**1. Preliminary **

Suppose is a Riemannian manifold with . is the Schouten tensor

and denote . Define

where . It is well known that is conformally invariant.

Suppose where is a symmetric 2-tensor. We want to calculate the first derivative of at . To that end, let us list some basic facts (see the book of Toppings). Firstly denote the divergence operator and

where is the Lichnerowicz Laplacian. Then the first variation of Ricci curvature and scalar curvature are

where we were using upper dot to denote the derivative with respect to .

**2. First variation of the sigma2 functional **

*Proof:*

where . Since we have

Plugging this into the derivative of to get

In order to simplify the above equation, we recall the definition of Lichnerowicz Laplacian

Apply (1) to get

Therefore we can simplify it to be

Let us denote . Using the fact and the definition of ,

where

Remark 1It is easy to verify , this is equivalent to say is invariant under conformal change. More precisely, letting , then

Remark 2If is an Einstein metric with , then , and

It is easy to verify that . In other words, Einstein metrics are critical points of .

**Are there any non Einstein metric which are critical points of ?**

Here is one example. Suppose , where is the sphere with standard round metric and is a two dimensional compact manifolds with sectional curvature . is endowed with the product metric. We can prove , , , and consequently .

Note that the above example is a locally conformally flat manifold. For this type of manifold, we have the following lemma which can say

Lemma 2Suppose is locally conformally flat and , then

*Proof:* When is locally conformally flat,

is equivalent to

Actually this is equivalent to the Bach tensor is zero.

**3. Another point of view **

We have the Euler Characteristic formula for four dimensional manifolds

therefore the critical points for will be the same as the critical points of . However, the functional

is well studied by Bach. The critical points of this functional satisfy Bach tensor equal to 0.

Obviously, for Einstein metric, but not all Bach flat metrics are Einstein. For example for any locally conformally flat manifolds.

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- For every and , define

It is well know that . Moreover is isometric to the standard sphere minus one point.

- For any and define

where is the geodesic distance of and on . Actually

- For each , define by

Both the second and third one satisfy

If we make , the third one will be changed to the second one.

To get the second one from the first one, let us deonte be the stereographic projection from point . Then

It should be able to see the third one from hyperbolic translation directly.

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Suppose is a smooth domain in , and is a harmonic function in . If there exists such that

for small, then . If , the same conclusion holds for the solutions of a general second order linear elliptic equation.

A borderline example for this theorem is be the real part of , . is harmonic in the right half plane and and consequently .

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when , Heinz proved

when , Pogorelov has a counter example. One can have a solution , but for some . See his book *The Minkowski Multidimensional Problem, *on page 83.

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One can define the th Newton transformation as the following

This means

By comparing coefficients of , we get the relations of

Induction shows

For example

One of the important property of Newton transformation is that: Suppose , then

The is because

If , then is positive definite and therefore is elliptic.

**Remark:** Hu, Z., Li, H. and Simon, U. . Schouten curvature functions on locally conformally flat Riemannian manifolds. Journal of Geometry, 88(12), (2008), 75100.

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Proof: Suppose

Then one can derive the following equation

Multiplying both sides by and integration on , which makes sense because of the polynomial volume growth, we get

However, integration by parts shows the LHS is zero. Thus , must be a plane.

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and Einstein tensor and gravitational tensor

Suppose is the elemantary symmetric function

Thinking of as a tensor of type . is defined as applied to eigenvalues of . Then

Notice . Easy calculation reveals that

Under conformal change of metric , we have

We want to solve the equation , which is equivalent to solve

This is an fully nonlinear equation of Monge-Ampere type. Under local coordinates, the above equation can be treated as

where . This equation is elliptic if the matrix is positive definite. In order to find that matrix, we need the linearized operator

Using the elementary identity

for any fixed matrix and . Now plug in is Schouten tensor and . One can calculate them as

Then we get

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for all real number . One can see that represents the mean curvature of , is the gauss-Kronecker curvature. can reflect the scalar curvature of on the condition that the ambient manifold is a space form.

We want to study the consequence of moving the hypersurface parallel. Namely, define to be

When is small enough, is well defined immersed hypersurface. Suppose are principle directions at a point of , then

here we identify as abbreviation. This implies that is also an unit normal field of . The area element will be

The second fundamental form of with respect to will be

for all tangent vector fields on . Plugging in and , we get

So are also principle directions for and principle curvatures are

Another way to see this is by choosing a geodesic local coordinates such that are the principle directions of at . Then

Since at . Therefore we get the principle curvature are .

Therefore the mean curvature for is

Since we have identity

which implies

Plugging in all the information,

Reorder the terms in the above identity by the order of , we get

One can use this to prove Heintze-Karcher inequality. There are Minkowski formula in Hyperbolic space and also.

**Remark:** S. Montiel and Anotnio Ros, compact hypersurfaces: the alexandrov theorem for higher order mean curvatures. Differential Geometry, 52, 279-296

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There is a BCN conjecture related to this

**BCN: **If is Lipschitz, is a smooth(in fact, Lipschitz) connected domain with connected where OEP admits a bounded solution, then must be either a ball, a half space, a generalized cylinder or the complement of one of them.

BCN is false in . Epsinar wih Mazet proved BCN when . This implies the Shiffer conjecture in dimension 2. In higher dimension of Shiffer conjecture, if we know the domain is contained in one hemisphere of , then one can use the equator or the great circle to perform the moving plane.

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**Theorem: **If and , then , which is the hardy space.

Given , it has harmonic extension

**Definition:** the non-tangential maximal function

It is easy to prove that the Hardy-Littlewood maximal function. From this we can Hardy norm as

Hardy space consists of all having finite hardy norm. There is well know fact that the dual space of is BMO, which is defined as the following.

Define , if for any cube ,

then . and but not in .

Let us see how do we use the main theorem. Suppose on , is the solution of the following elliptic equation

where , and is uniform elliptic. YanYan Li and Sagun Chanillo proved that the green function of this elliptic operator belongs to BMO. The right hand side of this equation can be rewritten as , where

therefore the right hand side belong to . Since

therefore from the theorem we stated at the beginning, we get

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