Quantification of C^0-convergence in Dimension 3 (I)

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 $\inf_{x \in B} R_g(x) \lt R_{g_0}(0) + C \|g-g_0\|_{C^0}^{1/2}$ provided $|g-g_0|_{C^0} \lt \epsilon_0$. 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.

Liam Mazurowski (Southern Connecticut State University)