Proves that the harmonic mass of a continuous asymptotically flat metric on R^3 is non-negative, with equality only when the metric is flat.
Quantification of $C^0$ Convergence in Dimension Three
6 Pith papers cite this work, alongside 11 external citations. Polarity classification is still indexing.
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Pith papers citing it
11
external citations · Pith
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.
fields
math.DG 6years
2026 6verdicts
UNVERDICTED 6representative citing papers
Scalar curvature lower bounds are preserved under weak limits of smooth closed 3-manifolds through μ-bubble comparisons when volumes and Lipschitz constants converge appropriately.
The refined quantitative scalar curvature lower bound under C^0 convergence holds in all dimensions greater than or equal to three.
Proves that a smooth complete metric g on R^3 with Scal(g) >= 0 and |g - g_Euc| = o(r^{-1}) as r -> infinity is isometric to Euclidean space.
Defines scalar curvature lower bounds for C^0 3-metrics via IMCF Hawking mass monotonicity and states a stability theorem for nonnegative scalar curvature in this sense.
Smooth metrics on R^3 with non-negative scalar curvature and |g - g_euc| = O(|x|^{-1-τ}) for τ>0 are necessarily flat.
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Defines scalar curvature lower bounds for C^0 3-metrics via IMCF Hawking mass monotonicity and states a stability theorem for nonnegative scalar curvature in this sense.
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Smooth metrics on R^3 with non-negative scalar curvature and |g - g_euc| = O(|x|^{-1-τ}) for τ>0 are necessarily flat.