This paper is one of a series
in which we generalize our earlier results on the equivalence of existence of Calabi
extremal metrics to the geodesic stability for any type I compact complex almost
homogeneous manifolds of cohomogeneity one. In this paper, we actually carry all the
earlier results to the type I cases. As requested by earlier referees of this series of
papers, in this third part, we shall first give an updated description of the geodesic
principles and the classification of compact almost homogeneous Kähler manifolds
of cohomogeneity one. Then, we shall give a proof of the equivalence of the geodesic
stability and the negativity of the integral in the first part. Finally, we shall
address the relation of our result to Ross–Thomas version of Donaldson’s
K-stability. One should easily see that their result is a partial generalization of our
integral condition in the first part. And we shall give some further comments
on the Fano manifolds with the Ricci classes. In Theorem 14, we give a
result of Nadel type. We define the strict slope stability. In our case, it is
stronger than Ross–Thomas slope stability. We strengthen two Ross–Thomas
results in Theorems 15 and 16. The similar proofs of the results other than
the existence for the type II cases are more complicated and will be done
elsewhere.
