The parabolic split-type Monge-Ampere on split tangent bundle surfaces

TL;DR

This paper introduces a parabolic analogue of the split-type Monge-Ampère equation on split tangent bundle surfaces, establishing long-time existence and convergence conditions with applications to Kähler geometry.

Contribution

It extends the elliptic split-type Monge-Ampère equation to a parabolic setting, providing new existence and convergence results for fully nonlinear, non-concave equations.

Findings

Proved long-time existence for certain exponent ranges.

Established convergence to twisted Monge-Ampère equations.

Applied results to Kähler geometry and curvature conditions.

Abstract

We introduce a parabolic analogue of the elliptic split-type Monge-Amp\`ere equation developed by Fang and the author, extending Streets' twisted Monge-Amp\`ere equation. The resulting equation is fully nonlinear and non-concave. We prove long-time existence for equations whose exponents are not too far apart and give conditions for convergence to the twisted Monge-Amp\`ere when the exponents approach each other. Applications include long-time convergence on K\"ahler backgrounds and reduction to the twisted Monge-Amp\`ere equation under curvature assumptions.