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Register a new userEisenstein series and equidistribution of Lebesgue probability measures on compact leaves of the horocycle foliations of Bianchi 3-orbifolds
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Inspired by the works of Zagier, we study the probability measures $ u(t)$ with support on the flat tori which are the compact orbits of the maximal unipotent subgroup acting holomorphically on the positive orthonormal frame bundle $mathcal{F}({M}_D)$ of 3-dimensional hyperbolic Bianchi orbifolds ${M}_D=mathbb{H}^3/widetilde{Gamma}_D$, of finite volume and with only one cusp. Here $Gamma_D=PSL(2, mathcal{O})$, where $mathcal{O}$ is the ring of integers of an imaginary quadratic field of class number one.
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A classic result due to Furstenberg is the strict ergodicity of the horocycle flow for a compact hyperbolic surface. Strict ergodicity is unique ergodicity with respect to a measure of full support, and therefore implies minimality. The horocycle flow has been previously studied on minimal foliations by hyperbolic surfaces on closed manifolds, where it is known not to be minimal in general. In this paper, we prove that for the special case of Riemannian foliations, strict ergodicity of the horocycle flow still holds. This in particular proves that this flow is minimal, which establishes a conjecture proposed by Matsumoto. The main tool is a theorem due to Coud`ene, which he presented as an alternative proof for the surface case. It applies to two continuous flows defining a measure-preserving action of the affine group of the line on a compact metric space, precisely matching the foliated setting. In addition, we briefly discuss the application of Coud`enes theorem to other kinds of foliations.
We show that the equidistribution theorem of C. Bonatti and X. Gomez-Mont for a special kind of foliations by hyperbolic surfaces does not hold in general, and seek for a weaker form valid for general foliations by hyperbolic surfaces.
We show that the horocycle flow associated with a foliation on a compact manifold by hyperbolic surfaces is minimal under certain conditions.
In this paper, we study stability for harmonic foliations on locally conformal Kahler manifolds with complex leaves. We also discuss instability for harmonic foliations on compact submanifolds immersed in Euclidean spaces and compact homogeneous spaces.
We combine classic stability results for foliations with recent results on deformations of Lie groupoids and Lie algebroids to provide a cohomological characterization for rigidity of compact foliations on compact manifolds.
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