Some Spectral Estimates For Manifolds And Their Applications

Spectral estimation is one of the important topics in the field of spectral analysis on manifolds,and there are some open conjectures(such as,Pólya’s conjecture about the lower bound of Dirichlet eigenvalues of the Laplacian on bounded domains in Euclidean space,Yau’s conjecture on the first non-zero closed eigenvalue of the Laplacian on the minimal hypersurfaces of a sphere,etc.)attracting the attention of geometers.In this theis,some important spectral estimates for the Laplacian,the drifting Laplacian and nonlinear p-Laplacian on bounded domains of manifolds satisfying some curvature assumptions have been obtained.More precisely,we got:(1)Let M be an n-dimensional(n≥2)simply connected Hadamard manifold.If the radial Ricci curvature of M is bounded from below by(n-1)k(t)with respect to some point p∈M,where t=d(·,p)is the Riemannian distance on M to p,k(t)is a nonpositive continuous function on(0,∞),then the first n nonzero Neumann eigenvalues of the Laplacian on the geodesic ball B(p,l),with center p and radius 0<l<∞,satisfy 1/μ1+1/μ2+…+1/μn≥(ln+2)/((n+2)∫0lfn-1(t)dt),where f(t)is the solution to(2)We investigate minimal submanifolds M immersed into warped products of type Nn×f Qq,where f∈C∞(N)is positive,and can give lower bounds for the weighted fundamental tone of the drifting Laplacian,the first eigenvalue of the pLaplacian on open domains in M.This achievement enables us to deal with spectral estimates for minimal submanifolds bounded by balls,cylinders,pseudo-hyperbolic and hyperbolic spaces,and cones,and meanwhile some interesting byproducts can be obtained.For instance,we can show that the fundamental tone of any cylindrically bounded minimal hypersurface in the Euclidean m-space Rm(m≥3)is positive.