Second variation of compact minimal Legendrian submanifolds of the sphere
The author computes the Jacobi operator \(L\) of a compact minimal Legendrian submanifold \(M\) of \({\mathbb S}^{2n+1}\) to prove that it is an intrinsic operator on the submanifold and that it can be written in terms of the exterior differential operator \(d,\) its codifferential operator \(\delta\) and the Laplacian \(\triangle\) as \(L(\alpha, f)=(\triangle \alpha+2(n-1)\alpha+2df, \triangle f-2\delta\alpha)\) where \(\alpha\) is an 1-form and \(f\) is a smooth function. He also obtains a formula for the index \(\text{Ind}(M)\) and the nullity \(\text{Null}(M)\) of \(L\) and in particular, he proves the following results for an orientable, compact, non-totally geodesic minimal Legendrian surface in \({\mathbb S}^5\): (1) \(\text{Ind}(M)\geq8\) and the equality holds if and only if \(M\) is an equilateral torus. (2) \(\text{Null}(M)\geq13\) and the equality holds if and only if \(M\) is an equilateral torus.
scientific article; zbMATH DE number 1997322
scientific article