Shortest closed geodesics on three-dimensional isospectral manifolds

Abstract

We prove that the spectrum of a smooth closed three-manifold gives a positive lower bound for the length of its shortest nonconstant closed geodesic. The argument uses three curvature heat coefficients and the first positive singularity of the retarded wave trace. The heat coefficients yield a curvature--length dichotomy: sufficiently large maximum curvature forces a closed geodesic shorter than the curvature scale. We then show that a shortest closed geodesic lying below the conjugate radius produces a singularity of the sine trace. Combining these estimates gives a lower bound depending only on the common spectrum. The compactness theorem of Anderson then implies smooth compactness of the full isospectral family modulo diffeomorphisms.