# Deforming convex real projective structures

## TL;DR

This paper introduces two new geometric flows on the deformation space of convex real projective structures on surfaces, expanding the understanding of the structure's deformation dynamics and their interactions.

## Contribution

It defines the internal bulging and eruption flows, and demonstrates their properties and interactions with existing flows, providing new tools for studying deformation spaces.

## Key findings

- Eruption flows form a half-dimensional family of commuting flows.
- Internal bulging flow deforms internal parameters of structures.
- Flows are associated to pairs of pants in a decomposition.

## Abstract

Let S be a closed, connected, orientable surface of genus at least 2, and let C(S) denote the deformation space of convex real projective structures S. In this article, we introduce two new flows on C(S), which we call the internal bulging flow and the eruption flow. These are geometrically defined flows associated to each pair of pants in a pants decomposition on S that deform the internal parameters. We show that the eruption flows, together with the generalized twist flows about the pants curves, give rise to a half-dimensional family of commuting flows on C(S).