n, g), n 5 Q-curvature: Q g = c n g (scal g ) + d n Ric g 2 + e n scal 2 g c n = 1 2(n 1) d n = 2 (n 2) 2 e n = n3 4n 2 +16n 16 8(n 1) 2 (n 2) 2 Paneitz operator: P g ψ = 2 gψ + α n div g ( Ricg ( ψ, e i )e i ) β n div g (scal g ψ) + γ n Q g ψ α n = 4 n 2 β n = n2 4n+8 2(n 1)(n 2) γ n = n 4 2

7 Constant Q-curvature Constant Q-curvature metric: g = u 4 n 4 g0 P g0 u = λ u n+4 n 4, λ = n 4 2 Q g. Variational formulation. Solutions are critical points of associated quadratic functional: E g0 (u) = 1 u P g0 u dm 2 in the space: M { u H 2 (M) : u L 2n n 4 } = const. Problems: non-compact embedding = minimizing sequences may converge weakly to 0; minimizers may not be positive.

8 Yamabe vs. Paneitz Yamabe problem (constant scal curvature) Non-compact embedding: H 1 L 2n n 2 Conf. invariant operator: conformal Laplacian L g = g + n 2 4(n 1) scal g scalar curvature scal g : g = u 4 n 2 g0 scal g = 4(n 1) n+2 n 2 u n 2 Lg0 (u) minimizers exist and have constant scal g. Constant Q-curvature Non-compact embedding: H 2 L 2n n 4 Conf. inv. operator: P g Q-curvature Q g : g = u 4 n 4 g0 Q g = 2 n+4 n 4u n 4 Pg0 (u) Positivity of minimizers? When so, they give constant Q-curvature. Very little known on the existence of positive minimizers: scal g0 > 0 and almost positive. Q g0

9 Yamabe-type invariants L g0 (u) = M u L g0 u dm E g0 (u) = Yamabe invariant: Y (M, g 0 ) = inf u 0 M L(u) u 2 L 2n n 2 u P g0 u dm Y 4 (M, g 0 ) = inf u 0 E(u) u 2 L 2n n 4 Y + 4 (M, g 0) = inf u>0 E(u) u 2 L 2n n 4 Theorem (Gursky Han Lin, 2016) If g [g 0 ] with scal g > 0 and Q g > 0, then: Y 4 (M, g 0 ) = Y + 4 (M, g 0)( 0), with infimum attained by some positive function. Not known if Y 4 > 0 when Y > 0 and Q almost positive.

10 A new invariant and Aubin inequality Aubin inequality: Y (M n, g 0 ) Y (S n, g round ) Theorem (Gursky Han Lin 2016) Assume Y (M, g 0 ) > 0 and Q g0 almost positive. Then: Ker(P g0 ) = {0} and P g0 > 0; Green function G Pg0 positive on M M. Inverse of P g0 : G g0 f (p) = M G P g0 (p, q)f (q) dq Quadratic functional: G g0 (f ) = M f G g 0 f dm G g0 (f ) New invariant: Θ 4 (M, g 0 ) = sup G(f ) = sup f L n+4 2n f L n+4 2n f 2n L n+4 Supremum always attained at some smooth positive f, 4 f n 4 g0 has constant Q-curvature. Θ 4 (M n, g 0 ) Θ 4 (S n, g round )

11 Multiple constant Q-curvature metrics on spheres Theorem For n 5 and 0 k < n 4 2, there are infinitely many pairwise nonhomothetic complete metrics with constant Q-curvature on S n \ S k that are conformal to the round metric. Proof. S n \ S k conf. equivalent to S n k 1 H k+1 scal = (n 2k 2)(n 1) Q = 1 8n(n 2k)(n 2k 4) are positive when k < n 4 2 ; topological argument + Aubin inequality.

12 The topological argument 1 H k+1 has compact quotients that give an infinite tower of finite-sheeted Riemannian coverings: (H k+1, g hyp )... (Σ 2, g 2 ) (Σ 1, g 1 ) (Σ 0, g 0 ) 2 Multiply by (S n k 1, g round ), product metrics:... (S n k 1 Σ 1, g round g 1 ) (S n k 1 Σ 0, g round g 0 ) 3 pull-back Θ 4 -metric in [ g round g 0 ] : energy G goes to 0! (uses scal > 0 and Q > 0) 4 By Aubin inequality, maximum of G must be attained at some other metric in the conformal class of the product. 5 Iterate.

13 Coverings of hyperbolic surfaces

14 Profinite completion and residually finite groups Infinite tower of finite-sheeted coverings:... M k M k 1... M 1 M 0 iff G = π 1 (M 0 ) has infinite profinite completion Ĝ. Def. Ĝ = lim G/Γ, Γ G, [ G : Γ ] < +. Canonical homomorphism ι : G Ĝ Ker(i) = Def. G is residually finite if: Γ G [G:Γ]<+ Γ G [G:Γ]<+ Γ Γ = {1}

15 Symmetric spaces Theorem (Borel) Symmetric spaces of noncompact type X admit irreducible compact quotients X/Γ. X/Γ loc. symmetric = constant scalar and Q-curvature Selberg Malcev lemma Finitely generated linear groups are residually finite. Corollary. Γ = π 1 (X/Γ) has infinite profinite completion. Theorem (M, g) closed, (X, h) as above scal and Q-curvature of g h positive. Then, there are infinitely many complete constant Q-curvature metrics in [g h].

16 Bifurcation theory for the Q-curvature problem Need a (compact) manifold M with a family (g t ) t [a,b] of constant Q-curvature metrics with computable Morse index. Ansatz: Riemannian submersions π t : (M, g t ) (B, g B ) with: scal gt and Q gt constant; minimal fibers; horizontally Einstein: Ric gt = κ t g t on the horizontal distribution. Typical example: Homogeneous fibration. H K G compact Lie groups K /H G/H G/K with bi-invariant metric g 1 on G g t obtained by rescaling metric of fibers.

17 A bifurcation criterion Theorem M n closed manifold with n 5 π t : (M, g t ) (B, g B ), t [t ε, t + ε], a 1-parameter family of horizontally Einstein (κ t ) Riemannian submersions with minimal fibers scal gt and Q gt constant for all t α t = (n2 4n+8)scal gt 8 κ t (n 1) 4(n 1)(n 2), β t = 2 Q gt. If for some λ spec( gb ): 1 2 λ2 + α t λ + β t = 0 α t λ + β t 0, then t is a bifurcation instant for (g t ) t. If λ scalg t n 1, then bifurcation metrics do not have constant scalar curvature.

18 Berger spheres S 2n+1 Upshot Hopf bundle S 1 S 2n+1 CP n Berger metric g t = t g V + g H ( g 1 unit round metric) homogeneous fibration corresponding to U(n) U(n)U(1) U(n + 1) horizontal space H = C n irreducible U(n)-representation, vertical space V = R does not contain copies of this irreducible For n 6, sequence t k + bifurcation instants for (S 2n+1, g t ). If 6 n 9, then infinitely many of these bifurcating branches issue from metrics on S 2n+1 that have scal < 0 and Q < 0.

19 The Berger spheres S 4n+3 Upshot Hopf bundle S 3 S 4n+3 HP n Berger metric g t = t g V + g H ( g 1 unit round metric) homogeneous fibration corresponding to Sp(n) Sp(n)Sp(1) Sps(n + 1) horizontal space H = H n irreducible Sp(n)-representation, vertical space V = R 3 does not contain copies of this irreducible For n 2, sequences t k + and t k 0 of bifurcation instants for (S 2n+3, g t ). Infinitely many of these bifurcating branches issue from metrics on S 2n+3 that have scal < 0 and Q < 0. Similar results for Hopf bundle CP 2n+1 HP n.

20 Thanks for your attention!! See you at ICM2018!