F-biharmonic maps into general Riemannian manifolds

Ронг Ми

doi:10.1515/math-2019-0112

Abstract

Abstract Let ψ:(M, g) → (N, h) be a map between Riemannian manifolds (M, g) and (N, h). We introduce the notion of the F-bienergy functional $$\begin{array}{} \displaystyle E_{F,2}(\psi)=\int\limits_{M}F\left(\frac{|\tau(\psi)|^{2}}{2}\right)\text{d}V_{g}, \end{array}$$ where F : [0, ∞) → [0, ∞) be C3 function such that F′ > 0 on (0, ∞), τ(ψ) is the tension field of ψ. Critical points of τF,2 are called F-biharmonic maps. In this paper, we prove a nonexistence result for F-biharmonic maps from a complete non-compact Riemannian manifold of dimension m = dimM ≥ 3 with infinite volume that admit an Euclidean type Sobolev inequality into general Riemannian manifold whose sectional curvature is bounded from above. Under these geometric assumptions we show that if the Lp-norm (p > 1) of the tension field is bounded and the m-energy of the maps is sufficiently small, then every F-biharmonic map must be harmonic. We also get a Liouville-type result under proper integral conditions which generalize the result of [Branding V., Luo Y., A nonexistence theorem for proper biharmonic maps into general Riemannian manifolds, 2018, arXiv: 1806.11441v2].

Highlights

We prove a nonexistence result for Fbiharmonic maps from a complete non-compact Riemannian manifold of dimension m = dimM ≥ with in nite volume that admit an Euclidean type Sobolev inequality into general Riemannian manifold whose sectional curvature is bounded from above
Under these geometric assumptions we show that if the Lp-norm (p > ) of the tension eld is bounded and the m-energy of the maps is su ciently small, every Fbiharmonic map must be harmonic
We get a Liouville-type result under proper integral conditions which generalize the result of [Branding V., Luo Y., A nonexistence theorem for proper biharmonic maps into general Riemannian manifolds, 2018, arXiv: 1806.11441v2]

Summary

Introduction

In the past several decades harmonic map plays a central role in geometry and analysis. In this case we can use the integration by parts argument, by choosing proper test functions Based on this idea, Baird, Fardoun and Ouakkas ([30]) showed that: If a non-compact manifold (M, g) is complete and has non-negative Ricci curvature and (N, h) has non-positive sectional curvature, every bienergy nite biharmonic map of (M, g) into (N, h) is harmonic. Branding and Luo [26] proved a nonexistence result for proper biharmonic maps from complete non-compact Riemannian manifolds, by assuming that the sectional curvature of the Riemannian manifold have an upper bound and give a more natural integrability condition, which generalized Branding’s. Suppose that (M, g) is a complete, connected non-compact Riemannian manifold of m = dimM ≥ with nonnegative Ricci curvature that admits an Euclidean type Sobolev inequality of the form (1.1).

Proof of the main result

Then we have