First variation of flat traces on negatively curved surfaces
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866910020884496384 |
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| author | Lam, Hy |
| author_facet | Lam, Hy |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_12230 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | First variation of flat traces on negatively curved surfaces Lam, Hy Dynamical Systems Differential Geometry Spectral Theory 37D40, 58J50, 37C30, 53C24 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. |
| title | First variation of flat traces on negatively curved surfaces |
| topic | Dynamical Systems Differential Geometry Spectral Theory 37D40, 58J50, 37C30, 53C24 |
| url | https://arxiv.org/abs/2602.12230 |