Horospherical means and uniform distribution of curves of constant geodesic curvature
Let \(\mathbb{R} H^n\) denote the \(n\)-dimensional real hyperbolic space of constant sectional curvature \(K\equiv 1\), and let \(S_p(r)\) denote the sphere of radius \(r\) at a fixed point \(p\) of \(\mathbb{R} H^n\). For a continuous function \(f:\mathbb{R} H^n\to\mathbb{R}\), let \(M_p(f)(r)= (1/\text{vol}(S_P(r))) \int_{S_p (r)}f dS\) for every \(r>0\), the spherical mean of \(f\) for radius \(r\). A result of P. Günther says that if \(M\) is any compact quotient manifold of \(\mathbb{R} H^n\) with covering projection \(\pi:\mathbb{R} H^n\to M\), then \((1/\text{vol} (M))\int_Mf dM= \lim_{r\to \infty}M_p(f \circ\pi)(r)\). A key feature of this result is that the spheres \(\{S_p(r): r>0\}\) are the leaves of a codimension 1 foliation \({\mathcal F}\) of \(\mathbb{R} H^n-\{p\}\) in which each leaf has constant mean curvature and such that the 1-dimensional orthogonal foliation \({\mathcal F}^\perp\) is integrable with leaves that are geodesics of \(\mathbb{R} H^n\). In the present article, the authors extend the result and method of proof of Günther to the case where \(\mathbb{R} H^n\) is replaced by an arbitrary complete rank 1 Riemannian symmetric space \(\widetilde M\) with \(K<0\), and the pair of foliations \({\mathcal F}\) and \({\mathcal F}^\perp\) has the same properties as above but the leaves of \({\mathcal F}\) are allowed to be noncompact hypersurfaces of \(\widetilde M\). In this case, each leaf of \({\mathcal F}\) must possess an admissible exhaustion \(\{K_j:j\geq 1\}\) by compact sets; that is, area \((\partial K_j)/ \text{volume} (K_j)\to 0\) as \(j \to\infty\). An admissible exhaustion \(\{K_j\}\) on one leaf \(F=F\times\{0\}\) of \({\mathcal F}\) induces an admissible exhaustion \(\{K_j(t)\}\) on the leaf \(F \times \{t\}\) by flowing along the unit speed geodesics orthogonal to \({\mathcal F}\) for a time \(t\). The main result says that \(\lim_{j\to\infty} (1/\text{vol} (K_j))\int_{K_j} (f\circ\pi) dK_j=(1/ \text{vol} (M)) \int_Mf dM\), where \(\{K_j\}\) is an admissible exhaustion of any fixed leaf \(F\) of \({\mathcal F}\), \(f:M\to\mathbb{R}\) is an arbitrary continuous function, and \(\pi:\widetilde M\to M\) is the projection. The main applications occur when either (a) \({\mathcal F}\) is a codimension 1 foliation by concentric horospheres of a complete rank 1 Riemannian symmetric space \(\widetilde M\) with \(K<0\) or (b) \({\mathcal F}\) is a foliation of the hyperbolic plane \(\mathbb{R} H^2\) by curves \(F\times\{t\}\) that lie at constant distance from a fixed core geodesic \(F\times\{0\}\). In case (b) \(F=\mathbb{R}\) and \(K_j=[-T_j, T_j]\) for some sequence \(T_j\to\infty\) as \(j\to\infty\). Moreover, the leaves of \({\mathcal F}\) have constant geodesic curvature, which is the 1-dimensional analogue of constant mean curvature for foliations with leaves of dimension \(k\geq 2\). The result in case (b) also requires that the core geodesic \(F\times \{0\}\) in the compact quotient \(M\) of \(\mathbb{R} H^2\) be ergodic. Ergodicity for a geodesic of \(M\) is a kind of strong uniform distribution condition that nevertheless is satisfied for geodesics whose initial tangent vector belongs to a subset \(A\) of full measure in the unit tangent bundle of \(M\). The results of this article also extend to codimension 1 foliations \({\mathcal F}\) defined by horospheres in certain Riemannian product manifolds, in particular for foliations defined by concentric ``diagonal'' horospheres of \(\mathbb{R} H^2\times \mathbb{R} H^2\). These examples are likely to be special cases of a more general phenomenon. The proof of the main result follows the method used by Günther. One first reduces from the case of an arbitrary continuous function \(f:M \to\mathbb{R}\) to the special case that \(f\) is a nonconstant eigenfunction of the Laplace operator \(\Delta\) on functions. Write the universal cover \(\widetilde M\) as a union of equidistant leaves \(F\times\{t\}\) with admissible exhaustions \(\{K_j(t)\}\) coming from an admissible exhaustion \(\{K_j\}\) on \(F\times \{0\}\). Let \(g_j(t)\) denote the average of the lifted eigenfunction \(f\circ\pi\) over the set \(K_j(t) \subseteq F\times\{t\}\), where \(\pi:\widetilde M\to M\) is the covering projection. Since the leaves \(F\times \{t\}\) have constant mean curvature the functions \(g_j(t)\) satisfy a certain ordinary differential equation in \(t\). Using this differential equation one shows that \(g_j(t)\to 0\) for all \(t\) as \(j\to\infty\), and the proof of the main result now concludes quickly.
scientific article; zbMATH DE number 1336869