The Initial Value Problem for Harmonic maps of Cohomogeneity One manifolds

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.

💡 Research Summary

The paper addresses the initial value problem for equivariant harmonic maps between cohomogeneity‑one manifolds, establishing local existence and uniqueness of solutions near a singular orbit. The authors begin by recalling the variational definition of harmonic maps: a smooth map φ : (M,g) → (N,h) is harmonic if it is a critical point of the energy functional E(φ)=½∫_Ω|dφ|² dv_g, which leads to the vanishing of the tension field τ(φ). In general, τ(φ)=0 forms a semilinear elliptic system of second‑order PDEs, for which no general solution theory exists.

To overcome this difficulty, the authors impose symmetry. A cohomogeneity‑one manifold M carries a smooth isometric action of a compact Lie group G such that the orbit space M/G is one‑dimensional. Regular (principal) orbits have codimension one, while singular orbits correspond to the boundary points of M/G. An equivariant map φ must send each G‑orbit to a G‑orbit; consequently, after fixing a unit‑speed normal geodesic γ intersecting the singular orbit at t=0, any equivariant map can be written as  ψ(g·γ(t)) = g·γ(r(t)), where r : (M/G)° → ℝ is a smooth scalar function satisfying appropriate boundary conditions at the singular points (r(0)=0, possibly r(L)=kL in the closed‑interval case). The harmonic condition τ(ψ)=0 reduces to a single ordinary differential equation  r̈(t) + h₁(t)·ṙ(t) + h₂(t, r(t)) = 0  (1.3), where the coefficients h₁, h₂ are determined by the geometry of M (in particular by the metric endomorphism P_t describing the G‑invariant metric along γ). At a singular orbit the ODE becomes singular because h₁ and h₂ typically contain terms like 1/t.

The central result (Theorem 1.1) states that for any prescribed initial velocity v∈ℝ there exists a unique smooth solution r defined on a neighbourhood of the singular orbit with r(0)=0, ṙ(0)=v, and that the solution depends continuously on v. The proof proceeds by rewriting (1.3) as a first‑order system and recognizing it as a regular‑singular system of the form t·Y′ = A(t)Y + t·F(t,Y). Section 6 collects the necessary theory of such systems: under the condition that the eigenvalues of A(0) are non‑negative integers, one can construct a formal power series solution and then apply a fixed‑point argument to obtain a genuine smooth solution. The authors verify that the geometric data of cohomogeneity‑one manifolds satisfy these spectral conditions, thereby guaranteeing existence and uniqueness.

Beyond harmonic maps, the paper also treats proper biharmonic maps, which are critical points of the bienergy functional E₂(φ)=½∫_M|τ(φ)|² dv_g. The Euler‑Lagrange equation τ₂(φ)=0 involves the bitension field τ₂ = Δ̄τ(φ)+Tr Rⁿ(τ(φ),dφ)dφ. Under the same equivariance assumption, τ₂=0 reduces to a fourth‑order ODE  r⁽⁴⁾(t) + h₁(t)·r⁽³⁾(t) + h₂(t, r, ṙ, r̈) = 0  (1.6). When the metric endomorphism P_t is diagonal (a technical hypothesis ensuring decoupling of the equations), the authors prove an analogue of Theorem 1.1 (Theorem 1.2): for any prescribed triple of initial data (v₁,v₂,v₃) there exists a unique smooth solution r with r(0)=0, ṙ(0)=v₁, r̈(0)=v₂, r⁽³⁾(0)=v₃, again depending continuously on the data.

Sections 2 and 3 provide the necessary background on cohomogeneity‑one manifolds, smoothness conditions for the invariant metric near a singular orbit, and explicit formulas for the tension and bitension fields of equivariant maps. The smoothness conditions are expressed in terms of the expansion of the metric components g_{ij}(t) along γ; the authors cite the work of Verdiani–Ziller and others to ensure that the metric can be written as a power series with prescribed parity, which in turn guarantees the regular‑singular structure of the ODEs.

Section 4 details the construction of the solution for the harmonic case: after transforming (1.3) into a first‑order system, the authors apply the regular‑singular theory to obtain a formal series, verify convergence using a contraction mapping argument on a weighted Banach space, and finally prove the continuous dependence on the initial velocity. Section 5 repeats the same strategy for the biharmonic case, noting the additional technicalities due to the higher order and the need for diagonal P_t.

Section 6 serves as a concise survey of regular‑singular first‑order systems, gathering scattered results from the literature (Coddington–Levinson, Wasow, etc.) and presenting them in a unified framework tailored to the geometric problems at hand.

In summary, the paper makes three main contributions: (1) it formulates the initial value problem for equivariant harmonic maps on cohomogeneity‑one manifolds as a regular‑singular ODE; (2) it proves local existence, uniqueness, and continuous dependence on initial data for both harmonic and proper biharmonic maps; (3) it provides a self‑contained exposition of regular‑singular system theory, filling a gap in the geometric analysis literature. These results open the way to construct new families of harmonic and biharmonic maps by prescribing initial data at singular orbits, and they lay a solid analytical foundation for further investigations of higher‑order variational problems on manifolds with symmetry.