PreprintCC BY 4.0arXiv
A note on σk maximal conformal volume
Abstract
We establish a sharp spherical upper bound for the σk-maximal conformal volume Λk introduced by Gursky and Viaclovsky. Let (Mn,[g]) be a smooth, closed, connected conformal manifold with n≥ 3 and n/2<k≤ n, and suppose that [g] contains a k-admissible metric, we prove that $ Λ_k(M,[g]) \leq \operatorname{Vol}(\mathbb S^n,g_{\mathrm{rd}}), $ with equality if and only if (M,[g]) is conformally equivalent to the standard sphere. Combined with the existence and compactness theorem of Gursky and Viaclovsky below the spherical threshold, this completes their approach to the σk-Yamabe problem through maximal conformal volume in the range k>n/2.
Related
BibTeX
@misc{lai2026note,
title = {{A note on $σ_k$ maximal conformal volume}},
author = {Mijia Lai},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.33986},
eprint = {2610.33986},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.33986},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A note on σ_k maximal conformal volume
AU - Mijia Lai
PY - 2026
DA - 2026/10/25
PB - arXiv
DO - 10.48550/arxiv.2610.33986
UR - https://arxiv.org/abs/2610.33986
AB - We establish a sharp spherical upper bound for the σ_k-maximal conformal volume Λ_k introduced by Gursky and Viaclovsky. Let (M^n,[g]) be a smooth, closed, connected conformal manifold with n≥ 3 and n/2<k≤ n, and suppose that [g] contains a k-admissible metric, we prove that $ Λ_k(M,[g]) \leq \operatorname{Vol}(\mathbb S^n,g_{\mathrm{rd}}), $ with equality if and only if (M,[g]) is conformally equivalent to the standard sphere. Combined with the existence and compactness theorem of Gursky and Viaclovsky below the spherical threshold, this completes their approach to the σ_k-Yamabe problem through maximal conformal volume in the range k>n/2.
ER -