Equivariant Harmonic Cylinders

Abstract

We prove that a primitive harmonic map is equivariant if and only if it admits a holomorphic potential of degree one. We investigate when the equivariant harmonic map is periodic, and as an application discuss constant mean curvature cylinders with screw motion symmetries.

A New Look at Equivariant Minimal Lagrangian Surfaces in $${\mathbb {C}} P^2$$ C P 2

“…Hence there exists a real function η ∈ C ∞ (U) locally such that idη = dF ·F . ThenF = e −iη F is a local horizontal lift of f to S 5 (1). We can therefore assume F z · F = Fz · F = 0.…”

“…In this section we assume that the frame is chosen as in Theorem 4.3. Then, following Burstall-Kilian ( [1]) and setting t = −x and z = x + it, we derive from (14),…”

Note on equivariant harmonic maps in complex projective spaces

“…Recently in [3,Theorem 3.4], it was shown that all equivariant primitive harmonic maps in k-symmetric spaces G/K (with an order k-automorphism τ ) are constructed from degree-one potentials:…”

“…Conversely, all extended framings of equivariant primitive harmonic maps are obtained in this way [3,Theorem 3.4]. We note if there exists a primitive lift ψ into a k-symmetric space G/K of an equivariant primitive harmonic map φ into ã k-symmetric space G/K such thatk ≤ k and K ⊂K , then ψ must be equivariant.…”

“…Using this criterion we give examples of equivariant harmonic cylinders-Example 1. Moreover, we give examples of non-equivariant harmonic cylinders as perturbations of equivariant harmonic cylinders-Example 2.For basic notions and results of primitive, equivariant and (non-)isotropic harmonic maps we refer to the readers[2,3,6]. Equivariant primitive harmonic maps in F r (CP n )2.1 Equivariant primitive harmonic maps…”