Initial value problems on cohomogeneity one manifolds, I

TL;DR

This paper investigates initial value problems for geometric equations on cohomogeneity one manifolds near singular orbits, establishing existence and uniqueness results under specific assumptions, with plans to generalize in future work.

Contribution

It introduces a simplified approach to solving geometric PDEs on cohomogeneity one manifolds near singular orbits, paving the way for more general solutions.

Findings

Existence of solutions near singular orbits for Ricci curvature, Einstein, and soliton equations.

Solutions are unique up to finitely many constants under certain assumptions.

Framework sets the stage for solving the general case in subsequent work.

Abstract

We study initial value problems for various geometric equations on a cohomogeneity manifold near a singular orbit. We show that when prescribing the Ricci curvature, or finding solutions to the Einstein and soliton equations, there exist solutions near the singular orbit, unique up to a finite number of constants. In part I we make a special assumption that significantly simplifies the proof, and will solve the general case in Part II.