A sharp diameter-mass inequality on Einstein manifolds and ADM mass
Abstract
Let (Mn,g), n≥ 4 be a close Einstein manifold with positive scalar curvature and G be the Green function of the conformal Laplacian. The ADM mass of the asymptotically flat metric ĝ:=G4/(n−2) g is an important geometric invariant. When either n=5 or n≥ 6 with Weyl vanishing condition, we prove a mass formula, generalizing Gursky-Malchiodi's formula in dimension $4$. An important step is an extension of Viaclovsky's Green function expansion in higher dimensions. Next we prove a sharp inequality related to the volume radius and the mass for Einstein manifolds. Finally we present two applications. First, we prove a mass gap theorem for Einstein manifolds. Second, for a volume collapsing family of normalized Einstein n-manifolds (Mi,gi) (satisfying the Weyl vanishing condition when n≥ 6), we prove that the mass sequence m(gî) blows up uniformly as i→∞. This in particular applies to Collins-Székelyhidi's family of Sasaki-Einstein metrics gi on 𝕊5, and to the recent $4$-dimensional examples of T. Ozuch.
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BibTeX
@misc{chen2026sharp,
title = {{A sharp diameter-mass inequality on Einstein manifolds and ADM mass}},
author = {Xuezhang Chen and Jiaxue Gan and Yalong Shi and Nan Wu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.31343},
eprint = {2610.31343},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.31343},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - A sharp diameter-mass inequality on Einstein manifolds and ADM mass
AU - Xuezhang Chen
AU - Jiaxue Gan
AU - Yalong Shi
AU - Nan Wu
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2610.31343
UR - https://arxiv.org/abs/2610.31343
AB - Let (M^n,g), n≥ 4 be a close Einstein manifold with positive scalar curvature and G be the Green function of the conformal Laplacian. The ADM mass of the asymptotically flat metric ĝ:=G^4/(n−2) g is an important geometric invariant. When either n=5 or n≥ 6 with Weyl vanishing condition, we prove a mass formula, generalizing Gursky-Malchiodi's formula in dimension $4$. An important step is an extension of Viaclovsky's Green function expansion in higher dimensions. Next we prove a sharp inequality related to the volume radius and the mass for Einstein manifolds. Finally we present two applications. First, we prove a mass gap theorem for Einstein manifolds. Second, for a volume collapsing family of normalized Einstein n-manifolds (M_i,g_i) (satisfying the Weyl vanishing condition when n≥ 6), we prove that the mass sequence m(g_î) blows up uniformly as i→∞. This in particular applies to Collins-Székelyhidi's family of Sasaki-Einstein metrics g_i on 𝕊^5, and to the recent $4$-dimensional examples of T. Ozuch.
ER -