Eigenvalue estimates and isoperimetric inequalities for cone-manifolds

Source abstract

This paper studies eigenvalue bounds and isoperimetric inequalities for Rieman-nian spaces with cone type singularities along a codimension-2 subcomplex. These “cone-manifolds” include orientable orbifolds, and singular geometric structures on 3-manifolds studied by W. Thurston and others. We first give a precise definition of “cone-manifold” and prove some basic results on the geometry of these spaces. We then generalise results of S.-Y. Cheng on upper bounds of eigenvalues of the Laplacian for disks in manifolds with Ricci curvature bounded from below to cone-manifolds, and characterise the case of equality in these estimates. We also establish a version of the Lévy-Gromov isoperimetric inequality for cone-manifolds. This is used to find lower bounds for eigenvalues of domains in cone-manifolds and to establish the Lichnerowicz inequality for cone-manifolds. These results enable us to characterise cone-manifolds with Ricci curvature bounded from below of maximal diameter.