Biharmonic maps from a complete Riemannian manifold into a non-positively curved manifold

Maeta, Shun

doi:10.1007/s10455-014-9410-8

Cited by 29 publications

(22 citation statements)

References 17 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Nakauchi et al [16], Maeta [14] and Luo [12] applied their nonexistence result for biharmonic maps to get conditions for which biharmonic submersions are harmonic morphisms. Here, we give another such result by using Theorem 1.2.…”

Section: )mentioning

confidence: 99%

A nonexistence theorem for proper biharmonic maps into general Riemannian manifolds

Branding

Luo

2020

Journal of Geometry and Physics

View full text Add to dashboard Cite

In this note we prove a nonexistence result for proper biharmonic maps from complete non-compact Riemannian manifolds of dimension m = dim M ≥ 3 with infinite volume that admit an Euclidean type Sobolev inequality into general Riemannian manifolds by assuming finiteness of τ (φ) L p (M) , p > 1 and smallness of dφ L m (M) . This is an improvement of a recent result of the first named author, where he assumed 2 < p < m. As applications we also get several nonexistence results for proper biharmonic submersions from complete non-compact manifolds into general Riemannian manifolds.

show abstract

Section: )mentioning

confidence: 99%

A nonexistence theorem for proper biharmonic maps into general Riemannian manifolds

Branding

Luo

2020

Journal of Geometry and Physics

View full text Add to dashboard Cite

show abstract

“…(27) We give an affirmative partial answer to BMO conjecture. By using (27), we show as follows. Therefore by the assumption, we have e j H = 0, at an arbitrary point x ∈ M and for all j = 1, · · · , m. So we have that H is constant at an arbitrary point x ∈ M. Therefore H is constant on M. Since H = 0, we have |A| 2 = cm − λ.…”

Section: Mh In Space Formsmentioning

confidence: 99%

Biharmonic submanifolds in manifolds with bounded curvature

Maeta

2016

Int. J. Math.

Self Cite

View full text Add to dashboard Cite

Abstract. We consider a complete biharmonic submanifold φ : (M, g) → (N, h) in a Riemannian manifold with sectional curvature bounded from above by a non-negative constant c. Assume that the mean curvature is bounded from below byfor some 0 < p < ∞, or (ii) the Ricci curvature of M is bounded from below, then the mean curvature is √ c. Furthermore, if M is compact, then we obtain the same result without the assumption (i) or (ii). IntroductionIn 1983, J. Eells and L. Lemaire [16] proposed the problem to consider biharmonic maps. Biharmonic maps are, by definition, a generalization of harmonic maps. As well known, harmonic maps have been applied into various fields in differential geometry. In 1964, J. Eells and J. H. Sampson considered the existence problem of harmonic maps between compact Riemannian manifolds. They showed that any continuous map from a compact Riemannian manifold into a compact Riemannian manifold of non-positive curvature is free homotopically deformable to harmonic maps. By using the existence and properties of harmonic maps, one can study the structure of Riemannian manifolds. On the other hand, non-existence results for harmonic maps are also known. For example, a map of degree ±1 from a 2-dimensional torus into a 2-dimensional sphere is not homotopic to any harmonic map. Therefore a generalization of harmonic maps seems an important subject instead. So far, it seems a biharmonic map. We would 2010 Mathematics Subject Classification. primary 58E20, secondary 53C43.

show abstract

“…Recently, N. Nakauchi, H. Urakawa and S. Gudmundsson [26] proved that biharmoic maps from a complete Riemannian manifold into a non-positive curved manifold with finite bienergy and energy are harmonic. S. Maeta [25] proved that biharmoic maps from a complete Riemannian manifold into a non-positive curved manifold with finite (a + 2)-bienergy M |τ (u)| a+2 dv g < ∞ (a ≥ 0) and energy are harmonic. The author and W. Zhang in [16] proved that pbiharmoinc maps from a complete manifold into a non-positive curved manifold with finite a + p-bienergy M |τ (u)| a+p dv g < ∞ and energy are harmonic.…”

Section: Introductionmentioning

confidence: 99%

Some Results of Exponentially Biharmonic Maps Into a Non-Positively Curved Manifold

Han¹

2016

Bulletin of the Korean Mathematical Society

View full text Add to dashboard Cite

Abstract. In this paper, we investigate exponentially biharmonic maps u : (M, g) → (N, h) from a Riemannian manifold into a Riemannian manifold with non-positive sectional curvature. We obtain that ifthen u is harmonic. When u is an isometric immersion, we get that|H| q dvg < ∞ for 2 ≤ p < ∞ and 0 < q ≤ p < ∞, then u is minimal. We also obtain that any weakly convex exponentially biharmonic hypersurface in space form N (c) with c ≤ 0 is minimal. These results give affirmative partial answer to conjecture 3 (generalized Chen's conjecture for exponentially biharmonic submanifolds).

show abstract

Biharmonic maps from a complete Riemannian manifold into a non-positively curved manifold

Cited by 29 publications

References 17 publications

A nonexistence theorem for proper biharmonic maps into general Riemannian manifolds

A nonexistence theorem for proper biharmonic maps into general Riemannian manifolds

Biharmonic submanifolds in manifolds with bounded curvature

Some Results of Exponentially Biharmonic Maps Into a Non-Positively Curved Manifold

Contact Info

Product

Resources

About