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# F-Willmore submanifold in space forms

Frontiers of Mathematics in China (Impact Factor: 0.45). 10/2011; 6(5):871-886. DOI: 10.1007/s11464-011-0140-y

ABSTRACT We introduce an F-Willmore functional of submanifold in space forms, which generalizes the well-known Willmore functional. Its critical point
is called the F-Willmore submanifold, for which the variational equation and Simons’ type integral inequality are obtained.

KeywordsMean curvature–Willmore submanifold–Simons’ type integral inequality

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ABSTRACT: We study Willmore immersed submanifoldsf: M m →S n into then-Möbius space, withm≥2, as critical points of a conformally invariant functionalW. We compute the Euler-Lagrange equation and relate this functional with another one applied to the conformal Gauss map of immersions intoS n . We solve a Bernestein-type problem for compact Willmore hypersurfaces ofS n , namely, if ∃a ∈ℝ n+2 such that <γf, a > ≠ 0 onM, whereγ f is the hyperbolic conformal Gauss map and <, > is the Lorentz inner product ofℝ n+2, and iff satisfies an additional condition, thenf(M) is an (n−1)-sphere.
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