# Capillary Immersions in E($\kappa$,${\tau}$)

## TL;DR

This paper classifies capillary disks in homogeneous spaces E(κ,τ) using Codazzi pairs, extending prior classifications of constant mean curvature disks in product spaces.

## Contribution

It introduces a classification of capillary disks in E(κ,τ) spaces based on Codazzi pairs, generalizing previous results in specific product spaces.

## Key findings

- Classification of capillary disks in E(κ,τ) spaces.
- Extension of constant mean curvature disk classifications.
- Use of Codazzi pairs related to Abresch-Rosenberg differential.

## Abstract

In [3] and [11] the authors showed the existence of a Codazzi pair defined on any constant mean curvature surface in the homogeneous spaces E($\kappa$,$\tau$) associated to the Abresch-Rosenberg differential. In this paper, we use the mentioned Codazzi pair to classify capillary disks in E($\kappa$,$\tau$). As a consequence, the results presented in this paper generalize the previous classification of constant mean curvature disks in the product spaces $\mathbb{S}^2 \times \mathbb{R}$ and $\mathbb{H}^2 \times \mathbb{R}$ in [4] and [5].