F-biharmonic maps into general Riemannian manifolds

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].

Stability of Biharmonic Legendrian Submanifolds in Sasakian Space Forms

Biharmonic maps are defined as critical points of the bienergy. Every harmonic map is a stable biharmonic map. In this article, the stability of nonharmonic biharmonic Legendrian submanifolds in Sasakian space forms is discussed.

TRANSVERSALLY BIHARMONIC MAPS BETWEEN FOLIATED RIEMANNIAN MANIFOLDS

We generalize the notions of transversally harmonic maps between foliated Riemannian manifolds into transversally biharmonic maps. We show that a transversally biharmonic map into a foliated manifold of non-positive transverse curvature is transversally harmonic. Then we construct examples of transversally biharmonic non-harmonic maps into foliated manifolds of positive transverse curvature. We also prove that if f is a stable transversally biharmonic map into a foliated manifold of positive constant transverse sectional curvature and f satisfies the transverse conservation law, then f is a transversally harmonic map.