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Sharp Hessian inequalities and monotonicity for Green's functions of positively curved Einstein manifolds
Abstract
We prove a sharp Hessian inequality for the natural comparison Green's function of an Einstein manifold with positive sectional curvature, which may be viewed as a Green's function analogue of the Hessian comparison theorem and as a positively-curved, elliptic counterpart to the matrix Li-Yau-Hamilton inequality for the heat equation. We also show that this inequality is closely related to a family of monotonicity formulae, extending previous results of Park to the setting of positive curvature. We further present several geometric applications, such as a sharp comparison inequality for the sum of two distinct Green's distance functions.
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BibTeX
@misc{manea2026sharp,
title = {{Sharp Hessian inequalities and monotonicity for Green's functions of positively curved Einstein manifolds}},
author = {Cosmin Manea and Jacob Reznikov},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.11295},
eprint = {2610.11295},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.11295},
}
RIS
TY - UNPB
TI - Sharp Hessian inequalities and monotonicity for Green's functions of positively curved Einstein manifolds
AU - Cosmin Manea
AU - Jacob Reznikov
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2610.11295
UR - https://arxiv.org/abs/2610.11295
AB - We prove a sharp Hessian inequality for the natural comparison Green's function of an Einstein manifold with positive sectional curvature, which may be viewed as a Green's function analogue of the Hessian comparison theorem and as a positively-curved, elliptic counterpart to the matrix Li-Yau-Hamilton inequality for the heat equation. We also show that this inequality is closely related to a family of monotonicity formulae, extending previous results of Park to the setting of positive curvature. We further present several geometric applications, such as a sharp comparison inequality for the sum of two distinct Green's distance functions.
ER -