Twisted and coupled constant scalar curvature K\"{a}hler metrics on minimal ruled surfaces

TL;DR

This paper investigates the existence and non-existence of twisted and coupled constant scalar curvature Kähler metrics on minimal ruled surfaces over genus 2 Riemann surfaces, providing bounds for related invariants.

Contribution

It presents new results on the existence criteria for twisted and coupled cscK metrics on minimal ruled surfaces, and establishes bounds for the Chen-Cheng invariant.

Findings

Existence of twisted cscK metrics under certain conditions

Non-existence of coupled cscK metrics on these surfaces

Bounds for the Chen-Cheng invariant related to the cscK problem

Abstract

In this paper, we study the existence of twisted constant scalar curvature K\"{a}hler (cscK) metrics and non-existence of coupled cscK metrics on minimal ruled surfaces over a Riemann surface of genus $2$. Moreover, we give a bound for the Chen-Cheng invariant related to Chen's continuity path for cscK problem on these ruled surfaces.