Relative Ding and K-stability of toric Fano manifolds in low dimensions

The purpose of this article is to clarify all of the uniformly relatively Ding stable toric Fano threefolds and fourfolds as well as unstable ones. The key player in our classification result is the Mabuchi constants, which can be calculated by combinatorial data of the associated moment polytopes due to the work of Yao (Int Math Res Not IMRN 2022(24):19790-19853, 2022). In this article, we give the list of uniform relative Ding stability of all toric Fano manifolds in dimension up to four with the values of the Mabuchi constants. As an application of our main theorem, we clarify the difference between relative K-stability and relative Ding stability by considering some specific toric Fano manifolds. In the proof, we used Bott tower structure of relatively Ding unstable toric Fano manifolds.