We prove that a stable minimal hypersurface of an open ball which is immersed away from a closed (singular) set of finite codimension 2 Hausdorff measure and weakly close to a multiplicity 2 hyperplane must in the interior be the graph over the hyperplane of a 2-valued function satisfying a local C1, estimate. This regularity is optimal under our hypotheses. As a consequence, we also establish compactness of the class of stable minimal hypersur- faces of an open ball which have volume density ratios uniformly bounded by 3−δ for any fixed δ ∈ (0, 1) and interior singular sets of vanishing co-dimension 2 Hausdorff measure.

In this article we show how the classical theory of Reeb and others extends to the case of codimension one singular foliations on closed oriented manifolds, provided that at the singular set sing(F), the foliation is locally defined by Bott-Morse functions which are transversely centers. We prove, in this setting, the equivalent of the local and the complete stability theorems of Reeb. We show that if F has a compact leaf with finite fundamental group, or if a component of sing(F) has codimension ≥ 3 and finite fundamental group, then all leaves of F are compact and diffeomorphic, sing(F) consists of two connected components, and there is a Bott-Morse function f : M → [0, 1] such that f : M \ sing(F) → (0, 1) is a fiber bundle defining F and sing(F) = f.1({0, 1}). This yields a topological description of the type of leaves that appear in these foliations, and also the type of manifolds admitting such foliations. These results unify and generalize well known results for cohomogeneity one isometric actions, and a theorem of Reeb for foliations with Morse singularities of center type.

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This is the very first paper to focus on the CR analogue of Yau’s uniformization conjecture in a complete noncompact pseudohermitian (2n+1)-manifold of vanishing torsion (i.e. Sasakian manifold) which is an odd dimensional counterpart of K¨ahler geometry. In this paper, we mainly deal with the problem of the sharp dimension estimate of CR holomorphic functions in a complete noncompact pseudohermitian manifold of vanishing torsion with nonnegative pseudohermitian bisectional curvature.