Pursuing an idea motivated by a question of S.-S. Chern from 1968 on the existence of
intrinsic Riemannian obstructions to minimality [Chern, S.-S.: Minimal submanifolds
in a Riemannian manifold (1968)], an important study of the very idea of curvature
was deepened after 1993 by B.-Y. Chen, then by other authors. In the last two decades,
B.-Y. Chen’s fundamental inequalities have been investigated by many authors in the
context of various geometric structures. In this talk, we start by presenting B.-Y. Chen’s
fundamental inequality for Kähler submanifolds in complex space forms, and we recall
why the case of Kähler surfaces in C^3 satisfying scal(p) = 4 inf sec(π^r ) appears
naturally and is important. Then we provide several characterizations of strongly minimal
complex surfaces in the complex three dimensional space. We focus our study on the question
of finding further examples of strongly minimal Kähler surfaces, as the question of a
complete classification of these geometric objects is still open.