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



Minimal Legendrian submanifolds and laplacian of particular functions
I think this is just for the sphere. Look at Theorem 3 in [this paper](http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.jmsj/1260480465) by Takahashi.


