Higher order obstructions to Riccati-type equations

TL;DR

This paper introduces new algebraic and geometric techniques to analyze Riccati-type equations on 3D Riemannian manifolds, revealing obstructions related to the metric and classifying certain harmonic manifolds.

Contribution

It develops a novel approach to identify obstructions in solving Riccati equations with algebraic constraints, leading to classification results for 3D harmonic manifolds.

Findings

Obstruction to solving Riccati equations is a degree 4 polynomial in metric coefficients.

Complete classification of 3D asymptotically harmonic manifolds as flat or hyperbolic.

New algebraic geometry methods applied to geometric analysis problems.

Abstract

We develop new techniques in order to deal with Riccati-type equations, subject to a further algebraic constraint, on Riemannian manifolds $(M^3,g)$. We find that the obstruction to solve the aforementioned equation has order $4$ in the metric coefficients and is fully described by an homogeneous polynomial in $\mathrm{Sym}^{16}TM$. Techniques from real algebraic geometry, reminiscent of those used for the "PositiveStellen-Satz " problem, allow determining the geometry in terms of the connection coefficients and a class of Hessian-type equations. Analysis of the latter shows flatness for the metric $g$; in particular we complete the classification of asymptotically harmonic manifolds of dimension $3$, establishing those are either flat or real hyperbolic spaces.