Stability estimates for the holonomy inverse problem (Q6595125)

The authors present several results on the holonomy inverse problem, which is: To what extent can a unitary connection \({\mathfrak a}\) (modulo the pull-back action) on a smooth vector bundle over a Riemannian manifold be determined by the Wilson loop operator \(\textbf{W}({\mathfrak a},\gamma)\), i.e., the traces of its parallel transport or holonomy over primitive closed geodesics \(\gamma\). Two conditions are considered on the unitary connection \({\mathfrak a}\) on the smooth vector bundle \(({\mathcal E},h)\to M\), where \((M,g)\) is a Riemannian manifold. The first condition \(\textbf{(A)}\) is that \({\mathfrak a}\) is opaque, and the second condition \(\textbf{(B)}\) is that \({\mathfrak a}\) has solenoidally injective generalized \(X\)-ray transform on \(1\)-forms with values in \(\mathrm{End}({\mathcal E})\). For each \(L>0\), the moduli space \({\mathbb A}^L_{\mathcal E}\) of connections with Hölder-Zygmund regularity \(L\) has a natural metric \(d_{C^*}^L\). The authors had previously shown that conditions \(\textbf{(A)}\) and \(\textbf{(B)}\) hold for an open and dense subset of \({\mathbb A}_{\mathcal E}^L\) for \(L\gg 1\) depending on \(n\).\N\NThe first main result is about local stability for connections on a smooth vector bundle over an Anosov \(n\)-manifold \((M,g)\). It states there exists \(N\gg 1\) (depending only on \(n\)) such that for a connection \({\mathfrak a}_0\in {\mathbb A}^N_{{\mathcal E}}\) satisfying conditions \(\textbf{(A)}\) and \(\textbf{(B)}\), and any \(\eta>0\) there exist \(\tau>0\) (depending only on \((M,g)\) and \(\eta\)) and \(\delta>0\) and \(C>0\) such that for all connections \({\mathfrak a_1},{\mathfrak a}_2 \in {\mathbb A}^N_{\mathcal E}\) satisfying \(d_{C^*}^N({\mathfrak a}_i,{\mathfrak a}_0)<\delta\) for \(i=1,2\) one has\N\[\Nd_{C^*}^{N-\eta}({\mathfrak a}_1,{\mathfrak a}_2) \leq C \left( \sup_{\gamma \in G^\#} \ell(\gamma)^{-1} \vert {\mathbf W}({\mathfrak a}_1,\gamma) - \textbf{W}({\mathfrak a}_2,\gamma)\vert \right)^\tau,\N\]\Nwhere \(G^{\#}\) is the set of primitive closed geodesics and \(\ell(\gamma)\) is the length of \(\gamma \in G^{\#}\).\N\NThe second main result is on global stability (with no conditions on the connections) and has a conclusion similar to the first main result, for Hermitian line bundles over Anosov \(n\)-manifolds.\N\NThe third main result extends the second main result to Hermitian vector bundles of low enough rank over a closed smooth Riemannian manifold \((M,g)\) with negative sectional curvature. For each \(n\) the upper bound \(q_{\mathbb C}(n)\) on the rank of the bundle is determined by the smallest positive integer \(r\) such that there is a nonconstant polynomial map from the unit sphere \({\mathbb S}^n\) in \({\mathbb R}^{n+1}\) to the unit sphere \({\mathbb S}^r\) in \({\mathbb R}^{r+1}\).\N\NA key element of the proofs of the main results is an approximate nonabelian version of the Livšic theorem for unitary cocyles. This result is stated as another main result, and is proved in the paper.