Two Results for Symphonic Maps Under Assumptions on m-Symphonic Energy

Abstract

We consider the $\mathrm {L}^2$-energy functional $E_{sym}(f)$ of pullbacks of metrics by smooth maps f between Riemannian manifolds. Stationary maps for $E_{sym}(f)$ are called symphonic maps and researched in Kawai and Nakauchi (Nonlinear Anal 74:2284–2295, 2011; Differ Geom Appl 44:161–177, 2016; Differ Geom Appl 65:147–159, 2019), Misawa and Nakauchi (Nonlinear Anal 75:5971–5974, 2012; Calc Var Partial Differ Equ 55:1–20, 2016; Adv Differ Equ 23:693–724, 2018; Part Differ Equ Appl 2:19, 2021) and Nakauchi and Takenaka (Ricerche Matematica 60:219–235, 2011). In this paper we are concerned with the m-symphonic energy $E_{sym}^m(f)$, i.e., the $\mathrm {L}^{\frac{m}{2}}$-version of the symphonic energy, where m denotes the dimension of the source manifold M. The m-symphonic energy is conformally invariant and is introduced in Misawa and Nakauchi (Part Differ Equ Appl 2:19, 2021). Under some conditions on this conformal energy $E_{sym}^m(f)$, we give two results—a gap theorem and a Liouville type theorem—for symphonic maps.