A new complete two-dimensional shrinking gradient K\"ahler-Ricci soliton

TL;DR

This paper establishes the existence and uniqueness of a complete shrinking gradient K"ahler-Ricci soliton with bounded scalar curvature on a specific complex surface, completing the classification in two dimensions.

Contribution

It proves the existence and uniqueness of a new complete shrinking gradient K"ahler-Ricci soliton on the blowup of 1, advancing the classification of such solitons in two complex dimensions.

Findings

Existence of a unique complete shrinking gradient K"ahler-Ricci soliton.

The soliton has bounded scalar curvature.

Classification of such solitons in two dimensions is complete.

Abstract

We prove the existence of a unique complete shrinking gradient K\"ahler-Ricci soliton with bounded scalar curvature on the blowup of $\mathbb{C}\times\mathbb{P}^{1}$ at one point. This completes the classification of such solitons in two complex dimensions.