# Superfluid Vortex Dynamics on Planar Sectors and Cones

## TL;DR

This paper investigates the behavior of superfluid vortices on curved conical surfaces by mapping the problem to planar sectors, revealing vortex precession and interactions influenced by the cone's geometry.

## Contribution

It introduces a conformal mapping approach to analyze vortex dynamics on cones, extending known planar and cylindrical results to conical geometries.

## Key findings

- Vortices precess uniformly at fixed distances from the cone apex.
- Interaction energy between two vortices on a cone is derived.
- Results recover known dynamics on cylinders and annuli in certain limits.

## Abstract

We study the dynamics of vortices formed in a superfluid film adsorbed on the curved two-dimensional surface of a cone. To this aim, we observe that a cone can be unrolled to a sector on a plane with periodic boundary conditions on the straight sides. The sector can then be mapped conformally to the whole plane, leading to the relevant stream function. In this way, we show that a superfluid vortex on the cone precesses uniformly at fixed distance from the apex. The stream function also yields directly the interaction energy of two vortices on the cone. We then study the vortex dynamics on unbounded and bounded cones. In suitable limits, we recover the known results for dynamics on cylinders and planar annuli.