On Interpolating Sesqui-Harmonic Maps Between Riemannian Manifolds

Abstract

Motivated from the action functional for bosonic strings with extrinsic curvature term we introduce an action functional for maps between Riemannian manifolds that interpolates between the actions for harmonic and biharmonic maps. Critical points of this functional will be called interpolating sesqui-harmonic maps. In this article we initiate a rigorous mathematical treatment of this functional and study various basic aspects of its critical points.

Some analytic results on interpolating sesqui-harmonic maps

“…In this article, we want to focus on the critical points of an energy functional that interpolates between the energy functionals for harmonic and biharmonic maps which is given by with . Various versions of this functional had already been studied by a number of mathematicians, a general study of ( 1.3 ) was recently initiated by the author [ 5 ].…”

On interpolating sesqui-harmonic Legendre curves in Sasakian space forms

“…The Euler-Lagrange equation of E 2 (ϕ) is τ 2 (ϕ) = tr(∇ ϕ ∇ ϕ − ∇ ϕ ∇ )τ (ϕ) − tr(R N (dϕ, τ (ϕ))dϕ) = 0, (1.2) and it is called the bitension field of ϕ [11]. In [3], Branding defined and considered interpolating sesqui-harmonic maps between Riemannian manifolds. The author introduced an action functional for maps between Riemannian manifolds that interpolated between the actions for harmonic and biharmonic maps.…”

“…where Ω is a compact domain of M and δ 1 , δ 2 ∈ R [3]. The interpolating sesquiharmonic map equation is τ δ 1 ,δ 2 (ϕ) = δ 2 τ 2 (ϕ) − δ 1 τ (ϕ) = 0 (1.4) for δ 1 , δ 2 ∈ R [3].…”

“…where Ω is a compact domain of M and δ 1 , δ 2 ∈ R [3]. The interpolating sesquiharmonic map equation is τ δ 1 ,δ 2 (ϕ) = δ 2 τ 2 (ϕ) − δ 1 τ (ϕ) = 0 (1.4) for δ 1 , δ 2 ∈ R [3]. An interpolating sesqui-harmonic map is biminimal if variations of (1.3) that are normal to the image ϕ(M) ⊂ N and δ 2 = 1, δ 1 > 0 [13].…”

Unique continuation theorems for biharmonic maps

“…Then, in contrast to the article at hand, the results follow directly from the classical results of Sampson [, Theorems 1,6] for harmonic maps. Here, our results are more general and the technique we are using is completely different. (2)Theorem was first proved in by simply applying [, Proposition 1.2.3], but here, the proof is more clear and based on the classical result from Aronszajn . (3)In [, Theorem 5.3], a unique continuation result for extrinsic biharmonic maps from to was proved. (4)Theorems – also hold for extrinsic biharmonic maps and interpolating sesqui‐harmonic maps, which were introduced in .…”