Remark on the paper "Eigenvalues of the Witten-Laplacian on compact Riemannian manifolds" Q.-M. Cheng and L. Zeng [arXiv:1304.3212]
Why the results of that article [arXiv:1304.3212] are immediate consequences of known ones.
math.DG↗
arXiv subjects
Saïd Ilias
Publications and source records attributed to Saïd Ilias.
Domain deformations and eigenvalues of the Dirichlet Laplacian in a Riemannian manifold
For any bounded regular domain $Ω$ of a real analytic Riemannian manifold $M$, we denote by $λ_{k}(Ω)$ the $k$-th eigenvalue of the Dirichlet Laplacian of $Ω$. In this paper, we consider $λ_k$ and as a functional upon the set of domains of fixed volume in $M$. We introduce and investigate a natural notion of critical domain for this functional. In particular, we obtain necessary and sufficient conditions for a domain to be critical, locally minimizing or locally maximizing for $λ_k$. These results rely on Hadamard type variational formulae that we establish in this general setting.
math.MG↗