Ergodicity in Geodesic Flow on Nonpositive Curvature Surfaces

The paper proves the ergodicity of the geodesic flow on closed, connected, oriented surfaces of genus at least two with nonpositive Gaussian curvature and $C^{3+ ext{Lip}}$ metrics. it verifies the Liouville probability measure is ergodic on surfaces by proving the vanishing of positive measure singular families of vectors whose geodesic trajectories lie entirely within the zero-curvature locus. The proof derives an incompatible lower bound on fundamental frequency modes, leveraging Guillemin-Kazhdan operators, connection Laplacians, and heat smoothing techniques. Main result and setting The paper proves that the geodesic flow on a closed, connected, oriented surface of genus at least two is ergodic with respect to Liouville measure whenever the metric is C3+LipC^{3+\mathrm{Lip}} and has nonpositive Gaussian curvature. More precisely, if MM is such a surface and gs:SM→SMg^s:SM\to SM denotes its geodesic flow, then the invariant probability measure dm=daˉ dθ2π,dm=\frac{d\bar a,d\theta}{2\pi}, where aˉ(M)=1\bar a(M)=1, is ergodic for gsg^s (2610.27697). The result concerns the non-Anosov case, in which the curvature is allowed to vanish on nontrivial sets. For strictly negative curvature, ergodicity follows from uniform hyperbolicity. In contrast, nonpositive curvature permits geodesics that remain entirely in the zero-curvature locus and therefore have vanishing transverse Lyapunov exponents. The central issue is consequently to show that this singular family has zero Liouville measure. The argument reduces the theorem to the following assertion. Let S={v∈SM:K(π(gsv))=0 for every s∈R}S={v\in SM:K(\pi(g^sv))=0\text{ for every }s\in\mathbb R} be the set of vectors whose entire geodesic remains in zero curvature. Pesin’s ergodicity theorem on the regular set implies that the restriction of the geodesic flow to SM∖SSM\setminus S is ergodic with respect to the restricted Liouville measure. Therefore, it suffices to establish m(S)=0m(S)=0. The paper proves this by assuming m(S)>0m(S)>0 and deriving an incompatible lower bound on the Fourier modes of the indicator function of MM0. Fourier decomposition of the singular set Let MM1. Since MM2 is closed and invariant under the geodesic flow, MM3 satisfies MM4 in the distributional sense, where MM5 is the geodesic vector field. Because MM6 is also invariant under velocity reversal, MM7, so only even fiberwise Fourier modes occur. Writing MM8 for the MM9th eigenspace of the vertical vector field …