Riccati Equation for Static Spaces and its Applications
Zhixin Wang
TL;DR
This work derives a Riccati-type inequality for (sub-)static Einstein spaces by introducing the conformal frame $\tilde{g}=\frac{1}{V^2}g$ and $\tilde{V}=\frac{1}{V}$, and defines $\theta=\frac{H}{V}$ from the mean curvature $H$ of level sets. The key result is the inequality $\frac{\partial \theta}{\partial s}\le -\frac{1}{n-1}\theta^2$ under a reparameterization $ds=V^2 d\tilde{r}$, which is interpreted via the Raychaudhuri equation in a lifted Lorentzian setting. This Riccati framework yields a robust splitting theorem for conformally compactifiable static spaces, showing their universal cover splits as $M^*=\mathbb{R}^k\times \Sigma$ with a product-type metric, and it enables proofs of the connectedness of conformal boundaries via superharmonic distance functions. In the positive scalar curvature regime, the results imply compact universal covers, plus structural and classification consequences for static triples, including a $H^1(M)=0$ identity and correspondence with known 3D models through an ODE for $\theta$, thereby providing a unified, Lorentz-free approach to several key geometric-analytic problems in static geometry.
Abstract
In this paper, we derive a Riccati-type equation applicable to (sub-)static Einstein spaces and examine its various applications. Specifically, within the framework of conformally compactifiable manifolds, we prove a splitting theorem for the Riemannian universal covering. Furthermore, we demonstrate two distinct methods by which the Riccati equation can establish the connectivity of the conformal boundary under the static Einstein equation. Additionally, for compact static triples possessing positive scalar curvature, we establish the compactness of the universal covering.