Asymptotics for the spectral function on Zoll manifolds

Abstract

Let $(M,g)$ be a Zoll manifold, i.e., a smooth, compact, Riemannian manifold without boundary all of whose geodesics are closed with a minimal common period $T$. The positive definite Laplace-Beltrami operator has eigenvalues $\{\hat{I}{»}_j^2{\}}_j$ which cluster around $\hat{I}{½}_{\ell }^2$ for some sequence $\hat{I}{½}_{\ell }\to \infty$. This article is concerned with the number of $\hat{I}{»}_j$ in a window of fixed size $\mathrm{w}$ around $\hat{I}{½}_{\ell }$, denoted by $\mathbf{N}(\hat{I}{½}_{\ell },\mathrm{w}):=\#\{j:\hat{I}{»}_j\in [\hat{I}{½}_{\ell }-\mathrm{w},\hat{I}{½}_{\ell }+\mathrm{w}]\}.$ When the set of trajectories with period smaller than $T$ has zero measure, there is $c_n>0$, depending only on $n=\dim M$, such that $$\mathbf{N}(\hat{I}{½}_{\ell },\mathrm{w})=c_n{\mathrm{vol}}_g(M)\hat{I}{½}_{\ell }^{n-1}+o(\hat{I}{½}_{\ell }^{n-1}),$$ as $\ell \to \infty$. However, for a general Zoll manifold this may not be the case. We show that, nevertheless, there is $N>0$, independent of $\ell$, such that $$\sum _{j=0}^{N-1}\mathbf{N}(\hat{I}{½}_{\ell +j},\mathrm{w})=c_{n}N{\mathrm{vol}}_g(M)\hat{I}{½}_{\ell }^{n-1}+o(\hat{I}{½}_{\ell }^{n-1}),$$ as $\ell \to \infty$. In addition to asymptotics for the counting function, we study the kernel of the spectral projector for the Laplacian, ${\hat{I}}_{\ell ,\mathrm{w}}(x,y)$ onto the spectrum in ${\bigcup }_{j=0}^{N-1}[\hat{I}{½}_{\ell +j}-\mathrm{w},\hat{I}{½}_{\ell +j}+\mathrm{w}]$. We show that for $x$ and $y$ in a shrinking neighborhood of a point with few loops of length smaller than $T$, ${\hat{I}}_{\ell ,\mathrm{w}}(x,y)$ and its derivatives have the same asymptotics as those on the round sphere and flat torus.

The main results in the current version are novel. Theorem 1 in this version is new. Theorem 1 in the previous version was misstated; specifically missing the assumption that the Zoll manifold be SC_T. Theorem 2 in the current posting replaces this assumption with a weaker one