Bach-pinched metrics on closed manifolds

TL;DR

This work proves that every closed $4$-manifold admits metrics with negative scalar-Bach curvature by extending Aubin's deformation method to the Bach tensor setting. It defines the scalar-Bach curvature $F_g^B = S_g + t|B_g|^{1/2}_g$ and the modified conformal Laplacian $\mathscr{L}^t_g$, establishing conformal covariance and a Yamabe-type variational framework via the invariant functional $\widehat{Y}^B$. The main result shows the existence of $g$ with $F_g^B \equiv -1$ (hence $S_g + t|B_g|^{1/2}_g \equiv -1$), leading to Bach-pinching with $S_g<0$ and $|B_g| < \varepsilon S_g^2$ for any $\varepsilon>0$. The proof combines Aubin-type metric deformations, precise estimates of the deformed Bach tensor, and a finite-ball construction to guarantee negativity of the global scalar-Bach functional, followed by a conformal reduction to the constant-$F^B$ case.

Abstract

Exploiting the deformation method introduced by Aubin in his seminal work to construct constant negative scalar curvature metrics, we show the existence, on every closed manifold of dimension four, of a metric whose Bach tensor is pinched by the scalar curvature.

Theorems & Definitions (12)

• Theorem 1.1
• Corollary 1.2
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 3.1
• proof
• Lemma 3.2
• Lemma 4.1