This paper concerns the problem of existence of taut foliations among $3$-manifolds. Since the contribution of David Gabai, we know that closed $3$-manifolds with non-trivial second homology group admit a taut foliation. The essential part of this paper focuses on Seifert fibered homology $3$-spheres. The result is quite different if they are integral or rational but non-integral homology $3$-spheres. Concerning integral homology $3$-spheres, we can see that all but the $3$-sphere and the Poincaré $3$-sphere admit a taut foliation. Concerning non-integral homology $3$-spheres, we prove there are infinitely many which admit a taut foliation, and infinitely many without taut foliation. Moreover, we show that the geometries do not determine the existence of taut foliations on non-integral Seifert fibered homology $3$-spheres.

In this paper we regularize the Kepler problem on $S^3$ in several different ways. First, we perform a Moser-type regularization. Then, we adapt the Ligon-Schaaf regularization to our problem. Finally, we show that the Moser regularization and the Ligon-Schaaf map we obtained can be understood as the composition of the corresponding maps for the Kepler problem in Euclidean space and the gnomonic transformation.

Based on the theory of spherical harmonics for measures invariant under a finite reflection group developed by Dunkl recently, we study orthogonal polynomials with respect to the weight functions $|x_1|^{\alpha_1}\cdots |x_d|^{\alpha_d}$ on the unit sphere $S^{d-1}$ in $\RR^d$. The results include explicit formulae for orthonormal polynomials, reproducing and Poisson kernel, as well as intertwining operator.