A sharp upper bound for the first eigenvalue of the Laplacian of compact hypersurfaces in rank-1 symmetric spaces

G. Santhanam

Home/Archive/arXiv

arXiv · 0709.3349

A sharp upper bound for the first eigenvalue of the Laplacian of compact hypersurfaces in rank-1 symmetric spaces

G. Santhanam

View original source ↗Read source document ↗Export metadata ↓

Abstract

Let $M$ be a closed hypersurface in a simply connected rank-1 symmetric space $\olm$. In this paper, we give an upper bound for the first eigenvalue of the Laplacian of $M$ in terms of the Ricci curvature of $\olm$ and the square of the length of the second fundamental form of the geodesic spheres with center at the center-of-mass of $M$.

Explore related subjects

math.DG math.MG

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

G. Santhanam. 2007-09-21. A sharp upper bound for the first eigenvalue of the Laplacian of compact hypersurfaces in rank-1 symmetric spaces. https://arxiv.org/abs/0709.3349

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Upper bound preservation of the total scalar curvature in a conformal class

We show that in an arbitrarily fixed conformal class with non-positive Yamabe constant on a closed manifold, the upper bound of the total scalar curvature is preserved under the $C^{0}$-convergence of metrics provided that the convergent sequence has a uniform Hölder bound. Moreover, if we consider the condition that the scalar curvature is bounded by some fixed continuous function from below in addition to the upper bound of the total scalar curvature, then such a condition is $C^{0}$-closed in the intersection of an arbitrarily fixed positive Yamabe conformal class and the space of metrics with a uniform Hölder bound.

math.DG↗

arXiv2026

On a new definition of the Bäcklund transformation in the isometric deformation of surfaces

We prove that a generic $4$-dimensional integrable rolling distribution of contact elements with the symmetry of the tangency configuration (excluding developable seed and isotropic developable leaves) splits into an $1$-dimensional family of generic $3$-dimensional integrable rolling distributions of contact elements with the symmetry of the tangency configuration, thus introducing a new definition of the Bäcklund transformation in the isometric deformation of surfaces.

math.DG↗

arXiv2026

Contact lifts and Holder lifts to central extension of Carnot groups

We consider the existence problem of lift F of a map f between Carnot group with different smoothness, where we use central extension to define lifting. Our main result is the existence of the contact lifts of Lipschitz and Sobolev maps and the rigidity result for the contact lift of quasiconformal maps: a quasiconformal map admits a contact lift then it is bi-Lipschitz. We also show a necessary criterion for the extension of γ-Holder lift when γ > 1/2 for step-n Carnot group.

math.DG↗