Numerical Solutions of Kahler-Einstein metrics on $P^2$ with conical singularities along a conic curve

TL;DR

This paper numerically constructs SO(3)-invariant Kahler-Einstein metrics on the complex projective plane with cone singularities along a conic, confirming theoretical angle bounds and the limit metric space.

Contribution

It provides the first numerical solutions for Kahler-Einstein metrics with conical singularities on P^2, validating theoretical angle constraints and the associated limit space.

Findings

Numerical solutions exist for cone angles in (rac{}, 2]

The limit metric space is identified as P(1,1,4)

Numerical results match theoretical predictions

Abstract

We solve for the SO(3)-invariant Kahler-Einstein metric on $\mathbb{P}^2$ with cone singularities along a smooth conic curve using numerical approach. The numerical results show the sharp range of angles ($(\pi/2,2\pi]$) for the solvability of equations, and the right limit metric space ($P(1,1,4)$). These results exactly match our theoretical conclusions.