Narrow Results

We establish some comparison theorems on Finsler manifolds with curvature quadratic decay. As their applications, we obtain some optimal Cheeger–Gromov–Taylor type compact theorems, volume growth and Mckean type estimate for the first Dirichlet eigenvalue for such manifolds. Although we present the results for Finsler manifolds, they are all new results for Riemannian manifolds.

This paper provides the optimal exponential decay rate of the lower bound for the first positive eigenvalue of the Laplacian operator on a compact Riemannian manifold with a negative lower bound on the Ricci curvature and with large diameter. For manifolds with boundary, suitable convexity conditions are assumed.

Given a closed symplectic manifold (M²ⁿ, ω) of dimension 2n ≥ 4, we consider all Riemannian metrics on M, which are compatible with the symplectic structure ω. For each such metric g, we look at the first eigenvalue λ₁ of the Laplacian associated with it. We show that λ₁ can be made arbitrarily large, when we vary g. This generalizes previous results of Polterovich, and of Mangoubi.