PreprintCC BY 4.0arXiv
Volume Entropy and Scalar Curvature near a Hyperbolic Metric
Abstract
Let (Mn,g0) be a closed hyperbolic manifold, n≥3. We compute the Hessian of volume entropy at g0 under the constraint Scg=−n(n−1) and express it as a quadratic form on transverse-traceless symmetric two-tensors. In dimension three, the Hessian is negative definite, and every sufficiently C∞-close metric with Scg≥−6 has entropy at most $2$, with equality precisely for pullbacks of g0. In dimensions n≥4, a universal spectral threshold determines the Hessian sign. Nonzero trace-free Codazzi tensors give positive directions, and hyperbolic bending provides smooth constrained entropy saddles in every such dimension.
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BibTeX
@misc{wang2026volume,
title = {{Volume Entropy and Scalar Curvature near a Hyperbolic Metric}},
author = {Xiaodong Wang and Bo Zhu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.06466},
eprint = {2610.06466},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.06466},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Volume Entropy and Scalar Curvature near a Hyperbolic Metric
AU - Xiaodong Wang
AU - Bo Zhu
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2610.06466
UR - https://arxiv.org/abs/2610.06466
AB - Let (M^n,g_0) be a closed hyperbolic manifold, n≥3. We compute the Hessian of volume entropy at g_0 under the constraint Sc_g=−n(n−1) and express it as a quadratic form on transverse-traceless symmetric two-tensors. In dimension three, the Hessian is negative definite, and every sufficiently C^∞-close metric with Sc_g≥−6 has entropy at most $2$, with equality precisely for pullbacks of g_0. In dimensions n≥4, a universal spectral threshold determines the Hessian sign. Nonzero trace-free Codazzi tensors give positive directions, and hyperbolic bending provides smooth constrained entropy saddles in every such dimension.
ER -