  We study the problem of extension of normal jets from a hypersurface, with
focus on the growth order of the constant. Using aspects of the standard,
twisted approach for $L^2$ extension and of the new approach to $L^2$ extension
introduced by Berndtsson and Lempert, we are able to obtain an extension
theorem with a constant $C^k$ where $C$ is universal and $k$ is the jet order.
We then use the jet extension theorem to extend positively curved singular
Hermitian metrics from smooth, deformably pseudoeffective hypersurfaces in
projective manifolds.
