Quantification of $C^0$ Convergence in Dimension Three

TL;DR

This paper proves a sharp quantitative relation between $C^0$ metric convergence and scalar curvature bounds in three dimensions, using harmonic functions and elliptic PDE estimates.

Contribution

It establishes the first explicit $C^0$ convergence rate for scalar curvature lower bounds in three dimensions, including sharpness and special cases.

Findings

Quantifies how scalar curvature bounds are preserved under $C^0$ metric convergence.

Constructs examples demonstrating the sharpness of the $1/2$ exponent.

Provides estimates for metrics with $C^2$ regularity and rotational symmetry.

Abstract

We address Gromov's Quantification of $C^0$ Convergence Conjecture in dimension three. Let $B$ be the unit ball in $\mathbb R^3$. 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 \[ \inf_{x\in B} R_g(x) \leq R_{g_0}(0) + C \|g-g_0\|_{C^0}^{1/2} \] provided $\|g-g_0\|_{C^0}\leq \epsilon_0$. 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. When $g_0$ is merely $C^2$ 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 $C^0$ 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.