Does the problem "partition a vertex-weighted graph into $k$ balanced connected parts" have a standard name? Consider the following problem:


  Given an integer $k$ and a vertex-weighted graph $G=(V,E)$, find a
  partition of $V$ into $V_1,\ldots,V_k$ such that each subgraph induced by $V_i$ is connected, and such that $\max_i w(V_i)$ is
  minimized, where $w(V_i)$ is the total weight of the vertices in
  $V_i$.


It can be easily seen that this problem is NP-complete. For hardness, note that we can solve Partition by using a complete graph.

Questions: Does this problem have a standard name? Are there any papers that study this problem, or maybe slight variations of it? Is the problem still hard for sparse graphs? I am particularly interested in references that could lead to an implementation with good performance in practice.
The paper [Chlebikova, Approximating the maximally balanced connected partition problem in graphs, 1996] studies a related problem: They have $k=2$, and they maximize $\min(|V_1|,|V_2|)$. It should be a good place to start searching for related results, by following the citation graph.