Article

# 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

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- Bollettino dell Unione Matematica Italiana 01/1974; 10:380-385.
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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.manuscripta mathematica 01/1993; 81(1):203-222. · 0.50 Impact Factor -
##### Article: L p Willmore functionals

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**ABSTRACT:**For immersed tori M⊂S 3 in the (metric) 3-sphere, the author introduces the family of functionals W p (M)=∫ M |κ 1 -κ 2 2| p dM, p∈ℝ, as a generalization of the Willmore functional, W p=2 . Here the κ i denote the principal curvatures of M. Two inequalities for these functions are given: Theorem 1.1. For p≥1+π 2, one has that W p ≥2π 2 , and equality holds if and only if M is the Clifford torus. Theorem 1.2. W 2 ≥∫ M [(1-K - ) 1 2+π 4 -(1-K - )]≥2π 2 , and equality holds if and only if M is the Clifford torus, where K - =min{K,0} and K is the Gauss curvature of M.Proceedings of the American Mathematical Society 01/1999; 127(2). · 0.63 Impact Factor

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