The Yamabe invariant and surgery

Transcription

1 The Yamabe invariant and surgery B. Ammann 1 M. Dahl 2 E. Humbert 3 1 Universität Regensburg Germany 2 Kungliga Tekniska Högskolan, Stockholm Sweden 3 Université François-Rabelais, Tours France Geometric Analysis in Geometry and Topology 2015 Tokyo 2015

2 Einstein-Hilbert functional Let M be a compact n-dimensional manifold, n 3. The renormalised Einstein-Hilbert functional is M E : M R, E(g) := scal g dv g vol(m, g) (n 2)/n M := {metrics on M}. [g] := {u 4/(n 2) g u > 0}. Stationary points of E : [g] R = metrics with constant scalar curvature Stationary points of E : M R = Einstein metrics

3 Conformal Yamabe constant Inside a conformal class Y (M, [g]) := inf E( g) >. g [g] This is the conformal Yamabe constant. Y (M, [g]) Y (S n ) where S n is the sphere with the standard structure. Solution of the Yamabe problem (Trudinger, Aubin, Schoen-Yau) E : [g] R attains its infimum. Remark Y (M, [g]) > 0 if and only if [g] contains a metric of positive scalar curvature.

4 Obata s theorem Theorem (Obata) Assume: M is connected and compact g 0 is an Einstein metric on M g = u 4/(n 2) g 0 with scal g constant (M, g 0 ) not conformal to S n Then u is constant. Conclusion E(g 0 ) = Y (M, [g 0 ]) This conclusion also holds if g 0 is a non-einstein metric with scal = const 0 (Maximum principle). So in these two cases, we have determined Y (M, [g 0 ]). However in general it is difficult to get explicit good lower bounds for Y (M, [g 0 ]).

5 On the set of conformal classes σ(m) := sup Y (M, [g]) (, Y (S n )] [g] M The smooth Yamabe invariant. Introduced by O. Kobayashi and R. Schoen. Remark σ(m) > 0 if and only if M caries a metric of positive scalar curvature. Supremum attained? Depends on M.

6 Example CP 2 The Fubini-Study g FS metric is Einstein and = E(g FS ) = Y (CP 2, [g FS ]) = σ(cp 2 ). Supremum attained in the Fubini-Study metric. LeBrun 97 Seiberg-Witten theory LeBrun & Gursky 98 Twisted Dirac operators

7 Similar examples σ(s n ) = n(n 1)ω 2/n n. Gromov & Lawson, Schoen & Yau 83: Tori R n /Z n. σ(r n /Z n ) = 0. Enlargeable Manifolds LeBrun 99: All Kähler-Einstein surfaces with non-positive scalar curvature. Seiberg-Witten methods Bray & Neves 04: RP 3. σ(rp 3 ) = 2 2/3 σ(s 3 ). Inverse mean curvature flow Perelman, M. Anderson 06 (sketch), Kleiner-Lott 08 compact quotients of 3-dimensional hyperbolic space Ricci flow Example where supremum is not attained Schoen: σ(s n 1 S 1 ) = σ(s n ). The supremum is not attained.

8 Some known values of σ All examples above. Akutagawa & Neves 07: Some non-prime 3-manifolds, e.g. σ(rp 3 #(S 2 S 1 )) = σ(rp 3 ). Seiberg-Witten methods have been extended to the Pin (2)-setting by Ishida, Matsuo and Nakamura 15 Compact quotients of nilpotent Lie groups: σ(m) = 0. Unknown cases Nontrivial quotients of spheres, except RP 3. S k S m, with k, m 2. No example of dimension 5 known with σ(m) 0 and σ(m) σ(s n ).

9 Positive scalar curvature psc σ(m) > 0 Suppose n σ(m) > 0 is a bordism invariant. 2. Bordism classes admitting psc metrics form a subgroup in the bordism group Ω spin n (Bπ 1 ). 3. If P p π B b is a fiber bundle, equipped with a family of vertical metrics (g p ) p B with Y (π 1 (p), [g p ]) > 0, then σ(p) > 0.

10 Guiding questions of our work, ɛ > 0 1. Is σ(m) > ɛ a bordism invariant? Yes for 0 < ɛ < Λ n, Λ 5 = 45.1, Λ 6 = 49.9, ADH 2. Do σ(m) > ɛ-classes form a subgroup? Yes for 0 < ɛ < Λ n, ADH 3. If P p π B b is a fiber bundle, equipped with a family of vertical metrics (g p ) p B with Y (π 1 (p), [g p ]) > 0, f = p b 3, b = dim B 3, then ADH + M. Streil ) f σ(p) p c b,f (min Y p B (π 1 (p), [g p ]).

11 Explicit values for Λ n Theorem (ADH) Let M be a compact simply connected manifold, n = dim M. Then n = 5 : Λ 5 = 45.1 < σ(m) σ(s 5 ) = n = 6 : Λ 6 = 49.9 < σ(m) σ(s 6 ) =

12 Gap theorems Theorem (ADH) Let M is a 2-connected compact manifold of dimension n 5. If α(m) 0, then σ(m) = 0. If α(m) = 0, then n = σ(m) σ(s n ) =

13 Theorem (ADH) Let Γ be group whose homology is finitely generated in each degree. In the case n 5, we know that {σ(m) π 1 (M) = Γ, dim M = n} [0, Λ n ] is a well-ordered set (with respect to the standard order ). In other words: there is no sequence of n-dimensional manifolds M i with π 1 (M i ) = Γ such that σ(m i ) [0, Λ n ] and such that σ(m i ) is strictly decreasing. On the other hand it is conjectured that σ(s n /Γ) 0 for #Γ

14 Techniques Key ingredients (1) A monotonicity formula for surgery, ADH (2) A lower bound for products, ADH Other techniques (3a) Rearranging functions on H r c S s to test functions on R r S s, ADH (3b) Conformal Yamabe constants of Y (R 2 S n 2 ), Petean-Ruiz (4) Are L p -solutions of the Yamabe equation on complete manifolds already L 2? Results by ADH (5) Obata s theorem about constant scalar metrics conformal to Einstein manifolds (6) Standard bordism techniques: Smale,..., Gromov-Lawson, Stolz

15 (1) A Monotonicity formula for surgery Let Mk Φ be obtained from M by k-dimensional surgery, 0 k n 3. Theorem (ADH, # 1) There is Λ n,k > 0 with Furthermore Λ n,0 = Y (S n ). σ(m Φ k ) min{σ(m), Λ n,k} Special cases were already proved by Gromov-Lawson, Schoen-Yau, Kobayashi, Petean. Thm # 1 follows directly from Thm # 2. Theorem (ADH, #2) For any metric g on M there is a sequence of metrics g i on M Φ k such that lim Y i (MΦ k, [g i]) = min { } Y (M, [g]), Λ n,k.

16 Construction of the metrics Let Φ : S k B n k M be an embedding. We write close to S := Φ(S k {0}), r(x) := d(x, S) g g S + dr 2 + r 2 g n k 1 round where g n k 1 round is the round metric on S n k 1. t := log r. 1 r 2 g e2t g S + dt 2 + g n k 1 round We define a metric g for r > r 1 1 g i = g for r (ρ, r r 2 0 ) f 2 (t)g S + dt 2 + g n k 1 for r < ρ that extends to a metric on M Φ k. round

17 r 1 r 0 2ρ ρ r 1 r 0 ρ g ρ = g g ρ = F 2 g 2ρ S n k 1 has constant length

18 Proof of Theorem #2, continued Any class [g i ] contains a minimizing metric written as u 4/(n 2) i g i. We obtain a PDE: u i > 0, This sequence might: 4 n 1 n 2 g i u i + scal g i u i = λ i ui n+2 n 2 u 2n/(n 2) i dv g i = 1, λ i = Y ([g i ]) Concentrate in at least one point. Then lim inf λ i Y (S n ). Concentrate on the old part M \ S. Then lim inf λ i Y ([g]). Concentrate on the new part. Gromov-Hausdorff convergence of pointed spaces. Limit spaces: H k+1 c S n k 1, c [0, 1] H k+1 c : simply connected, complete, K = c 2 Then lim inf λ i Y (M c ).

19 The numbers Λ n,k (Disclaimer: Additional conditions for k + 3 = n 7 See Ammann Große 2015 for some related questions) Λ n,k := Y (N) := inf c [0,1] Y (Hk+1 c S n k 1 ) inf u Cc (N) Note: H k+1 1 S n k 1 = S n \ S k. k = 0: Λ n,k = Y (R S n 1 ) = Y (S n ) k = 1,..., n 3: Λ n,k > 0 Λ n := min{λ n,2,..., Λ n,n 3 } N 4 n 1 n 2 du 2 + scal u 2 ( N up ) 2/p

20 Conjecture #1: Y (H k+1 c S n k 1 ) Y (R k+1 S n k 1 ) Conjecture #2: The infimum in the definition of Y (H k+1 c S n k 1 ) is attained by an O(k + 1) O(n k) invariant function if 0 c < 1. O(n k)-invariance is difficult, O(k + 1)-invariance follows from standard reflection methods Comments If we assume Conjecture #2, then Conjecture #1 reduces to an ODE and Y (H k+1 c S n k 1 ) can be calculated numerically. Assuming Conjecture #2, a maple calculation confirmed Conjecture #1 for all tested n, k and c. The conjecture would imply: Compare this to σ(s 2 S 2 ) Λ 4,1 = Y (S 4 ) = Y (S 2 S 2 ) = σ(cp 2 ) =

21 Values for Λ n,k More values of Λ n,k n k Λ n,k Λ n,k = Y (S n ) known conjectured > > >

22 (2) A lower bound for products a n := 4(n 1)/(n 2) Theorem (ADH) Let (V, g) and (W, h) be Riemannian manifolds of dimensions v, w 3. Assume that Y (V, [g]) 0, Y (W, [h]) 0 and that Scal g + Scal h a v+w Scalg a v + Scalh a w. (1) Then, Y (V W, [g + h]) (v + w)a v+w ( Y (V, [g]) va v ) v Main technique: Iterated Hölder inequality. m ( Y (W, [h]) wa w ) w m.

23 How good is this bound? If (V, g) and (W, h) are compact manifolds with psc and of dimension at least 3, then b v,w inf λ (0, ) Y (V W, [g + λh]) (v + w) ( Y (V,[g]) v ) v v+w ( Y (W,[h]) w ) w v+w 1, b v,w := a v+w a v v/(v+w) a w w/(v+w) < 1. b v,w w=3 w=4 w=5 w=6 w=7 v=

24 Application to Λ n,k H k+1 c conformal to a subset of S k+1 Y (H k+1 c ) = Y (S k+1 ) Thus for 2 k n k 4: Λ n,k = inf c [0,1] Y (Hk+1 c S n k 1 ) n b k+1,n k 1 ( Y (S k+1 ) k + 1 ) (k+1)/n ( Y (S n k 1 ) (n k 1)/n ) n k 1

25 Application to fiber bundles Assume F f P n B b is a fiber bundle, σ(f) > 0, f = dim F 3. Shrink a psc metric g F on F. We see: σ(p) Y ((F, g F ) R b ) (M. Streil, in preparation). For b 3: Y ((F, g F ) R b ) n b f,b ( Y (F, [gf ]) f ) f /n ( Y (S b ) b ) b/n If g F carries an Einstein metric, then Petean-Ruiz can provide lower bounds for Y (F R, [g F + dt 2 ]) and Y (F R 2, [g F + dt 2 + ds 2 ]).

26 Important building blocks For the following manifolds we have lower bounds on the smooth Yamabe invariant and the conformal Yamabe constant. Smooth Yamabe invariant of total spaces of bundles with fiber CP 2. These total spaces generate the oriented bordism classes. Smooth Yamabe invariant of total spaces of bundles with fiber HP 2. These total spaces generate the kernel of α : Ω spin n KO n Conformal Yamabe constant of Einstein manifolds: SU(3)/SO(3), CP 2, HP 2 HP 2 R, HP 2 R 2, CP 2 R, CP 2 R Petean-Ruiz Conformal Yamabe constant of R 2 S n 2. Particularly important for n = 4, 5, 9, 10. Petean-Ruiz Conformal Yamabe constant of R 3 S 2. Petean-Ruiz

27 Thanks for your attention!

28 Literature by the authors B. Ammann, M. Dahl, E. Humbert, Smooth Yamabe invariant and surgery, J. Diff. Geom. 94, 1 58 (2013), The conformal Yamabe constant of product manifolds, Proc. AMS (2013), Square-integrability of solutions of the Yamabe equation, Commun. Anal. Geom. 21, (2013), Low-dimensional surgery and the Yamabe invariant, J. Math. Soc. Japan 67, (2015) B. Ammann, N- Große, Relations between threshold constants for Yamabe type bordism invariants, to appear in J. Geom. Anal.

29 Related Literature K. Akutagawa, L. Florit, J. Petean, On the Yamabe constant of Riemannian products, Comm. Anal. Geom. 2007, N. Große, The Yamabe equation on manifolds of bounded geometry, J. Petean, J. M. Ruiz, Isoperimetric profile comparisons and Yamabe constants, AGAG 2011, On the Yamabe constants of S 2 R 3 and S 3 R 2,

30 Possible application to CP 3 Lemma Assume that the surgery monotoncity formula holds for the conjectured values Λ 6.2 = Λ 6.3 = Then σ(cp 3 ) min{λ 6.2, Λ 6.3 } Compare to the Fubini-Study metric g FS µ(cp 3, [g FS ]) = Proof. CP 3 is spin-bordant to S 6. Find such a bordism W such that that W is 2-connected. Then one can obtain CP 3 by surgeries of dimension 2 and 3 out of S 6.

31 Application to connected sums Assume that M is compact, connected of dimension at least 5 with 0 < σ(m) < min{λ n,1,... Λ n,n 3 } =: Λ n. Let p, q N be relatively prime. Then σ(m# #M) = σ(m) }{{} p times or σ(m# #M) = σ(m). }{{} q times Are there such manifolds M? Schoen conjectured: σ(s n /Γ) = σ(s n )/(#Γ) 2/n (0, Λ n ) for #Γ large.

32 Application to connected sums M#N Assume that M and N are compact, connected of dimension at least 5 with 0 < σ(n) > σ(m) < Λ n. Then σ(m) = σ(m#n).

33 More values of Λ n,k Back n k Λ n,k Λ n,k = Y (S n ) known conjectured > > >

34 Back n k Λ n,k Λ n,k = Y (S n ) known conjectured > > >