Schrödinger operators and topological pressure on manifolds of negative curvature (Q2778797)

Let \(N\) be a complete simply-connected manifold of negative curvature and \(B_v:N\to \mathbb{R}\) the Busemann function on the unit tangent bundle \(UN\ni v\). Given a co-compact lattice group \(\Gamma\) of isometries of \(N\), the function \(UN\ni v\to (\Delta^jB_v) (\pi v)\), \(\pi: UN\to N\), descends to a function \({\mathcal B}^{(j)}\) on \(UM\), \(M=N/ \Gamma\). \({\mathcal B}^{(1)}\) at \(v\) is the mean curvature at \(\pi v\) of the \(v\)-horosphere. The authors prove that \({\mathcal B}^{(2)}\) is not cohomologous to a negative continuous function, in particular, the topological pressure of \({\mathcal B}^{(2)}\) is nonnegative. This supports the following conjecture: A compact manifold of negative curvature is locally symmetric iff its bilaplacian cocycle is cohomologous to a nonpositive function.NEWLINENEWLINEFor the entire collection see [Zbl 0973.00044].