In this note, we give a new proof for Perelman's scalar curvature and diameter estimates for the K & auml;hler-Ricci flow on Fano manifolds. The proof relies on a new Harnack estimate for a special family of functions in space-time. Our new approach initiates the work in Jian (Finite time singularities of the K & auml;hler-Ricci flow, preprint) for general finite time solutions of the K & auml;hler-Ricci flow.

Publication:

MATHEMATISCHE ANNALEN

http://dx.doi.org/10.1007/s00208-026-03487-5

Author:

Jian, Wangjian(corresponding author)

Chinese Acad Sci, Inst Math, Acad Math & Syst Sci, Beijing 100190, Peoples R China

Email address:wangjian@amss.ac.cn

Song, Jian

Rutgers State Univ, Dept Math, Piscataway, NJ 08854 USA

Email address:jiansong@math.rutgers.edu

Tian, Gang

Peking Univ, BICMR & SMS, Beijing 100871, Peoples R China

Email address:gtian@math.pku.edu.cn