# Equivariant Schr\"odinger maps from two dimensional hyperbolic space

## TL;DR

This paper studies the existence and global behavior of equivariant Schr"odinger maps from two-dimensional hyperbolic space to the sphere, establishing local existence for energies below 4π and global solutions for sufficiently small energies.

## Contribution

It proves local existence for Schr"odinger maps with energy less than 4π and global existence for small initial data in hyperbolic space, extending previous results to this setting.

## Key findings

- Local existence for energy < 4π
- Global solutions for sufficiently small energy
- Convergence to the north pole at origin and infinity

## Abstract

In this article, we consider the equivariant Schr\"odinger map from $\Bbb H^2$ to $\Bbb S^2$ which converges to the north pole of $\Bbb S^2$ at the origin and spatial infinity of the hyperbolic space. If the energy of the data is less than $4\pi$, we show that the local existence of Schr\"odinger map. Furthermore, if the energy of the data sufficiently small, we prove the solutions are global in time.