Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 16 (2020), 090, 8 pages      arXiv:2002.03698
Contribution to the Special Issue on Scalar and Ricci Curvature in honor of Misha Gromov on his 75th Birthday

About Bounds for Eigenvalues of the Laplacian with Density

Aïssatou Mossèle Ndiaye
Institut de Mathématiques, Université de Neuchâtel, Switzerland

Received February 13, 2020, in final form September 01, 2020; Published online September 25, 2020

Let $M$ denote a compact, connected Riemannian manifold of dimension $n\in{\mathbb N}$. We assume that $ M$ has a smooth and connected boundary. Denote by $g$ and ${\rm d}v_g$ respectively, the Riemannian metric on $M$ and the associated volume element. Let $\Delta$ be the Laplace operator on $M$ equipped with the weighted volume form ${\rm d}m:= {\rm e}^{-h}\,{\rm d}v_g$. We are interested in the operator $L_h\cdot:={\rm e}^{-h(\alpha-1)} (\Delta\cdot +\alpha g(\nabla h,\nabla\cdot))$, where $\alpha > 1$ and $h\in C^2(M)$ are given. The main result in this paper states about the existence of upper bounds for the eigenvalues of the weighted Laplacian $L_h$ with the Neumann boundary condition if the boundary is non-empty.

Key words:eigenvalue; Laplacian; density; Cheeger inequality; upper bounds.

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