# Analyticity and spectral properties of noncommutative Ricci flow in a matrix geometry

## TL;DR

This paper investigates the spectral behavior of the Laplacian under Ricci flow within a noncommutative matrix geometry, demonstrating the flow's analyticity in this setting and analyzing eigenvalue variations.

## Contribution

It establishes the analyticity of Ricci flow in a noncommutative matrix geometry and derives a first variation formula for Laplacian eigenvalues.

## Key findings

- Ricci flow is analytic in the noncommutative matrix geometry setting
- Derived a first variation formula for Laplacian eigenvalues
- Provided insights into spectral properties under noncommutative Ricci flow

## Abstract

We study a first variation formula for the eigenvalues of the Laplacian evolving under the Ricci flow in a simple example of a noncommutative matrix geometry, namely a finite dimensional representation of a noncommutative torus. In order to do so, we first show that the Ricci flow in this matrix geometry is analytic.