Biharmonic Submanifolds in a Riemannian Manifold with Non-Positive Curvature

By Nobumitsu Nakauchi, Hajime Urakawa. Results in Mathematics, 2011

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In this paper, we show that, for every biharmonic submanifold (M, g) of a Riemannian manifold (N, h) with non-positive sectional curvature, if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\int_M\vert \eta \vert^2 v_g < \infty}$$\end{document} , then (M, g) is minimal in (N, h), i.e., \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\eta\equiv0}$$\end{document}, where η is the mean curvature tensor field of (M, g) in (N, h). This result gives an affirmative answer under the condition \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\int_M\vert \eta \vert^2 v_g < \infty}$$\end{document} to the following generalized Chen’s conjecture: every biharmonic submanifold of a Riemannian manifold with non-positive sectional curvature must be minimal. The conjecture turned out false in case of an incomplete Riemannian manifold (M, g) by a counter example of Ou and Tang (in The generalized Chen’s conjecture on biharmonic sub-manifolds is false, a preprint, 2010).

Subfields: Applied Mathematics, Astronomy and Astrophysics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/ec9212cc0d07852f0dbe1e59fc2592c1ad0479d0

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In Applied Mathematics

In Astronomy and Astrophysics

In Geometry and Topology

Citations in the corpus

Biharmonic properly immersed submanifolds in Euclidean spaces (2011, background): “submanifolds in Conjecture 1, in a sense it is a problem in local diﬀerential geometry. In this article, we reformulate Conjecture 1 into a problem in globaldiﬀerential geometry as the following (cf. [8, 9]): Conjecture2. Anycomplete biharmonicimmersedsubmanifoldinEN isminimal. Date: June, 2011. ∗ s”
Some open problems and conjectures on submanifolds of finite type: recent development (2014, background): “• Biharmonic submanifolds with finite total mean curvature in a Riemannian manifold of non-positive sectional curvature [148].”
Biharmonic maps into a Riemannian manifold of non-positive curvature (2012, methodology): “ﬁnite. For example, f(x) = r(x)2 = Pm i=1(xi) 2,x = (x1,··· ,xm) ∈ Rm is biharmonic, but not harmonic, and have inﬁnite energy and bienergy. As the ﬁrst bi-product of our method, we obtain (cf. [22], [23]) Theorem 1.2. (cf. Theorem 3.1) Assume that (M,g) is a complete Riemannian manifold, and let ϕ ”