SIGMA 16 (2020), 090, 8 pages      arXiv:2002.03698      https://doi.org/10.3842/SIGMA.2020.090
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

Abstract
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.

pdf (336 kb)   tex (13 kb)  

References

1. Berger M., A panoramic view of Riemannian geometry, Springer-Verlag, Berlin, 2003.
2. Blacker C., Seto S., First eigenvalue of the $p$-Laplacian on Kähler manifolds, Proc. Amer. Math. Soc. 147 (2019), 2197-2206, arXiv:1804.10876.
3. Bochner S., Vector fields and Ricci curvature, Bull. Amer. Math. Soc. 52 (1946), 776-797.
4. Colbois B., El Soufi A., Spectrum of the Laplacian with weights, Ann. Global Anal. Geom. 55 (2019), 149-180, arXiv:1606.04095.
5. Colbois B., El Soufi A., Savo A., Eigenvalues of the Laplacian on a compact manifold with density, Comm. Anal. Geom. 23 (2015), 639-670, arXiv:1310.1490.
6. Du F., Bezerra A.C., Estimates for eigenvalues of a system of elliptic equations with drift and of bi-drifting Laplacian, Commun. Pure Appl. Anal. 16 (2017), 475-491.
7. Du F., Mao J., Wang Q., Wu C., Eigenvalue inequalities for the buckling problem of the drifting Laplacian on Ricci solitons, J. Differential Equations 260 (2016), 5533-5564.
8. Helffer B., Nier F., Hypoelliptic estimates and spectral theory for Fokker-Planck operators and Witten Laplacians, Lecture Notes in Math., Vol. 1862, Springer-Verlag, Berlin, 2005.
9. Huang G., Zhang C., Zhang J., Liouville-type theorem for the drifting Laplacian operator, Arch. Math. (Basel) 96 (2011), 379-385.
10. Kouzayha S., Pétiard L., Eigenvalues of the Laplacian with density, arXiv:1908.05051.
11. Li X.-D., Perelman's entropy formula for the Witten Laplacian on Riemannian manifolds via Bakry-Emery Ricci curvature, Math. Ann. 353 (2012), 403-437.
12. Lu Z., Rowlett J., Eigenvalues of collapsing domains and drift Laplacians, Math. Res. Lett. 19 (2012), 627-648, arXiv:1003.0191.
13. Ma L., Liu B., Convexity of the first eigenfunction of the drifting Laplacian operator and its applications, New York J. Math. 14 (2008), 393-401.
14. Xia C., Xu H., Inequalities for eigenvalues of the drifting Laplacian on Riemannian manifolds, Ann. Global Anal. Geom. 45 (2014), 155-166.