Appears in collection : Not Only Scalar Curvature Seminar

We address Gromov's Quantification of $C^0$-Convergence Conjecture in dimension three. Let $B$ be the unit ball in $R^3$. Let g and $g_0$ be smooth metrics on $B$. We prove there are constants $C$ and $\epsilon_0$ depending only on $g_0$ so that $\inf_{x \in B} R_g(x) < R_{g_0}(0) + C |g-g_0|_{C^0}^{1/2}$ provided $|g-g_0|_{C^0} < \epsilon_0$. 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 $C^0$-convergence of metrics. The proof is based on an analysis of harmonic functions. We will also briefly discuss some further applications of this method to other problems.