Quantification of $C^0$ Convergence in Dimension Three

Abstract: We address Gromov's Quantification of $C<sup>0$ Convergence Conjecture in dimension three. Let $B$ be the unit ball in $\mathbb R<sup>3$. Let $g$ and $g_0$ be smooth metrics on $B$. We prove there are constants $C$ and $ε<em>0$ depending only on $g_0$ so that [ \inf{x\in B} R_g(x) \leq R_{g_0}(0) + C |g-g_0|{C^0}^{1/2} ] provided $|g-g_0|</em>{C^0}\leq ε_0$. We also construct examples to show that the exponent $1/2$ is sharp. This explicitly quantifies the fact that scalar curvature lower bounds are preserved under $C<sup>0$ convergence of metrics. When $g_0$ is merely $C<sup>2$ we prove a related estimate with a slightly weaker rate, and when $g_0$ has rotational symmetry we prove a related estimate with a stronger linear rate. To prove these results, we use harmonic functions to define a local quantity that detects the scalar curvature. Then we use classical elliptic PDE estimates to show that this quantity is stable under $C<sup>0$ perturbations of the metric. As a further application of this method, we give a partial answer to a question of Gromov on the preservation of scalar curvature lower bounds for metrics that are converging in measure.