Optimal \(L^2\) extensions of openness type

We study the following optimal \(L^2\) $L^2$ extension problem of openness type: given a complex manifold M, a closed subvariety \(S\subset M\) $S\subset M$ and a holomorphic vector bundle \(E\rightarrow M\) $E\to M$ , for any \(L^2\) $L^2$ holomorphic section f defined on some open neighborhood U of S, find an \(L^2\) $L^2$ holomorphic section F on M such that \(F|_S = f|_S\) ${F|}_S{=f|}_S$ , and the \(L^2\) $L^2$ norm of F on M is optimally controlled by the \(L^2\) $L^2$ norm of f on U. Answering the above problem, we prove an optimal \(L^2\) $L^2$ extension theorem of openness type on weakly pseudoconvex Kähler manifolds, which generalizes a couple of known results on such a problem. Moreover, we prove a product property for certain minimal \(L^2\) $L^2$ extensions and give an alternative proof to a version of the above \(L^2\) $L^2$ extension theorem. We also present some applications to the usual optimal \(L^2\) $L^2$ extension problem and the equality part of Suita’s conjecture.