Convergence of Closed Pseudo-Hermitian Manifolds

TL;DR

This paper proves the compactness and convergence properties of certain classes of pseudo-Hermitian and Sasakian manifolds under geometric bounds, using Sobolev inequalities and subelliptic estimates, with applications to Kähler cones.

Contribution

It establishes new compactness results for pseudo-Einstein and Sasakian $ ext{η}$-Einstein manifolds under uniform geometric conditions, extending convergence theory in pseudo-Hermitian geometry.

Findings

Set of pseudo-Einstein manifolds is compact under geometric bounds.

Set of Sasakian $ ext{η}$-Einstein manifolds is $C^ ext{∞}$ compact with specified bounds.

Pointed convergence of Kähler cones with Sasakian links is achieved.

Abstract

Based on uniform CR Sobolev inequality and Moser iteration, this paper investigates the convergence of closed pseudo-Hermitian manifolds. In terms of the subelliptic inequality, the set of closed normalized pseudo-Einstein manifolds with some uniform geometric conditions is compact. Moreover, the set of closed normalized Sasakian $\eta$-Einstein $(2n+1)$-manifolds with Carnot-Carath\'eodory distance bounded from above, volume bounded from below and $L^{n + \frac{1}{2}}$ norm of pseudo-Hermitian curvature bounded is $C^\infty$ compact. As an application, we will deduce some pointed convergence of complete K\"ahler cones with Sasakian manifolds as their links.