Triharmonic isometric immersions into a manifold of non-positively constant curvature

Abstract

A triharmonic map is a critical point of the 3-energy in the space of smooth maps between two Riemannian manifold. We study the generalized Chen's conjecture for a triharmonic isometric immersion ϕ into a space form of non-positively constant curvature. We show that if the domain is complete and both the 4-energy of ϕ, and the L 4 -norm of the tension field τ (ϕ), are finite, then such an immersion ϕ is minimal.2000 Mathematics Subject Classification. primary 58E20, secondary 53C43.

Unique continuation properties for polyharmonic maps between Riemannian manifolds

“…Moreover, for any k ≥ 2, proper k-harmonic immersions into spheres, ellipsoids and rotation hypersurfaces, which are not k -harmonic for any k = k, were constructed in [20,23,24]. Other results on triharmonic maps were achieved in [17,19]. The Euler-Lagrange equations of (1.5), (1.6) were calculated in [14,16] and can be described as follows (note that we set∆ −1 = 0): (1) The critical points of (1.5) (1.8)…”

Triharmonic CMC hypersurfaces in space forms with 4 distinct principal curvatures

“…In this paper, we focus on the case k = 3, which is a particularly active subject recently (cf. [12,[14][15][16][17]25]). For instance, Maeta-Nakauchi-Urakawa [14] proved that a triharmonic isometric immersion into a Riemannian manifold of non-positive curvature must be minimal under certain suitable conditions.…”

“…[12,[14][15][16][17]25]). For instance, Maeta-Nakauchi-Urakawa [14] proved that a triharmonic isometric immersion into a Riemannian manifold of non-positive curvature must be minimal under certain suitable conditions. Maeta proved that any compact constant mean curvature (CMC in short) triharmonic hypersurface M n in R n+1 (c)(c ≤ 0) is minimal (see [10,Proposition 4.3]), and Montaldo-Oniciuc-Ratto removed the compactness assumption for n = 2 very recently since there are at most two distinct principal curvatures for a surface M 2 (see [15,Theorem 1.3]).…”

Triharmonic CMC hypersurfaces in space forms with at most 3 distinct principal curvatures

“…Nowadays, the study of triharmonic maps is a particularly active subject (cf. [21,[23][24][25][26]35]). In 2015, Maeta-Nakauchi-Urakawa [23] proved that under suitable conditions (e.g.…”