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I'm reading the paper

Lê, Hôngvân(D-MPI-NS); Wang, Guofang(PRC-ASBJ-MSY) A characterization of minimal Legendrian submanifolds in $S^{2n+1}$. Compositio Math. 129 (2001), no. 1, 87–93.

Let $x: L^n \rightarrow S^{2n+2}$ be a minimal embedded submanifold and for fixed $a \in \mathbb R^{2n+2}$ consider the function on $L$ given by $f(x) = \langle x, a \rangle$ on $L$, where $\langle, \rangle$ is the standard scalar product on $\mathbb R^{2n+2}$.

The authors say that minimality implies $\Delta_L f = nf$ where $\Delta_L$ is the Laplacian of the induced metric on $L$.

About that they cite the book

Chern, S. S.: Minimal Submanifolds in a Riemannian Manifolds, University of Kansas, Lawrence, 1968.

to which unfortunately I have no access so I cannot find an aswer to the following:

Question: Is this result specific for minimal submanifolds in the sphere or is it more general holding, for instance, for minimal Legendrian submanifolds in (homogeneous) contact manifolds and taking as $f$ some other particular function?

Thank you

David

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I think this is just for the sphere. Look at Theorem 3 in this paper by Takahashi.

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  • $\begingroup$ Thank you! Anyway is anyone aware of more general results about the Laplacian on minimal (Legendrian) submanifolds of (contact) manifolds? $\endgroup$
    – David P
    Commented Aug 5, 2013 at 9:53
  • $\begingroup$ Not me, but maybe the experts in submanifold theory know about this (B. Chen, L. Verstraelen, F. Dillen, L. Vrancken, ...). $\endgroup$
    – alvarezpaiva
    Commented Aug 5, 2013 at 10:08

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