Skip to main content
placeholder image

Chen's conjecture and e-superbiharmonic submanifolds of Riemannian manifolds

Journal Article


Download full-text (Open Access)

Abstract


  • B.-Y. Chen famously conjectured that every submanifold of Eu-

    clidean space with harmonic mean curvature vector is minimal. In this note

    we prove a much more general statement for a large class of submanifolds sat-

    isfying a growth condition at innity. We discuss in particular two popular

    competing natural interpretations of the conjecture when the Euclidean back-

    ground space is replaced by an arbitrary Riemannian manifold. Introducing

    the notion of "-superbiharmonic submanifolds, which contains each of the pre-

    vious notions as special cases, we prove that "-superbiharmonic submanifolds

    of a complete Riemannian manifold which satisfy a growth condition at innity

    are minimal.

Publication Date


  • 2013

Geographic Focus


Citation


  • Wheeler, G. (2013). Chen''s conjecture and e-superbiharmonic submanifolds of Riemannian manifolds. International Journal of Mathematics, 24 (4), 1-4.

Scopus Eid


  • 2-s2.0-84877353281

Ro Full-text Url


  • http://ro.uow.edu.au/cgi/viewcontent.cgi?article=1579&context=eispapers

Ro Metadata Url


  • http://ro.uow.edu.au/eispapers/573

Has Global Citation Frequency


Number Of Pages


  • 3

Start Page


  • 1

End Page


  • 4

Volume


  • 24

Issue


  • 4

Place Of Publication


  • Singapore

Abstract


  • B.-Y. Chen famously conjectured that every submanifold of Eu-

    clidean space with harmonic mean curvature vector is minimal. In this note

    we prove a much more general statement for a large class of submanifolds sat-

    isfying a growth condition at innity. We discuss in particular two popular

    competing natural interpretations of the conjecture when the Euclidean back-

    ground space is replaced by an arbitrary Riemannian manifold. Introducing

    the notion of "-superbiharmonic submanifolds, which contains each of the pre-

    vious notions as special cases, we prove that "-superbiharmonic submanifolds

    of a complete Riemannian manifold which satisfy a growth condition at innity

    are minimal.

Publication Date


  • 2013

Geographic Focus


Citation


  • Wheeler, G. (2013). Chen''s conjecture and e-superbiharmonic submanifolds of Riemannian manifolds. International Journal of Mathematics, 24 (4), 1-4.

Scopus Eid


  • 2-s2.0-84877353281

Ro Full-text Url


  • http://ro.uow.edu.au/cgi/viewcontent.cgi?article=1579&context=eispapers

Ro Metadata Url


  • http://ro.uow.edu.au/eispapers/573

Has Global Citation Frequency


Number Of Pages


  • 3

Start Page


  • 1

End Page


  • 4

Volume


  • 24

Issue


  • 4

Place Of Publication


  • Singapore