Short-Time Existence of the Second Order Renormalization Group Flow in Dimension Three

TL;DR

This paper proves the short-time existence of a second-order renormalization group flow on compact three-manifolds under certain curvature conditions, advancing understanding of geometric evolution equations.

Contribution

It establishes the short-time existence of the second-order RG flow in three dimensions with specific curvature assumptions, a novel result in geometric analysis.

Findings

Short-time existence of the flow is proven under curvature conditions.

The flow's behavior is characterized for three-dimensional manifolds.

The result extends previous first-order flow analyses.

Abstract

Given a compact three-manifold together with a Riemannian metric, we prove the short-time existence of a solution to the renormalization group flow, truncated at the second order term, under a suitable hypothesis on the sectional curvature of the initial metric.