Abstract
In [17] the third author presented Moebius geometry for sub-manifolds in Sn and calculated the first variational formula of the Willmore functional by using Moebius invariants. In this paper we present the second variational formula for Willmore submanifolds. As an application of these variational formulas we give the standard examples of Willmore hypersurfaces \( \lbrace W_{k}^{m}:= S^{k}(\sqrt {(m-k)/m}) \times S^{m-k}(\sqrt {k/m}), 1 \leq k \leq m-1 \rbrace \) in Sm+1 (which can be obtained by exchanging radii in the Clifford tori \( S^{k}(\sqrt {k/m}) \times S^{m-k}(\sqrt {(m-k)/m)})\) and show that they are stable Willmore hypersurfaces. In case of surfaces in S3, the stability of the Clifford torus \( S^{1}{({1\over \sqrt {2}})}\times S^{1}{({1\over \sqrt {2}})} \) was proved by J. L. Weiner in [18]. We give also some examples of m-dimensional Willmore submanifolds in an n-dimensional unit sphere Sn.
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Dedicated to Professor S. S. Chern at the occasion of his 90th birthday
The second author is partially supported by the Alexander von Humboldt Research Fellowship and the US-China Cooperative Research NSF Grant INT-9906856 and the third author is partially supported by 973-project, DFG466-CHV-II3/127/0, Qiushi Award and RFDP.
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Guo, Z., Wang, C. & Li, H. The second variational formula for Willmore submanifolds in Sn . Results. Math. 40, 205–225 (2001). https://doi.org/10.1007/BF03322706
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DOI: https://doi.org/10.1007/BF03322706