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arXiv · 2610.11295

Sharp Hessian inequalities and monotonicity for Green's functions of positively curved Einstein manifolds

Cosmin Manea·Jacob Reznikov
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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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Cosmin Manea, Jacob Reznikov. 2026-10-08. Sharp Hessian inequalities and monotonicity for Green's functions of positively curved Einstein manifolds. https://arxiv.org/abs/2610.11295

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