FUNDAMENTAL TONE ESTIMATES ON FINSLER MANIFOLDS

We study the fundamental tone of Laplacian operators on Finsler manifold $M$ evolved by a special function $u:\Omega\subset M \rightarrow \mathbb{R}$, and we give some geometric estimates of the first eigenvalue of p-laplace and (p,q)-Laplace operators depend on this function for simply connected manifolds, a class of warped product manifolds, and a class of Finsler submersions. Under a similar setting, we also study these results on a quasi-linear operator $Lu=-\Delta_{p}u+X\vert u\vert^{p-2}u$.