Scalar Curvature Flexibility in the Riemannian Burnett Compactness Class

Abstract

Let $M$ be a connected smooth $n$-manifold without boundary, where $n\ge 3$, and let $\kappa \in \mathbb{R}$, with $\kappa \le 0$ if $M$ is open. We prove that every smooth Riemannian metric $g_0$ with ${\mathrm{Scal}}_{g_0}\ge \kappa$ is a locally uniform limit of smooth Riemannian metrics $g_i$ with ${\mathrm{Scal}}_{g_i}=\kappa$ that are locally uniformly bounded in $W^{1,\infty }$. As a corollary, combining this with Gromov's $C^0$-stability theorem, we obtain the perhaps surprising identity \[ \overline{\{g:\mathrm{Scal}_g=κ\}}^{\,C^{0,α}_{\mathrm{loc}}}=\{g:\mathrm{Scal}_g\geqκ\}, \quad \forall α\in(0,1). \] The restriction $\alpha <1$ is sharp. At $\kappa =0$, this proves and strengthens the Riemannian reverse-Burnett conjecture of Huneau and Luk.

26 pages, all comments welcome!