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Chen's conjecture and epsilon-superbiharmonic submanifolds of Riemannian manifolds | | Glen Wheeler
; | | Date: |
23 May 2013 | | Abstract: | B.-Y. Chen famously conjectured that every submanifold of Euclidean space
with harmonic mean curvature vector is minimal. In this note we establish a
much more general statement for a large class of submanifolds satisfying a
growth condition at infinity. We discuss in particular two popular competing
natural interpretations of the conjecture when the Euclidean background space
is replaced by an arbitrary Riemannian manifold. Introducing the notion of
epsilon-superbiharmonic submanifolds, which contains each of the previous
notions as special cases, we prove that epsilon-superbiharmonic submanifolds of
a complete Riemannian manifold which satisfy a growth condition at infinity are
minimal. | | Source: | arXiv, 1305.5294 | | Services: | Forum | Review | PDF | Favorites | |
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