# Harmonic maps with free boundary from degenerating bordered Riemann surfaces

## TL;DR

This paper analyzes the behavior of harmonic maps with free boundary conditions from degenerating bordered Riemann surfaces, establishing a generalized energy identity using Pohozaev constants, which advances understanding of their blow-up phenomena.

## Contribution

It introduces a new energy identity for harmonic maps with free boundary on degenerating surfaces, utilizing Pohozaev constants on collars, a novel approach in this context.

## Key findings

- Established a generalized energy identity for harmonic maps with free boundary.
- Analyzed blow-up behavior on degenerating collars of Riemann surfaces.
- Utilized Pohozaev type constants to understand energy concentration.

## Abstract

We study the blow-up analysis and qualitative behavior for a sequence of harmonic maps with free boundary from degenerating bordered Riemann surfaces with uniformly bounded energy. With the help of Pohozaev type constants associated to harmonic maps defined on degenerating collars, including vertical boundary collars and horizontal boundary collars, we establish a generalized energy identity.