# Flow method to prove existence of certain metrics

## Contents

## Description

### Problem statement

We are given a differential manifold , and we want to know whether there exists a Riemannian metric on such that satisfies a certain property of interest to us. This property could, for instance, be:

- The property of being a constant-curvature metric
- The property of being an Einstein metric
- The property of being a Ricci soliton

### Approach

The idea is as follows:

- Construct a differential operator on the space of sections of (0,2)-tensor fields such that the points which map to zero under that operator are precisely those satisfying the given property.
- Consider the flow equation associated with that differential operator. Note that this will be a vector flow equation, rather than a scalar one.
- Try to locate a trajectory for that flow equation for which the limit at infinity is well-defined.

## The Ricci-flat case

### The differential operator

A Ricci-flat metric is defined as a metric whose Ricci curvature is zero everywhere. Thus, a natural choice of differential operator is the Ricci curvature tensor viewed as an operator that inputs the metric tensor and outputs the Ricci curvature tensor, both living in the space of (0,2)-tensor fields on the differential manifold.

However, for various reasons of normalization, the differential operator that we actually take is (-2) times the Ricci curvature tensor.

### The flow equation

The flow equation we get from the above is:

This is termed the Ricci flow.

## The Einstein case

*Fill this in later*