First variation of flat traces on negatively curved surfaces

Abstract

For a closed negatively curved surface $(X,g)$ the flat trace of the geodesic Koopman operators $V_g^{\tau }f=f\circ G_g^{\tau }$ is the periodic orbit distribution \[ \mathrm{Tr}^{\flat} V_{g}(τ)=\sum_γ\frac{L_γ^{\#}}{\lvert\det(I-P_γ)\rvert}\,δ(τ-L_γ), \qquad τ>0, \] supported on the length spectrum and weighted by the linearized Poincaré maps $P_{\gamma }$. For a smooth family of negatively curved metrics $g_t$ we compute the first variation ${\partial }_t{|}_0{\mathrm{Tr}}^{♭}{V}_{g_t}$ as a distribution. At an isolated length $\ell$ the leading singularity is a multiple of ${\delta }^{\prime }(\tau -\ell )$, and its coefficient is an explicit linear functional of the length variations ${\dot{L}}_{{\gamma }^m}$ of the closed geodesics with $L_{{\gamma }^m}=\ell$. This transport coefficient forces the marked lengths to be locally constant along any deformation with constant flat trace. As an application, if ${\mathrm{Tr}}^{♭}{V}_{g_t}={\mathrm{Tr}}^{♭}{V}_{g_0}$ for all $t$ then $g_t$ is isometric to $g_0$ for all $t$. Together with Sunada-type constructions of non isometric pairs with equal flat traces, this shows that the flat trace is globally non-unique yet locally complete along smooth families.

arXiv comment15 pages