scholarly journals Biharmonic submanifolds of Kaehler product manifolds

2021 ◽  
Vol 6 (9) ◽  
pp. 9309-9321
Author(s):  
Yanlin Li ◽  
◽  
Mehraj Ahmad Lone ◽  
Umair Ali Wani ◽  
Keyword(s):  
Biharmonic Submanifolds

scholarly journals Chen’s Biharmonic Conjecture and Submanifolds with Parallel Normalized Mean Curvature Vector

Mathematics ◽  
2019 ◽  
Vol 7 (8) ◽  
pp. 710 ◽  
Author(s):  
Bang-Yen Chen
Keyword(s):  
Euclidean Space ◽  
Mean Curvature ◽  
Curvature Vector ◽  
Mean Curvature Vector ◽  
Biharmonic Submanifolds ◽  
Euclidean Spaces ◽  
Principal Curvatures ◽  
Chen’S Conjecture ◽  
Biharmonic Submanifold

The well known Chen’s conjecture on biharmonic submanifolds in Euclidean spaces states that every biharmonic submanifold in a Euclidean space is a minimal one. For hypersurfaces, we know from Chen and Jiang that the conjecture is true for biharmonic surfaces in E 3 . Also, Hasanis and Vlachos proved that biharmonic hypersurfaces in E 4 ; and Dimitric proved that biharmonic hypersurfaces in E m with at most two distinct principal curvatures. Chen and Munteanu showed that the conjecture is true for δ ( 2 ) -ideal and δ ( 3 ) -ideal hypersurfaces in E m . Further, Fu proved that the conjecture is true for hypersurfaces with three distinct principal curvatures in E m with arbitrary m. In this article, we provide another solution to the conjecture, namely, we prove that biharmonic surfaces do not exist in any Euclidean space with parallel normalized mean curvature vectors.

scholarly journals Biharmonic submanifolds of $${\mathbb{C}P^n}$$

2009 ◽  
Vol 266 (3) ◽  
pp. 505-531 ◽  
Author(s):  
D. Fetcu ◽  
E. Loubeau ◽  
S. Montaldo ◽  
C. Oniciuc
Keyword(s):  
Biharmonic Submanifolds

scholarly journals Biharmonic submanifolds in manifolds with bounded curvature

2016 ◽  
Vol 27 (11) ◽  
pp. 1650089
Author(s):  
Shun Maeta
Keyword(s):  
Riemannian Manifold ◽  
Sectional Curvature ◽  
Ricci Curvature ◽  
Mean Curvature ◽  
Biharmonic Submanifolds ◽  
Bounded Curvature ◽  
Negative Constant ◽  
The Mean ◽  
Biharmonic Submanifold

We consider a complete biharmonic submanifold [Formula: see text] in a Riemannian manifold with sectional curvature bounded from above by a non-negative constant [Formula: see text]. Assume that the mean curvature is bounded from below by [Formula: see text]. If (i) [Formula: see text], for some [Formula: see text], or (ii) the Ricci curvature of [Formula: see text] is bounded from below, then the mean curvature is [Formula: see text]. Furthermore, if [Formula: see text] is compact, then we obtain the same result without the assumption (i) or (ii). These are affirmative partial answers to Balmuş–Montaldo–Oniciuc conjecture.

Proper biharmonic submanifolds in a sphere

2011 ◽  
Vol 28 (1) ◽  
pp. 205-218 ◽  
Author(s):  
Xian Feng Wang ◽  
Lan Wu
Keyword(s):  
Biharmonic Submanifolds

Biharmonic submanifolds in metallic Riemannian manifolds

2021 ◽  
Author(s):  
Mohamd Saleem Lone ◽  
Siraj Uddin ◽  
Mohammad Hasan Shahid
Keyword(s):  
Riemannian Manifolds ◽  
Mean Curvature ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Biharmonic Submanifolds ◽  
Complex Surfaces ◽  
Metallic Structures ◽  
Necessary And Sufficient

In this paper, we study the biharmonic submanifolds of Riemannian manifolds endowed with metallic and complex metallic structures. In case of both the structures, we obtain the necessary and sufficient conditions for a submanifold to be biharmonic. Particularly, we find the estimates for mean curvature of Lagrangian and complex surfaces.

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