# Integral representation for fractional Laplace-Beltrami operators

## TL;DR

This paper derives an integral representation for the fractional Laplace-Beltrami operator on general Riemannian manifolds, with proofs for both compact and non-compact cases, enabling diverse applications in geometric analysis.

## Contribution

It introduces a novel integral representation of the fractional Laplace-Beltrami operator applicable to broad classes of Riemannian manifolds, with multiple proof techniques.

## Key findings

- Provides integral formulas for fractional Laplace-Beltrami operators
- Establishes proofs for compact and non-compact manifolds
- Enables new applications in geometric analysis

## Abstract

In this paper we provide an integral representation of the fractional Laplace-Beltrami operator for general riemannian manifolds which has several interesting applications. We give two different proofs, in two different scenarios, of essentially the same result. One of them deals with compact manifolds with or without boundary, while the other approach treats the case of riemannian manifolds without boundary whose Ricci curvature is uniformly bounded below.