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First variation of flat traces on negatively curved surfaces

Hy Lam

TL;DR

This work analyzes the flat trace of the geodesic Koopman operator on a closed negatively curved surface and its behavior under smooth metric deformations. By developing a microlocal framework with a clean fixed-set structure and applying a parameter-dependent stationary phase, the authors identify a leading $\delta'(\tau-\ell)$ term in the first variation whose coefficient transports the marked length spectrum via $\dot L_{\gamma^m}$. Using a Guillemin–Kazhdan $SO(2)$ calculus together with Livšic theory, they convert the resulting constraint into an infinitesimal triviality: if the flat traces remain constant along a deformation, the metric deformation is generated by a diffeomorphism, yielding $g_t=\varphi_t^* g_0$. Consequently, the flat trace is locally complete for smooth deformations (providing pathwise rigidity) but globally non-unique (as Sunada-type examples show). The paper thus links dynamical zeta-type data to local geometric rigidity while highlighting global non-uniqueness of the flat trace.

Abstract

For a closed negatively curved surface $(X,g)$ the flat trace of the geodesic Koopman operators $V_g^τf=f\circ G_g^τ$ 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 $P_γ$. 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 $δ'(τ-\ell)$, and its coefficient is an explicit linear functional of the length variations $\dot L_{γ^m}$ of the closed geodesics with $L_{γ^m}=\ell$. 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.

First variation of flat traces on negatively curved surfaces

TL;DR

This work analyzes the flat trace of the geodesic Koopman operator on a closed negatively curved surface and its behavior under smooth metric deformations. By developing a microlocal framework with a clean fixed-set structure and applying a parameter-dependent stationary phase, the authors identify a leading term in the first variation whose coefficient transports the marked length spectrum via . Using a Guillemin–Kazhdan calculus together with Livšic theory, they convert the resulting constraint into an infinitesimal triviality: if the flat traces remain constant along a deformation, the metric deformation is generated by a diffeomorphism, yielding . Consequently, the flat trace is locally complete for smooth deformations (providing pathwise rigidity) but globally non-unique (as Sunada-type examples show). The paper thus links dynamical zeta-type data to local geometric rigidity while highlighting global non-uniqueness of the flat trace.

Abstract

For a closed negatively curved surface the flat trace of the geodesic Koopman operators is the periodic orbit distribution supported on the length spectrum and weighted by the linearized Poincaré maps . For a smooth family of negatively curved metrics we compute the first variation as a distribution. At an isolated length the leading singularity is a multiple of , and its coefficient is an explicit linear functional of the length variations of the closed geodesics with . This transport coefficient forces the marked lengths to be locally constant along any deformation with constant flat trace. As an application, if for all then is isometric to for all . 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.
Paper Structure (18 sections, 3 theorems, 123 equations)

This paper contains 18 sections, 3 theorems, 123 equations.

Table of Contents

  1. Introduction
  2. Flat trace, clean fixed sets, and first variation
  3. Kernel and flat trace as a pullback and pushforward
  4. Normal form near a closed orbit
  5. Stationary phase with a clean one dimensional critical set
  6. Parameter dependent phase and $t$ variation of the clean contribution
  7. Isolation of the most singular terms and the weighted identities
  8. The variations $\dot L_\gamma$ and $\dot p$ in terms of $\dot g$
  9. Variation of length for a closed geodesic
  10. Variation of the Hamiltonian on $T^*X$
  11. Variation of the flat trace with explicit coefficients
  12. Triviality from the vanishing of $\delta'$
  13. Guillemin-Kazhdan variational identity
  14. Deducing constant lengths from the $\delta'$ vanishing
  15. Proof of Theorem \ref{['theoremA']}
  16. ...and 3 more sections

Key Result

Theorem 1.2

If $\mathrm{Tr}^{\flat}V_{g_t}=\mathrm{Tr}^{\flat}V_{g_0}$ as distributions on $(0,\infty)$ for all $t$, then there exists a smooth one parameter family of diffeomorphisms $\{\varphi_t\}$ of $X$ with $\varphi_0=\mathrm{id}$ such that Equivalently, $(X,g_t)$ is isometric to $(X,g_0)$ for every $t$.

Theorems & Definitions (9)

  • Remark 1.1
  • Theorem 1.2
  • Remark 1.3
  • Remark 2.1
  • Proposition 2.2
  • Remark 2.3
  • Lemma 5.1: Guillemin-Kazhdan
  • proof
  • proof