Sharp Hessian inequalities and monotonicity for Green's functions of positively curved Einstein manifolds
Year: 2026
Type: preprint
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.
Source: arXiv (Cornell University)
Authors: Manea, Cosmin, Reznikov, Jacob
Institution: Massachusetts Institute of Technology
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7221092559