A surgery formula for the (smooth) Yamabe invariant

Transcription

1 A surgery formula for the (smooth) Yamabe invariant B. Ammann 1 M. Dahl 2 E. Humbert 3 1 Universität Regensburg Germany 2 Kungliga Tekniska Högskolan, Stockholm Sweden 3 Université Henri Poincaré, Nancy France Conference on scalar curvature, Münster 2010

2 (Smooth) Yamabe invariant Let M be a compact n-dimensional manifold, n 3. Einstein-Hilbert functional E : M 1 R, g scal g dv g M 1 := {metrics on M of volume 1}. [g] := {u 4/(n 2) g vol(u 4/(n 2) g) = 1}. Stationary points of E : [g] R = metrics with constant scalar curvature Stationary points of E : M 1 R = Einstein metrics M

3 Conformal and smooth invariants Inside a conformal class Y (M, [g]) := inf E( g) >. g [g] 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 Conformal and smooth invariants Inside a conformal class Y (M, [g]) := inf E( g) >. g [g] 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.

5 Conformal and smooth invariants Inside a conformal class Y (M, [g]) := inf E( g) >. g [g] 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.

6 On the set of conformal classes σ(m) := sup [g] M 1 Y (M, [g]) (, Y (S n )] The smooth Yamabe invariant, also called Schoen s σ-constant. Remark σ(m) > 0 if and only if M caries a metric of positive scalar curvature. Supremum attained? Depends on M.

7 On the set of conformal classes σ(m) := sup [g] M 1 Y (M, [g]) (, Y (S n )] The smooth Yamabe invariant, also called Schoen s σ-constant. Remark σ(m) > 0 if and only if M caries a metric of positive scalar curvature. Supremum attained? Depends on M.

8 On the set of conformal classes σ(m) := sup [g] M 1 Y (M, [g]) (, Y (S n )] The smooth Yamabe invariant, also called Schoen s σ-constant. Remark σ(m) > 0 if and only if M caries a metric of positive scalar curvature. Supremum attained? Depends on M.

9 Example CP 2 The Fubini-Study g FS metric is Einstein and 53, = E(g FS ) = Y (CP 2, [g FS ]) = σ(cp 2 ). Supremum attained in the Fubini-Study metric. Claude LeBrun 97 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. LeBrun 99: All Kähler-Einstein surfaces with non-positive scalar curvature. Bray & Neves 04: RP 3. σ(rp 3 ) = 2 2/3 σ(s 3 ). Perelman, M. Anderson 06: compact quotients of 3-dimensional hyperbolic space

10 Example CP 2 The Fubini-Study g FS metric is Einstein and 53, = E(g FS ) = Y (CP 2, [g FS ]) = σ(cp 2 ). Supremum attained in the Fubini-Study metric. Claude LeBrun 97 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. LeBrun 99: All Kähler-Einstein surfaces with non-positive scalar curvature. Bray & Neves 04: RP 3. σ(rp 3 ) = 2 2/3 σ(s 3 ). Perelman, M. Anderson 06: compact quotients of 3-dimensional hyperbolic space

11 Example CP 2 The Fubini-Study g FS metric is Einstein and 53, = E(g FS ) = Y (CP 2, [g FS ]) = σ(cp 2 ). Supremum attained in the Fubini-Study metric. Claude LeBrun 97 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. LeBrun 99: All Kähler-Einstein surfaces with non-positive scalar curvature. Bray & Neves 04: RP 3. σ(rp 3 ) = 2 2/3 σ(s 3 ). Perelman, M. Anderson 06: compact quotients of 3-dimensional hyperbolic space

12 Example CP 2 The Fubini-Study g FS metric is Einstein and 53, = E(g FS ) = Y (CP 2, [g FS ]) = σ(cp 2 ). Supremum attained in the Fubini-Study metric. Claude LeBrun 97 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. LeBrun 99: All Kähler-Einstein surfaces with non-positive scalar curvature. Bray & Neves 04: RP 3. σ(rp 3 ) = 2 2/3 σ(s 3 ). Perelman, M. Anderson 06: compact quotients of 3-dimensional hyperbolic space

13 Example where supremum is not attained Schoen: σ(s n 1 S 1 ) = σ(s n ). The supremum is not attained. Connected sum of several S n 1 S 1. Attention In the case σ(m) 0 the minimizers are unique. Hence, if a maximizing conformal class exists, then the unique minimizing metric in that class is Einstein. However, in the case σ(m) > 0, a minimizing metric in a maximizing class may be non-einstein!

14 Example where supremum is not attained Schoen: σ(s n 1 S 1 ) = σ(s n ). The supremum is not attained. Connected sum of several S n 1 S 1. Attention In the case σ(m) 0 the minimizers are unique. Hence, if a maximizing conformal class exists, then the unique minimizing metric in that class is Einstein. However, in the case σ(m) > 0, a minimizing metric in a maximizing class may be non-einstein!

15 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 ). 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 ).

16 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 ). 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 ).

17 Relations to similar invariants to the spectrum of the conformal Laplacian ( inf λ 1 4 n 1 ) { g + scal g Y (M, [g]) if Y (M, [g]) 0 = g [g] n 2 if Y (M, [g]) < 0... to the Perelman-invariant (Akutagawa, Ishida, LeBrun 06)... to the L n/2 -norm of scal λ(m) := sup λ 1 (4 g + scal g ) g M 1 { σ(m) if σ(m) 0 = + if σ(m) > 0 inf scal g L g M n/2 ( g) = 1 { σ(m) if σ(m) 0 0 if σ(m) > 0 = Hence min{σ(m), 0} is determined by an infimum.

18 Positive scalar curvature σ(m) > 0 is a bordism invariant. Bordism classes admitting positive scalar curvature metrics form an ideal in the bordism ring. Questions for given ɛ > 0 1. Is σ(m) > ɛ a bordism invariant? Yes for small ɛ > 0 2. Do σ(m) > ɛ-classes form a subgroup? Yes for small ɛ > a subring, an ideal? Unkown 4. σ(m) > ɛ > 0 = σ(m N) f (σ(m), σ(n), dim M, dim N) f > 0 Unknown, partial results by Akutagawa and Petean t Main tool: Surgery formula for σ(m)

19 Positive scalar curvature σ(m) > 0 is a bordism invariant. Bordism classes admitting positive scalar curvature metrics form an ideal in the bordism ring. Questions for given ɛ > 0 1. Is σ(m) > ɛ a bordism invariant? Yes for small ɛ > 0 2. Do σ(m) > ɛ-classes form a subgroup? Yes for small ɛ > a subring, an ideal? Unkown 4. σ(m) > ɛ > 0 = σ(m N) f (σ(m), σ(n), dim M, dim N) f > 0 Unknown, partial results by Akutagawa and Petean t Main tool: Surgery formula for σ(m)

20 Positive scalar curvature σ(m) > 0 is a bordism invariant. Bordism classes admitting positive scalar curvature metrics form an ideal in the bordism ring. Questions for given ɛ > 0 1. Is σ(m) > ɛ a bordism invariant? Yes for small ɛ > 0, ADH 2. Do σ(m) > ɛ-classes form a subgroup? Yes for small ɛ > 0, ADH a subring, an ideal? Unkown 4. σ(m) > ɛ > 0 = σ(m N) f (σ(m), σ(n), dim M, dim N) f > 0 Unknown, partial results by Akutagawa and Petean t Main tool: Surgery formula for σ(m)

21 Positive scalar curvature σ(m) > 0 is a bordism invariant. Bordism classes admitting positive scalar curvature metrics form an ideal in the bordism ring. Questions for given ɛ > 0 1. Is σ(m) > ɛ a bordism invariant? Yes for small ɛ > 0, ADH 2. Do σ(m) > ɛ-classes form a subgroup? Yes for small ɛ > 0, ADH a subring, an ideal? Unknown 4. σ(m) > ɛ > 0 = σ(m N) f (σ(m), σ(n), dim M, dim N) f > 0 Unknown, partial results by Akutagawa and Petean t Main tool: Surgery formula for σ(m)

22 Positive scalar curvature σ(m) > 0 is a bordism invariant. Bordism classes admitting positive scalar curvature metrics form an ideal in the bordism ring. Questions for given ɛ > 0 1. Is σ(m) > ɛ a bordism invariant? Yes for small ɛ > 0, ADH 2. Do σ(m) > ɛ-classes form a subgroup? Yes for small ɛ > 0, ADH a subring, an ideal? Unknown 4. σ(m) ɛ > 0 = σ(m N) f (σ(m), σ(n), dim M, dim N) f > 0 Unknown, partial results by Akutagawa and Petean Main tool: Surgery formula for σ(m)

23 Positive scalar curvature σ(m) > 0 is a bordism invariant. Bordism classes admitting positive scalar curvature metrics form an ideal in the bordism ring. Questions for given ɛ > 0 1. Is σ(m) > ɛ a bordism invariant? Yes for small ɛ > 0, ADH 2. Do σ(m) > ɛ-classes form a subgroup? Yes for small ɛ > 0, ADH a subring, an ideal? Unknown 4. σ(m) ɛ > 0 = σ(m N) f (σ(m), σ(n), dim M, dim N) f > 0 Unknown, partial results by Akutagawa and Petean Main tool for 1. and 2.: Surgery formula for σ(m)

24 Surgery Let Φ : S k B n k M n be an embedding. We define M Φ k := M \ Φ(Sk B n k ) (B k+1 S n k 1 )/ where / means gluing the boundaries via M Φ(x, y) (x, y) S k S n k 1. We say that M Φ k is obtained from M by surgery of dimension k. Example: 0-dimensional surgery on a surface.

25 Known surgery formulas Theorem (Gromov & Lawson 80, Schoen & Yau 79) If 0 k n 3, then σ(m) > 0 = σ(m Φ k ) > 0. Theorem (Kobayashi 87) If k = 0, then σ(m Φ 0 ) σ(m). Theorem (Petean, Petean & Yun 99) If 0 k n 3, then σ(mk Φ ) min{σ(m), 0}. The proof uses the characterization of min{σ(m), 0} as an infimum.

26 Known surgery formulas Theorem (Gromov & Lawson 80, Schoen & Yau 79) If 0 k n 3, then σ(m) > 0 = σ(m Φ k ) > 0. Theorem (Kobayashi 87) If k = 0, then σ(m Φ 0 ) σ(m). Theorem (Petean, Petean & Yun 99) If 0 k n 3, then σ(mk Φ ) min{σ(m), 0}. The proof uses the characterization of min{σ(m), 0} as an infimum.

27 Known surgery formulas Theorem (Gromov & Lawson 80, Schoen & Yau 79) If 0 k n 3, then σ(m) > 0 = σ(m Φ k ) > 0. Theorem (Kobayashi 87) If k = 0, then σ(m Φ 0 ) σ(m). Theorem (Petean, Petean & Yun 99) If 0 k n 3, then σ(mk Φ ) min{σ(m), 0}. The proof uses the characterization of min{σ(m), 0} as an infimum.

28 Main results Theorem (ADH, # 1) Let 0 k n 3. There is a positive constant L(n, k) depending only on n and k such that σ(mk Φ ) min{σ(m), L(n, k)} if Mk Φ is obtained from M by k-dimensional surgery. Furthermore L(n, 0) = Y (S n ). This theorem implies all three previously known surgery formulas. Thm # 1 follows directly from Thm # 2. Theorem (ADH, #2) Suppose that Mk Φ is obtained from M by k-dimensional surgery. Then for any metric g on M there is a sequence of metrics g i on such that M Φ k min {Y (M, [g]), L(n, k)} lim inf i Y (MΦ k, [g i]) Y (M, [g]).

29 Main results Theorem (ADH, # 1) Let 0 k n 3. There is a positive constant L(n, k) depending only on n and k such that σ(mk Φ ) min{σ(m), L(n, k)} if Mk Φ is obtained from M by k-dimensional surgery. Furthermore L(n, 0) = Y (S n ). This theorem implies all three previously known surgery formulas. Thm # 1 follows directly from Thm # 2. Theorem (ADH, #2) Suppose that Mk Φ is obtained from M by k-dimensional surgery. Then for any metric g on M there is a sequence of metrics g i on such that M Φ k min {Y (M, [g]), L(n, k)} lim inf i Y (MΦ k, [g i]) Y (M, [g]).

30 Main results Theorem (ADH, # 1) Let 0 k n 3. There is a positive constant L(n, k) depending only on n and k such that σ(mk Φ ) min{σ(m), L(n, k)} if Mk Φ is obtained from M by k-dimensional surgery. Furthermore L(n, 0) = Y (S n ). This theorem implies all three previously known surgery formulas. Thm # 1 follows directly from Thm # 2. Theorem (ADH, #2) Suppose that Mk Φ is obtained from M by k-dimensional surgery. Then for any metric g on M there is a sequence of metrics g i on such that M Φ k min {Y (M, [g]), L(n, k)} lim inf i Y (MΦ k, [g i]) Y (M, [g]).

31 Topological conclusions L(n) := min{l(n, 1), L(n, 2),..., L(n, n 3)} > 0, σ(m) := min{σ(m), L(n)}. Let Mk Φ be obtained from M by surgery of dimension k {2, 3,..., n 3}, then σ(m) = σ(m k Φ). Goal: Find a bordism invariant!

32 Topological conclusions L(n) := min{l(n, 1), L(n, 2),..., L(n, n 3)} > 0, σ(m) := min{σ(m), L(n)}. Let Mk Φ be obtained from M by surgery of dimension k {2, 3,..., n 3}, then σ(m) = σ(m k Φ). Goal: Find a bordism invariant!

33 Let Γ be a finitely presented group. Let Ω spin n (BΓ) the spin cobordism group over BΓ. Any class in Ω spin n (BΓ) has a π 1 -bijective representative, i.e. it is represented by (M, f ), where M is a connected compact spin manifold, and where f : M BΓ induces an isomorphism π 1 (M) π 1 (BΓ) = Γ. Lemma n 5. Let (M 1, f 1 ) and (M 2, f 2 ) be spin cobordant over BΓ and let (M 2, f 2 ) be π 1 -bijective. Then σ(m 2 ) σ(m 1 ). We define s Γ ([M, f ]) := max σ(n) where the maximum runs over all (N, h) with [M, f ] = [N, h] Ω spin n (BΓ).

34 Let Γ be a finitely presented group. Let Ω spin n (BΓ) the spin cobordism group over BΓ. Any class in Ω spin n (BΓ) has a π 1 -bijective representative, i.e. it is represented by (M, f ), where M is a connected compact spin manifold, and where f : M BΓ induces an isomorphism π 1 (M) π 1 (BΓ) = Γ. Lemma n 5. Let (M 1, f 1 ) and (M 2, f 2 ) be spin cobordant over BΓ and let (M 2, f 2 ) be π 1 -bijective. Then σ(m 2 ) σ(m 1 ). We define s Γ ([M, f ]) := max σ(n) where the maximum runs over all (N, h) with [M, f ] = [N, h] Ω spin n (BΓ).

35 σ := min{σ(m), L(n)} s Γ ([M, f ]) = σ(m) if (M, f ) is π 1 -bijective s Γ : Ω spin n (BΓ) (, L(n)] Corollary Let n 5 and ɛ (0, L(n)). The groups {x Ω spin n (BΓ) s Γ (x) > ɛ} and {x Ω spin n (BΓ) s Γ (x) ɛ} are subgroups. Corollary Let n 5 and M n simply connected. Then σ(m) {0} [ɛ n, Y (S n )]. We even get that image(σ : Ω spin n R) (0, L(n)) has no accumulation points from above.

36 σ := min{σ(m), L(n)} s Γ ([M, f ]) = σ(m) if (M, f ) is π 1 -bijective s Γ : Ω spin n (BΓ) (, L(n)] Corollary Let n 5 and ɛ (0, L(n)). The groups {x Ω spin n (BΓ) s Γ (x) > ɛ} and {x Ω spin n (BΓ) s Γ (x) ɛ} are subgroups. Corollary Let n 5 and M n simply connected. Then σ(m) {0} [ɛ n, Y (S n )]. We even get that image(σ : Ω spin n R) (0, L(n)) has no accumulation points from above.

37 Application Let n 5. Take M n with σ(m) (0, L(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, L(n)) for #Γ large.

38 Application Let n 5. Take M n with σ(m) (0, L(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, L(n)) for #Γ large.

39 Application Let n 5. Take M n with σ(m) (0, L(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, L(n)) for #Γ large.

40 Application Let n 5. Take M n with σ(m) (0, L(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, L(n)) for #Γ large.

41 Proof of Theorem # 2 Theorem (ADH, #2) Suppose that Mk Φ is obtained from M by k-dimensional surgery. Then for any metric g on M there is a sequence of metrics g i on such that M Φ k min {Y (M, [g]), L(n, k)} lim inf i Y (MΦ k, [g i]) Y (M, [g]). Easy part: Difficult and important part lim inf i Y (MΦ k, [g i]) Y (M, [g]) min {Y (M, [g]), L(n, k)} lim inf i Y (MΦ k, [g i]) For this, we construct a sequence explicitly and study the behavior of the minimizers in [g i ] for i.

42 Proof of Theorem # 2 Theorem (ADH, #2) Suppose that Mk Φ is obtained from M by k-dimensional surgery. Then for any metric g on M there is a sequence of metrics g i on such that M Φ k min {Y (M, [g]), L(n, k)} lim inf i Y (MΦ k, [g i]) Y (M, [g]). Easy part: Difficult and important part lim inf i Y (MΦ k, [g i]) Y (M, [g]) min {Y (M, [g]), L(n, k)} lim inf i Y (MΦ k, [g i]) For this, we construct a sequence explicitly and study the behavior of the minimizers in [g i ] for i.

43 Proof of Theorem # 2 Theorem (ADH, #2) Suppose that Mk Φ is obtained from M by k-dimensional surgery. Then for any metric g on M there is a sequence of metrics g i on such that M Φ k min {Y (M, [g]), L(n, k)} lim inf i Y (MΦ k, [g i]) Y (M, [g]). Easy part: Difficult and important part lim inf i Y (MΦ k, [g i]) Y (M, [g]) min {Y (M, [g]), L(n, k)} lim inf i Y (MΦ k, [g i]) For this, we construct a sequence explicitly and study the behavior of the minimizers in [g i ] for i.

44 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 (2ρ, r r 2 0 ) f 2 (t)g S + dt 2 + g n k 1 for r < 2ρ that extends to a metric on M Φ k. round

45 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 (2ρ, r r 2 0 ) f 2 (t)g S + dt 2 + g n k 1 for r < 2ρ that extends to a metric on M Φ k. round

46 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 (2ρ, r r 2 0 ) f 2 (t)g S + dt 2 + g n k 1 for r < 2ρ that extends to a metric on M Φ k. round

47 g i = g g i = F 2 g S n k 1 has constant length

48 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, 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 ]) This sequence might: 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. Then study pointed Gromov-Hausdorff limits. Limit spaces: M c := H k+1 c S n k 1 H k+1 c : simply connected, complete, K = c 2

49 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, 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 ]) This sequence might: 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. Then study pointed Gromov-Hausdorff limits. Limit spaces: M c := H k+1 c S n k 1 H k+1 c : simply connected, complete, K = c 2

50 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, 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 ]) This sequence might: 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. Then study pointed Gromov-Hausdorff limits. Limit spaces: M c := H k+1 c S n k 1 H k+1 c : simply connected, complete, K = c 2

51 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, 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 ]) This sequence might: 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. Then study pointed Gromov-Hausdorff limits. Limit spaces: M c := H k+1 c S n k 1 H k+1 c : simply connected, complete, K = c 2

52 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, 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 ]) This sequence might: 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. Then study pointed Gromov-Hausdorff limits. Limit spaces: M c := H k+1 c S n k 1 H k+1 c : simply connected, complete, K = c 2

53 The numbers L(n, k) Additional conditions for k + 3 = n 6 L(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: L(n, 0) = Y (R S n 1 ) = Y (S n ) k = 1,..., n 3: L(n, k) > 0 N 4 n 1 n 2 du 2 + scal u 2 ( N up ) 2/p Conjecture #1: L(n, k) = 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 1 < c < 1. O(n k)-invariance is difficult, O(k + 1)-invariance follows from standard reflection methods

54 The numbers L(n, k) Additional conditions for k + 3 = n 6 L(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: L(n, 0) = Y (R S n 1 ) = Y (S n ) k = 1,..., n 3: L(n, k) > 0 N 4 n 1 n 2 du 2 + scal u 2 ( N up ) 2/p Conjecture #1: L(n, k) = 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 1 < c < 1. O(n k)-invariance is difficult, O(k + 1)-invariance follows from standard reflection methods

55 The numbers L(n, k) Additional conditions for k + 3 = n 6 L(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: L(n, 0) = Y (R S n 1 ) = Y (S n ) k = 1,..., n 3: L(n, k) > 0 N 4 n 1 n 2 du 2 + scal u 2 ( N up ) 2/p Conjecture #1: L(n, k) = 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 1 < c < 1. O(n k)-invariance is difficult, O(k + 1)-invariance follows from standard reflection methods

56 The numbers L(n, k) Additional conditions for k + 3 = n 6 L(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: L(n, 0) = Y (R S n 1 ) = Y (S n ) k = 1,..., n 3: L(n, k) > 0 N 4 n 1 n 2 du 2 + scal u 2 ( N up ) 2/p Conjecture #1: L(n, k) = 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 1 < c < 1. O(n k)-invariance is difficult, O(k + 1)-invariance follows from standard reflection methods

57 The numbers L(n, k) Additional conditions for k + 3 = n 6 L(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: L(n, 0) = Y (R S n 1 ) = Y (S n ) k = 1,..., n 3: L(n, k) > 0 N 4 n 1 n 2 du 2 + scal u 2 ( N up ) 2/p Conjecture #1: L(n, k) = 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 1 < c < 1. O(n k)-invariance is difficult, O(k + 1)-invariance follows from standard reflection methods

58 Simplest non-trivial case n = 4, k = 1. If Conjecture #2 holds, then Conjecture # 1 as well. L(4, 1) = 59, 4... Y (S 4 ) = 61, 5... Y (S 2 S 2 ) = 50, 2... One obtains S 2 S 2 via 1-dimensional surgery from S 4. Hence σ(s 2 S 2 ) 59, 4 > 50, 2... = Y (S 2 S 2 ).

59 Simplest non-trivial case n = 4, k = 1. If Conjecture #2 holds, then Conjecture # 1 as well. L(4, 1) = 59, 4... Y (S 4 ) = 61, 5... Y (S 2 S 2 ) = 50, 2... One obtains S 2 S 2 via 1-dimensional surgery from S 4. Hence σ(s 2 S 2 ) 59, 4 > 50, 2... = Y (S 2 S 2 ).

60 First ingredient to solve the conjectures: Theorem (Akutagawa, Große) If the scalar curvature of H k+1 c S n k 1 is positive, i.e. if c 2 k(k + 1) < (n k 1)(n k 2), then the infimum in the definition of Y (H k+1 c attained by a smooth positive L p -function. S n k 1 ) is