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

by Wheeler, G..

Series: 2012-09, Preprints

53C42 Immersions (minimal, prescribed curvature, tight, etc.)
53A07 Higher-dimensional and -codimensional surfaces in Euclidean $n$-space
35J60 Nonlinear elliptic equations

B.-Y. Chen famously conjectured that every submanifold of Euclidean space with harmonic mean curvature vector is minimal. In this note we prove 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 ε-superbiharmonic submanifolds, which contains each of the previous notions as special cases, we prove that ε-superbiharmonic submanifolds of a complete Riemannian manifold which satisfy a growth conditions at infinity are minimal.

Chen's conjecture, biharmonic submanifolds, geometric analysis, higher order system of quasilinear elliptic partial differential equations