Sub-Laplacian eigenvalue bounds on sub-Riemannian manifolds

Abstract

We study eigenvalue problems for intrinsic sub-Laplacians on regular sub-Riemannian manifolds. We prove upper bounds for sub-Laplacian eigenvalues λ k of conformal sub-Riemannian metrics that are asymptotically sharp as k → +∞. For Sasakian manifolds with a lower Ricci curvature bound, and more generally, for contact metric manifolds conformal to such Sasakian manifolds, we obtain eigenvalue inequalities that can be viewed as versions of the classical results by Korevaar and Buser in Riemannian geometry.

Berger inequality for Riemannian manifolds with an upper sectional curvature bound

“…The key ingredient is a construction of disjoint sets whose measure is carefully controlled by our geometric hypotheses. Though similar ideas, originating in the work by Buser [8] and Korevaar [23], have been used in a few papers recently, see for example [21,24,22], and [25], our hypotheses are rather different from the previous work. In particular, we do not use a lower Ricci curvature bound for a background or auxiliary metric, which is so essential in most of the past papers.…”

Systolic Inequalities for Compact Quotients of Carnot Groups with Popp's Volume

“…The difficulty compared from the 2-step case is the non-equality between y({X ni }) and Y ({X ni }). We will compute its difference by using the BCH formula (7).…”

“…Define the map P l : g → V l to be the linear projection. Then by the BCH formula (7), P l (y({X ni })) = 0 for l ≤ j − 1, P j (y({X ni })) = Y ({X ni }) = Z j , and P j (y({X ni })) does not vanish for i ≥ l + 1. We label the error term as follows.…”

“…The volume of the unit ball in Carnot group is more complicated than the Euclidean one. In [7], Hassannezhad-Kokarev estimated the volume of the unit ball of corank 1 Carnot groups, and gave an upper bound of the volume which depends on the dimension. It seems hard to apply their method to general Carnot groups.…”

High-Precision Chebyshev Spectral to Approximate Laplace Eigenvalue Problem

“…Laplace eigenvalue problem has the characteristics of strong practicability, continuable extension, and relatively complete theoretical system. This problem usually applied to particle motion, thermal magnetic radiation, lattice vibration [1], light thin film vibvicvib ration, graph theory [2], and Riemann manifolds theory [3].…”