This is the fourth part of [6] on the existence of K¨ahler Einstein metrics of the general type I almost homogeneous manifolds of cohomogeneity one. We actually carry out all the results in [8] to the type I cases. In part II [14], we obtained a lot of new K¨ahler-Einstein manifolds as well as Fano manifolds without K¨ahler-Einstein metrics. In particular, by applying Theorem 15 therein, we have complete results in the Theorems 3 and 4 in that paper. However, we only have some partial results in Theorem 5 there. In this note, we shall give a report of recent progress on the Fano manifolds Nn,m when n > 15 and N′n,m when n > 4. We actually give two nice pictures for these two classes of manifolds. See our Theorems 1 and 2 in the last section. Moreover, we post two conjectures. Once we could solve these two conjectures, the question for these two classes of manifolds would be completely solved. With applying our results to the canonical circle bundles we also obtain Sasakian manifolds with or without Sasakian-Einstein metrics. That also give some open Calabi-Yau manifolds.

Fruit softening is an important characteristic of peach fruit ripening. The auxin receptor TIR1 (Transport Inhibitor Response 1) plays an important role in plant growth and fruit maturation, but little research has been conducted on the relation of TIR1 to the softening of peach fruits. In this study, the hardness of isolated peach fruits was reduced under exogenous NAA treatment of low concentration, at the same time, the low concentration of NAA treatment reduced the transcription level of PpPG and Ppβ-GAL genes related to cell wall softening, and PpACS1 genes related to ethylene synthesis. The transient over-expression of the PpTIR1 gene in peach fruit blocks caused significant down-regulation of the expression of early auxin-responsive genes, ethylene synthesis and cell wall metabolic genes related to fruit firmness. Through yeast two-hybrid technology, bimolecular fluorescence complementary technology, and firefly luciferase complementation imaging assay, an interaction between PpTIR1 and PpIAA1/3/5/9/27 proteins was revealed, and the interaction depended on auxin and its type and concentration. These results show that PpTIR1-Aux/IAA module has a possible regulatory effect on fruit ripening and softening.