# Boundary integral formula for harmonic functions on Riemann surfaces

## TL;DR

This paper develops a boundary integral formula for harmonic functions on Riemann surfaces embedded in complex projective space, extending classical Green's formula to more complex geometries.

## Contribution

It introduces a novel boundary integral representation for harmonic functions on Riemann surfaces embedded in ^2, generalizing Green's formula beyond planar domains.

## Key findings

- Provides an explicit boundary integral formula for harmonic functions on Riemann surfaces.
- Extends classical potential theory to complex Riemann surface settings.
- Enables new analytical techniques for harmonic functions on complex manifolds.

## Abstract

We construct a boundary integral formula for harmonic functions on open, smoothly-bordered subdomains of Riemann surfaces embeddable into $\C\P^2$. The formula may be considered as an analogue of the Green's formula for domains in $\C$.