Since the celebrated work by Cartan, distributions with small growth vector $(2,3,5)$ have been studied extensively. In the holomorphic setting, there is a natural correspondence between holomorphic $(2,3,5)$-distributions and nondegenerate lines on holomorphic contact manifolds of dimension 5. We generalize this correspondence to higher dimensions by studying nondegenerate lines on holomorphic contact manifolds and the corresponding class of distributions of small growth vector $(2m, 3m, 3m+2)$ for any positive integer m.

In this paper, we prove that a strongly convex and Kähler-Finsler metric is a complex Berwald metric with zero holomorphic sectional curvature if and only if it is a complex locally Minkowski metric.

Contractivity and hypercontractivity properties of semigroups are now well understood when the generator, $A$, is a Dirichlet form operator. It has been shown that in some holomorphic function spaces the semigroup operators, ${{e}^{-tA}}$, can be bounded below from ${{L}^{p}}$ to ${{L}^{q}}$ when $p,\,q$ and $t$ are suitably related. We will show that such lower boundedness occurs also in spaces of subharmonic functions.