A surgery formula for the smooth Yamabe invariant Bernd Ammann (joint work with M. Dahl, E. Humbert)

1. Historical overview

For a compact Riemannian manifold (M, g_{0}) of dimension n ≥ 3 we define the conformal Yamabe invariant to be

Y (M, [g_{0}]) := inf_{g∈[g_{0}]} ∫_{M} Scal_{g} dv^{g}

where [g_{0}] denotes the set of all metrics conformal to g and of volume 1. The solution of the Yamabe problem (see e.g. [10] for a good overview) states that this infimum is always attained. The scalar curvature of the mimimzing metric satisfies Scal_{g} ≡ Y (M, [g_{0}]). It is not difficult to show that

Y (M, [g_{0}]) ≤ Y (𝕊^{n}) = n(n − 1)ω_{n}

where 𝕊^{n} denotes the sphere with the standard metric, and where ω_{n} = Vol(𝕊^{n}).

The smooth Yamabe invariant, also called Yamabe number or Schoen’s σinvariant is then defined as

σ(M) := sup_{[g_{0}]} Y (M, [g_{0}]) ∈ (−∞, Y (𝕊^{n})]

where the supremum runs over all conformal classes [g_{0}] on M.

The σ-invariant is positive if and only if M carries a metric of positive scalar curvature. Calculating σ-invariants in the positive case is also difficult. The first examples have been calculated by Schoen [12, Section 2] where one sees that σ(S^{n−1} × S^{1}) = Y (𝕊^{n}), and the same method yields for S^{n−1} ˜×S^{1}, the nonorientable S^{n−1} bundle over S^{1}, that σ(S^{n−1} ˜×S^{1}) = Y (𝕊^{n}). By Kobayashi’s surgery formula [8] it is also clear that the σ-invariant of connected sums of above manifolds have the value Y (𝕊^{n}). It is conjectured [12, Page 10, lines 6–11] that all quotient of round spheres satisfy σ(S^{n}/Γ) = (#Γ)^{−2/n}Y (𝕊^{n}), but this conjecture is only verified for ℝP^{3} [5]. An extended Aubin’s Lemma enabled the complete characterization of all 3-dimensional manifolds with σ(M) > σ(ℝP^{3}), see [1]. The smooth Yamabe invariant of σ(ℂP^{2}) was calculated in [9] and [7]. With similar methods the σ-invariants of some several other Kähler manifolds could be determined.

As smooth Yamabe invariants are difficult to calculate explicitly, estimates are very helpful. An important class of estimates can be proven with the help of surgery formulas. Let Φ : S^{k} × \overline{B^{n−k}} → M be an embedding. We define

M_{k}^{Φ} := M \ Φ(S^{k} × B^{n−k}) ∪_{∼} \overline{B^{k+1}} × S^{n−k−1}

and we say that M_{k}^{Φ} is obtained from M by k-dimensional surgery along Φ. Connected sums are examples for 0-dimensional surgery along non-connected manifolds.

Theorem 1 ([8]). If M_{0}^{Φ} is obtained from M by 0-dimensional surgery, then

σ(M_{0}^{Φ}) ≥ σ(M).

The next theorem states that monotonicity still holds in the case σ(M) ≤ 0 for k ∈ {1, …, n − 3}.

Theorem 2 ([11]). If M_{k}^{Φ} is obtained from M by k-dimensional surgery, 1 ≤ k ≤ n − 3, then

σ(M_{k}^{Φ}) ≥ min{σ(M), 0}.

Using results from bordism theory and from [14] this implies that all simply connected manifolds of dimension ≥ 5 have σ(M) ≥ 0.

On the other hand positivity of σ(M) is preserved under surgeries of dimension k ∈ {1, …, n − 3}.

Theorem 3 ([6],[13]). If M_{k}^{Φ} is obtained from M by k-dimensional surgery, 1 ≤ k ≤ n − 3, and σ(M) > 0, then we also have σ(M_{k}^{Φ}) > 0.

2. New surgery formula for σ

In [4] we prove a surgery formula for the σ-invariant that is stronger than Theorems 1, 2 and 3 together.

Theorem 4. Let 0 ≤ k ≤ n − 3. There is a positive constant Λ_{n,k} depending only on n and k such that

σ(M_{k}^{Φ}) ≥ min{σ(M), Λ_{n,k}}

if M_{k}^{Φ} is obtained from M by k-dimensional surgery. Furthermore Λ_{n,0} = Y (𝕊^{n}).

We will not give a detailed proof in this short announcement, but restrict to a short sketch of the method used herein. Details will be published in [4].

The theorem directly follows from the following:

Theorem 5. Suppose that M_{k}^{Φ} is obtained from M by k-dimensional surgery. Then for any metric g on M there is a family of metrics g_{θ}, θ ∈ [0, 1], on M_{k}^{Φ} such that

min {Y (M, [g]), Λ_{n,k}} ≤ lim inf_{θ→0} Y (M_{k}^{Φ}, [g_{θ}]) ≤ lim sup_{θ→0} Y (M_{k}^{Φ}, [g_{θ}]) ≤ Y (M, [g]).

These metrics are very similar to the metrics g_{ρ} described in [2] and the metrics g_{τ} described in [3].

The inequality

lim sup_{θ→0} Y (M_{k}^{Φ}, [g_{θ}]) ≤ Y (M, [g])

in Theorem 5 can be shown by standard cut-off arguments. The other inequality, namely

min {Y (M, [g]), Λ_{n,k}} ≤ lim inf_{θ→0} Y (M_{k}^{Φ}, [g_{θ}])

is much harder to show, but it is the important one for proving Theorem 4.

For showing this inequality we study minimzers of the Yamabe functional of (M_{k}^{Φ}, [g_{θ}]) for a sequence of θ → 0. A sophisticated blow-up analysis for such minimizers is worked out. The arguments of [3] can only be applied for 2k < n−2, hence completely new tools have to be developped to treat the general case. We see that the sequence either concentrates in the part (M_{k}^{Φ}, [g_{θ}]) that is locally conformal to (M \Φ(S^{k} ×{0}), g). In this case we obtain convergence to a solution of the Yamabe equation on (M \Φ(S^{k} ×{0}), g) that can be extended to a solution on (M, g) and we derive

Y (M, [g]) ≤ lim inf_{θ→0} Y (M_{k}^{Φ}, [g_{θ}]).

Or the minimizers concentrate on the new “handle” and then blowup analysis provides a solution of

Lu = λ|u|^{\frac{2}{n−2}} u (1)

on (P (n, k), H_{K}) := 𝕊^{n−k−1} × (B^{k+1}, h_{K}) for a suitable K ∈ [−1, 0]. Here h_{K} is “the” complete metric of sectional curvature K ∈ [−1, 0] on the ball B^{k+1}. For example (B^{k+1}, h_{0}) is isometric to euclidean space and (B^{k+1}, h_{−1}) is isometric to the hyperbolic space.

We define

Y (P (n, k), H_{K}) := inf λ ∈ [0, ∞]

where λ runs over all values λ ∈ (0, ∞) for which there is a positive function u ∈ L^{∞}(P (n, k)) ∩ C_{loc}^{2}(P (n, k)), ∥u∥_{L^{\frac{2n}{n−2}}} = 1 solving (1) on (P (n, k), H_{K}). and we obtain

Y (P (n, k), H_{K}) ≤ lim inf_{θ→0} Y (M_{k}^{Φ}, [g_{θ}]).

Other blowup arguments show

Λ_{n,k} := inf_{K∈[−1,0]} Y (P (n, k), H_{K}) > 0

and Theorem 5 follows.

As (P (n, k), H_{K}) is conformal to 𝕊^{n} \ {p_{1}, p_{2}}, one can see that Λ_{n,0} = Y (𝕊^{n}).

3. Topological applications

We consider pairs (M, f) where M is a compact spin manifold and where f : M → BΓ is continuous. Two such pairs (M_{1}, f_{1}) and (M_{2}, f_{2}) are spin bordant if there exists an n + 1-dimensional manifold W with boundary −M_{1} ∪ M_{2} with a map F : W → BΓ such that the restriction of F to the boundary yields f_{1} and f_{2}. It is implicitly required that the boundary carries the induced orientation and spin structure and −M_{1} denotes M_{1} with reversed orientation. Bordance is an equivalence relation, the corresponding equivalence classes are denoted by [(M, f)] and the set of equivalence classes is Ω_{n}(BΓ). Disjoint union defines a sum on Ω_{n}(BΓ) which turns it into a group.

We say that a tuple (M, f) with f : M → BΓ is a π_{1}-bijective representative of [(M, f)] if M is connected and if the induced map f_{∗} : π_{1}(M) → Γ is a bijection. Any equivalence class in Ω_{n}(BΓ) has a π_{1}-bijective representative.

Now we define

Λ_{n} := min{Λ_{n,1}, … Λ_{n,n−3}} > 0,

σ̄(M) := min{σ(M), Λ_{n}}.

Proposition. Let n ≥ 5. Let (M_{1}, f_{1}) and (M_{2}, f_{2}) be compact spin manifolds with maps f_{i} : M_{i} → BΓ. If (M_{1}, f_{1}) and (M_{2}, f_{2}) are spin bordant and if f_{2} is π_{1}-bijective, then

σ̄(M_{1}) ≤ σ̄(M_{2}).

Hence σ̄ defines a well-defined map Ω_{n}(BΓ) → ℝ. We conclude from the surgery formula.

Theorem 6. Let t ≥ 0. n ≥ 5. Then the sets

G(t) := {[x] ∈ Ω_{n}(BΓ) | σ̄(x) > t}

and

\overline{G}(t) := \{[x] ∈ Ω_{n}(BΓ) | σ̄(x) ≥ t\}

are subgroups of Ω_{n}(BΓ).

The theorem admits — among other interesting conclusions — the following application. For p ∈ ℕ^{∗} we write p#M for M# ⋯ #M where M appears p times. We already know σ̄(p#M) ≥ σ̄(M) if σ̄(M) ≥ 0.

Corollary. Suppose M is a compact spin manifold of dimension at least 5 with σ(M) ∈ (0, Λ_{n}). Let p and q be two relatively prime positive integers. If σ(p#M) > σ(M), then σ(q#M) = σ(M).

References

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