First variation of flat traces on negatively curved surfaces

TL;DR

This paper computes the first variation of the flat trace of geodesic Koopman operators on negatively curved surfaces, revealing how length spectrum variations influence the trace and implications for metric rigidity.

Contribution

It provides an explicit formula for the first variation of the flat trace under metric deformations, connecting length variations to trace singularities and metric rigidity.

Findings

First variation of flat trace involves delta' distributions at length spectrum points.

Leading singularity coefficient relates to linearized length variations of closed geodesics.

Flat trace is locally complete but globally non-unique among negatively curved metrics.

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 \[ \mathrm{Tr}^{\flat} V_{g}(\tau)=\sum_{\gamma}\frac{L_\gamma^{\#}}{\lvert\det(I-P_\gamma)\rvert}\,\delta(\tau-L_\gamma), \qquad \tau>0, \] supported on the length spectrum and weighted by the linearized Poincar\'e maps $P_\gamma$. For a smooth family of negatively curved metrics $g_t$ we compute the first variation $\partial_t\vert_{0}\,\mathrm{Tr}^{\flat} V_{g_t}$ as a distribution. At an isolated length $\ell$ the leading singularity is a multiple of $\delta'(\tau-\ell)$, 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 $\mathrm{Tr}^{\flat} V_{g_t}=\mathrm{Tr}^{\flat} V_{g_0}$ 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.