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The Initial Value Problem for Harmonic maps of Cohomogeneity One manifolds

Anna Siffert

TL;DR

This work develops a rigorous framework for the initial value problem of equivariant harmonic maps on cohomogeneity one manifolds by reducing the harmonic map equation to a singular ODE along a normal geodesic and proving local existence and uniqueness of smooth radius functions near a singular orbit. Central to the analysis is a comprehensive treatment of regular-singular first-order systems, including linear and nonlinear cases, monodromy maps, and their geometric interpretation via holomorphic vector and affine bundles. The authors establish a systematic method, via a modified Picard iteration, to solve the IVP and lay out precise smoothness and symmetry conditions that ensure well-posedness of the reduced problem. The paper further provides a detailed theory of regular-singular systems, covering scalar and vector power functions, fundamental solutions, monodromy, and normalization techniques, which collectively underpin the construction of equivariant harmonic maps on cohomogeneity-one spaces. This framework advances both the existence theory for equivariant harmonic maps and the broader toolbox for regular-singular differential systems with potential applications to geometric analysis on manifolds with symmetry.

Abstract

We set up and solve the initial value problem for equivariant harmonic maps of cohomogeneity one manifolds, i.e. we show the local existence of a harmonic map in the neighborhood of a singular orbit. Furthermore, we present some theory of regular-singular systems of first order.

The Initial Value Problem for Harmonic maps of Cohomogeneity One manifolds

TL;DR

This work develops a rigorous framework for the initial value problem of equivariant harmonic maps on cohomogeneity one manifolds by reducing the harmonic map equation to a singular ODE along a normal geodesic and proving local existence and uniqueness of smooth radius functions near a singular orbit. Central to the analysis is a comprehensive treatment of regular-singular first-order systems, including linear and nonlinear cases, monodromy maps, and their geometric interpretation via holomorphic vector and affine bundles. The authors establish a systematic method, via a modified Picard iteration, to solve the IVP and lay out precise smoothness and symmetry conditions that ensure well-posedness of the reduced problem. The paper further provides a detailed theory of regular-singular systems, covering scalar and vector power functions, fundamental solutions, monodromy, and normalization techniques, which collectively underpin the construction of equivariant harmonic maps on cohomogeneity-one spaces. This framework advances both the existence theory for equivariant harmonic maps and the broader toolbox for regular-singular differential systems with potential applications to geometric analysis on manifolds with symmetry.

Abstract

We set up and solve the initial value problem for equivariant harmonic maps of cohomogeneity one manifolds, i.e. we show the local existence of a harmonic map in the neighborhood of a singular orbit. Furthermore, we present some theory of regular-singular systems of first order.
Paper Structure (19 sections, 15 theorems, 111 equations)

This paper contains 19 sections, 15 theorems, 111 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Cohomogeneity one manifolds
  4. Smoothness conditions of cohomogeneity one metrics
  5. Harmonic maps between cohomogeneity one manifolds
  6. The initial value problem for harmonic maps
  7. Regular-singular systems of first order
  8. Scalar power functions
  9. Vector-valued generalizations of the scalar power functions
  10. Linear regular-singular system of first order in $n$ variables
  11. Fundamental solution of the homogeneous system
  12. Solutions of the inhomogeneous equation
  13. Discussion of the monodromy map
  14. Geometric properties of the monodromy map
  15. Non-linear regular-singular systems
  16. ...and 4 more sections

Key Result

Theorem 1.1

Let a cohomogeneity one manifold $M$ with a singular orbit $N$ be given. Assume that $N$ is the fibre of $0\in M/G$ under the natural projection map $\pi:M\rightarrow M/G$. For each $v$ there exists a unique smooth solution $r:(M/G)^{\circ}\cup\{0\}\rightarrow\mathbb{R}$ of (ode0) with $r(0)=0$ and

Theorems & Definitions (28)

  • Theorem 1.1
  • Theorem 2.1: MR4400726
  • Theorem 2.2
  • Theorem 2.3: Theorem 3.4 in MR4000241
  • Theorem 2.4
  • Theorem 3.1
  • proof : Proof of Theorem \ref{['main']}
  • Remark 3.3
  • Remark 4.1
  • Definition 4.2
  • ...and 18 more