On The Ellipticity of Generalised Monge-Amp\`ere Equations on Vector Bundles

TL;DR

This paper investigates the ellipticity properties of generalized Monge-Ampère equations on vector bundles over compact Kähler manifolds, revealing conditions under which ellipticity is preserved or lost.

Contribution

It provides a detailed analysis of ellipticity preservation for vector bundle Monge-Ampère type equations, highlighting the special case of the -equation.

Findings

Ellipticity is not preserved along continuity paths for most equations when dimension and rank are  or higher.

The -equation uniquely preserves ellipticity along continuity paths.

The study advances understanding of nonlinear geometric PDEs on vector bundles.

Abstract

In this paper, we study the ellipticity of the vector bundle versions of the Monge-Amp\`ere, $J$, dHYM and $\sigma_{k}$-equations at a point. These are nonlinear geometric partial differential equations defined on a holomorphic vector bundle over a compact K\"ahler manifold. We show that when both the dimension of the manifold and the rank of the bundle are greater than or equal to three, these equations do not preserve ellipticity along continuity paths in the connected component of the trivial solution. However, the $\sigma_{2}$-equation does preserve ellipticity along continuity paths.