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 Contributions in differential geometry Round table for the 65th birthday of Lionel Bérard Bergery Luxembourg 2010

2 (Smooth) Yamabe invariant Let M be a compact n-dimensional manifold, n 3. The renormalised Einstein-Hilbert functional is M E : M R, 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 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 Y (M, [g]) (, Y (S n )] [g] M 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 Y (M, [g]) (, Y (S n )] [g] M 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 Y (M, [g]) (, Y (S n )] [g] M 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. LeBrun 97 Seiberg-Witten theory LeBrun & Gursky 98 Twisted Dirac operators

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. LeBrun 97 Seiberg-Witten theory LeBrun & Gursky 98 Twisted Dirac operators

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. LeBrun 97 Seiberg-Witten theory LeBrun & Gursky 98 Twisted Dirac operators

12 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. Bray & Neves 04: RP 3. σ(rp 3 ) = 2 2/3 σ(s 3 ). Inverse mean curvature flow Perelman, M. Anderson 06: compact quotients of 3-dimensional hyperbolic space Ricci flow

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 copies of S n 1 S 1. Attention σ(m) = Y (M, [g]) = E(g) 0 = g Einstein σ(m) = Y (M, [g]) = E(g) g Einstein 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 copies of S n 1 S 1. Attention σ(m) = Y (M, [g]) = E(g) 0 = g Einstein σ(m) = Y (M, [g]) = E(g) g Einstein 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 Positive scalar curvature σ(m) > 0 is a bordism invariant. Bordism classes admitting positive scalar curvature metrics form an ideal in the bordism ring Ω spin n (Bπ 1 ). It is the preimage of an ideal of the map D : Ω spin (Bπ 1 ) ko (Bπ 1 )

18 Questions for fixed ɛ > 0 and π 1 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 5. Is the above subgroup a preimage of Ω spin n (Bπ 1 ) ko n (Bπ 1 )? Unknown, good project, a lot of work to do. Main tool for 1. and 2.: Surgery formula for σ(m)

19 Questions for fixed ɛ > 0 and π 1 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 5. Is the above subgroup a preimage of Ω spin n (Bπ 1 ) ko n (Bπ 1 )? Unknown, good project, a lot of work to do. Main tool for 1. and 2.: Surgery formula for σ(m)

20 Questions for fixed ɛ > 0 and π 1 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 5. Is the above subgroup a preimage of Ω spin n (Bπ 1 ) ko n (Bπ 1 )? Unknown, good project, a lot of work to do. Main tool for 1. and 2.: Surgery formula for σ(m)

21 Questions for fixed ɛ > 0 and π 1 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 5. Is the above subgroup a preimage of Ω spin n (Bπ 1 ) ko n (Bπ 1 )? Unknown, good project, a lot of work to do. Main tool for 1. and 2.: Surgery formula for σ(m)

22 Questions for fixed ɛ > 0 and π 1 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 5. Is the above subgroup a preimage of Ω spin n (Bπ 1 ) ko n (Bπ 1 )? Unknown, good project, a lot of work to do. Main tool for 1. and 2.: Surgery formula for σ(m)

23 Questions for fixed ɛ > 0 and π 1 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 5. Is the above subgroup a preimage of Ω spin n (Bπ 1 ) ko n (Bπ 1 )? Unknown, good project, a lot of work to do. Main tool for 1. and 2.: Surgery formula for σ(m)

24 Questions for fixed ɛ > 0 and π 1 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 5. Is the above subgroup a preimage of Ω spin n (Bπ 1 ) ko n (Bπ 1 )? Unknown, good project, a lot of work to do. Main tool for 1. and 2.: Surgery formula for σ(m)

25 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.

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 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.

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]).

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]).

31 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]).

32 Compare different results Gromov-Lawson-construction not compatible with conformal structure Kobayashi-construction special case of our construction no techniques for dealing with k > 0 Petean-construction Uses the formula inf scal g L g M n/2 ( g) = { σ(m) if σ(m) 0 0 if σ(m) > 0 Hence min{σ(m), 0} is determined by an infimum. ADH: We had to work with the sup-inf-characterization of σ. Our knowledge before: Similar surgery formula holds for a spinorial analogue. The proof of the spinorial analogue directly lead to a proof in the case k < (n/2) 1. Our proof needed a new engine for

33 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! Problems with k {0, 1, n 2, n 1}. Consider [M, f ] where f : M Bπ 1 = Bπ 1 (M) is the classifying map. Bordisms with maps to Bπ 1.

34 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! Problems with k {0, 1, n 2, n 1}. Consider [M, f ] where f : M Bπ 1 = Bπ 1 (M) is the classifying map. Bordisms with maps to Bπ 1.

35 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! Problems with k {0, 1, n 2, n 1}. Consider [M, f ] where f : M Bπ 1 = Bπ 1 (M) is the classifying map. Bordisms with maps to Bπ 1.

36 Let Ω spin n (Bπ 1 ) the spin cobordism group over Bπ 1. Lemma n 5. Let (M 1, f 1 ) and (M 2, f 2 ) be spin cobordant over Bπ 1 and let f 2 : M 2 Bπ 1 (M 2 ) be classifying. Then σ(m 2 ) σ(m 1 ). σ := min{σ(m), L(n)} s π1 ([M, f ]) := max{ σ(m), 0} if f is classifying. s π1 : Ω spin n (Bπ 1 ) (, L(n)]

37 Let Ω spin n (Bπ 1 ) the spin cobordism group over Bπ 1. Lemma n 5. Let (M 1, f 1 ) and (M 2, f 2 ) be spin cobordant over Bπ 1 and let f 2 : M 2 Bπ 1 (M 2 ) be classifying. Then σ(m 2 ) σ(m 1 ). σ := min{σ(m), L(n)} s π1 ([M, f ]) := max{ σ(m), 0} if f is classifying. s π1 : Ω spin n (Bπ 1 ) (, L(n)]

38 s π1 : Ω spin n (Bπ 1 ) (, 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 {σ(m) : dim M = n, π 1 (M) = 1} (0, L(n)) has no accumulation points from above.

39 s π1 : Ω spin n (Bπ 1 ) (, 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 {σ(m) : dim M = n, π 1 (M) = 1} (0, L(n)) has no accumulation points from above.

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 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.

42 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.

43 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.

44 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]). We construct a sequence explicitly and study the behavior of the minimizers in [g i ] for i.

45 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]). We construct a sequence explicitly and study the behavior of the minimizers in [g i ] for i.

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 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

48 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

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

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 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

54 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

55 Major problem Note that vol(m ϕ k, g i). And scal < 0 in large regions for k > (n/2) 1. A priori it might happen that u i L p = 1 u i L 0 Thus all weak limits and blow-up limits would vanish. Define w(t 0 ) := ui 2 dvol n 1 t=t 0 w (t) cw(t). Subtle estimates on warped products. Help from Arthur Besse s book on Einstein manifolds!

56 Major problem Note that vol(m ϕ k, g i). And scal < 0 in large regions for k > (n/2) 1. A priori it might happen that u i L p = 1 u i L 0 Thus all weak limits and blow-up limits would vanish. Define w(t 0 ) := ui 2 dvol n 1 t=t 0 w (t) cw(t). Subtle estimates on warped products. Help from Arthur Besse s book on Einstein manifolds!

57 The numbers L(n, k) (Disclaimer: 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 The numbers L(n, k) (Disclaimer: 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

59 The numbers L(n, k) (Disclaimer: 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

60 The numbers L(n, k) (Disclaimer: 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

61 The numbers L(n, k) (Disclaimer: 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

62 Simplest non-trivial case n = 4, k = 1. If Conjecture #2 holds, then Conjecture #1 will probably follow. 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 ).

63 Simplest non-trivial case n = 4, k = 1. If Conjecture #2 holds, then Conjecture #1 will probably follow. 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 ).

64 First ingredients 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 Other progress Isoperimetric lower bounds for Y (M R 2 ) by Petean assuming a lower bound on ric M (work in progress).

65 First ingredients 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 Other progress Isoperimetric lower bounds for Y (M R 2 ) by Petean assuming a lower bound on ric M (work in progress).