On the extension of $s{{h}_{m}}$ functions into higher-dimensional complex spaces

In this talk, we study the problem of extending $m$-subharmonic functions from a domain $D \subset \mathbb{C}^n$ to a neighborhood in the higher-dimensional space $\mathbb{C}^{n+1}$. We consider two distinct classes of functions: strongly $m$-subharmonic ($s{{h}_{m}}$) and weakly $m$-subharmonic. It is shown that while the extension of weakly $m$-subharmonic functions is a relatively straightforward task, strongly $m$-subharmonic functions, despite being a subclass, exhibit a fundamentally different behavior in this regard. This approach allows us to bridge the gap between local $m$-potential theory and the global geometric structure of Stein manifolds, highlighting the specific constraints that arise from the strong $m$-subharmonicity condition.