Low-dimensional surgery and the Yamabe invariant

Abstract

Assume that M is a compact n-dimensional manifold and that N is obtained by surgery along a k-dimensional sphere, k ≤ n − 3. The smooth Yamabe invariants σ(M ) and σ(N ) satisfy σ(N ) ≥ min(σ(M ), Λ) for a constant Λ > 0 depending only on n and k. We derive explicit lower bounds for Λ in dimensions where previous methods failed, namely for (n, k) ∈

Relations Between Threshold Constants for Yamabe Type Bordism Invariants

“…, Λ spin m,m−3 }. From Theorem 3.1, Section 11, and results in [6] and [7] we obtain explicit positive lower bounds for Λ spin m , summarized in Table 1 for low dimensions. Using standard techniques from bordism theory (see Section 12 for details) one obtains several conclusions: A similar bound also exists for non-simply connected manifolds, namely in this case for m = 9, 10…”

“…Proof. We start analogously as in the spin case from above with a nonnegative solution v ∈ L ∞ ∩ L 2 ∩ C 2 of Lv = µv p−1 on M m,k 1 and use the conformal map u in (7) to obtainṽ on S m \ S k . Analogous as in the proof of Lemma 7.4 we see that Uε(S k ) gives rise to a nonnegative solutionṽ on S m .…”

“…In Section 2.6 we pointed out that Λ spin m,m−3 = Λ spin, * m,m−3 . As a first step we estimate Q c : It was derived in [7,Thm. 4.1] for all c ∈ (0, 1) that…”

“…and use the conformal map u in (7) to obtain ṽ on S m \ S k . Analogous as in the proof of Lemma 7.4 we see that Uε(S k ) 1 ρ 2 ṽ2 dvol σ m < ∞ which allows to use Lemma 7.3 for k ≤ m − 2.…”

Non-existence of Yamabe minimizers on singular spheres

“…where M(M ) denotes the set of smooth Riemannian metrics on M . It is a very difficult problem to explicitly compute, or even give estimates for the Yamabe invariant, especially when it is positive (see for example Section 1.2 in [9], or [21], for reviews of the known results about the Yamabe invariant). In an upcoming work [6], we address to the question of obtaining a positive lower bound for the Yamabe invariant, given by the Yamabe constant of a singular Einstein metric with a edgecone singularity of codimension two and cone angle smaller than 2π.…”