**Case 1:** Suppose we have a compact manifold without boundary. For any metric , define

where , and are the scalar curvatures under metric and . If , using the conformal invariance, one can verify that

Consider a special set of metrics which preserve the volume,

If we view all conformal metrics form a Banach manifold, then is a hypersurface. The tangent space at is

Choose the inner product on tangent space as

We will restrict functional on and still use to denote . We want to find a flow for every time which converges to some special metric. Then necessarily must belong to for .

Fix any . Suppose , , , small enough. Then

where . We will get

Note that

One can calculate

Here is the average scalar curvature. Note that . On the other hand, for any ,

So

Then we can construct a negative gradient flow, which is

By scaling on time variable, one immediately have the yamabe flow

**Case 2:** without boundary. Consider the following functional

equivalently

where is some fixed constant. One can verify that for any , there exists a unique such that

So there exists a notion of Nehari manifold, , converting to metric sense, we get a hypersurface

The tangent space at is

We also use the inner product as before

Fix , ,

**Fact**: for , if metric was changed to , then , .

Using the above fact, , small enough. Suppose

For simplicity on notations, let us use , recall that . Plugging in back to ,

where

Differentiating this, we get

Considering the middle two terms

from the defnition of . This implies

On the other hand we have

So we want and also holds for every .

It is easy to verify that if . So we have

Let us assume has the form , then the above equation is equivalent to

Recall , which means

using this relation one can simplify to be

The restirction will give us

This is equivalent to

combining and , we get

** Remark: For the second case, it is firstly Professor Yan Yan Li told me the idea to construct flow on Nehari manifold.**

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