Lagrangian self-similar solutions in gradient shrinking K\"ahler-Ricci solitons

TL;DR

This paper establishes diameter bounds and non-existence results for Lagrangian self-shrinkers and self-expanders within gradient shrinking Kähler-Ricci solitons, extending classical Euclidean space results to this geometric setting.

Contribution

It provides the first diameter estimate for Lagrangian self-shrinkers and proves non-existence of compact self-expanders in gradient shrinking Kähler-Ricci solitons.

Findings

Lower bound estimate for the diameter of Lagrangian self-shrinkers.

Non-existence of compact self-expanders in the setting.

Extension of Euclidean space results to Kähler-Ricci solitons.

Abstract

In this paper, we give a lower bound estimate for the diameter of a Lagrangian self-shrinker in a gradient shrinking K\"ahler-Ricci soliton as an analog of a result of A. Futaki, H. Li and X.-D. Li for a self-shrinker in a Euclidean space. We also prove an analog of a result of H.-D. Cao and H. Li about the non-existence of compact self-expanders in a Euclidean space.