Mabuchi solitons and Mabuchi constants on Fano admissible manifolds

TL;DR

This paper characterizes when Fano admissible manifolds admit Mabuchi solitons, providing an explicit formula for the Mabuchi constant and determining existence conditions over complex projective spaces.

Contribution

It establishes a precise criterion for the existence of Mabuchi solitons on Fano admissible manifolds based on the Mabuchi constant and generalizes Mabuchi's formula.

Findings

A Fano admissible manifold admits a Mabuchi soliton iff the Mabuchi constant is less than 1.

An explicit formula for the Mabuchi constant on Fano admissible manifolds is derived.

Complete classification of Mabuchi soliton existence on Fano admissible manifolds over complex projective spaces.

Abstract

In this paper, we study the existence of Mabuchi solitons on admissible manifolds as defined by Apostolov--Calderbank--Gauduchon--T\o nnesen-Friedman. We prove that a Fano admissible manifold admits a Mabuchi soliton if and only if the Mabuchi constant is less than 1. We also provide an explicit formula for the Mabuchi constant on Fano admissible manifolds, which generalizes that of Mabuchi. Using this formula, we completely determine the existence and non-existence of Mabuchi solitons on Fano admissible manifolds over the complex projective space $\mathbf{P}^{n}$.