# Weak Geodesics for the deformed Hermitian-Yang-Mills equation

## TL;DR

This paper investigates weak geodesics in the space of potentials for the deformed Hermitian-Yang-Mills equation, using nonlinear duality theory to construct continuous solutions to a degenerate elliptic equation.

## Contribution

It introduces a novel approach employing nonlinear Dirichlet duality theory to solve the degenerate elliptic geodesic equation in this context.

## Key findings

- Constructed continuous solutions to the Dirichlet problem for weak geodesics.
- Utilized convexity of level sets of the Lagrangian angle operator.
- Applied nonlinear duality theory to a degenerate elliptic equation.

## Abstract

We study weak geodesics in the space of potentials for the deformed Hermitian-Yang-Mills equation. The geodesic equation can be formulated as a degenerate elliptic equation, allowing us to employ nonlinear Dirichlet duality theory, as developed by Harvey-Lawson. By exploiting the convexity of the level sets of the Lagrangian angle operator in the highest branch, we are able to construct continuous solutions of the associated Dirichlet problem.