The Weyl functional near the Yamabe invariant

Abstract

For a compact manifold M of dim M = n ≥ 4, we study two conformal invariants of a conformal class C on M . These are the Yamabe constant Y C (M ) and the L n 2norm W C (M ) of the Weyl curvature. We prove that for any manifold M there exists a conformal class C such that the Yamabe constant Y C (M ) is arbitrarily close to the Yamabe invariant Y (M ), and, at the same time, the constant W C (M ) is arbitrarily large. We study the image of the map YW :We also apply our results to certain classes of 4-manifolds, in particular, minimal compact Kähler surfaces of Kodaira dimension 0, 1 or 2.Akutagawa, Botvinnik, Kobayashi, Seshadri, The Weyl functional near the Yamabe invariant

Aubin’s Lemma for the Yamabe constants of infinite coverings and a positive mass theorem

“…where h 0 denotes the standard metric of constant curvature one on RP 3 . We now explain briefly how this inequality is obtained.…”

“…[1][2][3][4]6,7,39]). In this article, we will present some extensions to Aubin's Lemma which will include, as a particular case, the first equality in the above working hypothesis (4).…”

“…In this case, the renormalization argument in Case 1 does not work, and hence we cannot get a similar uniform estimate to (7) for all v k . However, combining the following approximation lemma (see [3,Proposition 3.4] for the proof) with Corollary 1.6, we will be able to overcome this difficulty. …”

“…Since each g( j) k for k ≥ 1 is a lifting metric of g( j) ∈ M(M), the proof of Proposition 3.4 in [3] implies that, there exist a constant K ≥ 1 (independent of j, k and ε > 0) and a large integer j 2 (independent of k but depending on ε > 0) such that…”

“…3 We start by noting that Y (X, C) < Y n because, otherwise, we could apply Corollary 1.6 to the normal infinite conformal covering (X, C) → (M,Č) and conclude that X is simply connected. This contradicts that X is a non-trivial covering of X .…”