therefore

If we use the polar coordinates , and and the following fact

then one can calculate the Hessian of under this coordinates

Then

]]>**Propsotion 2.4** Let be a domain in . is a smooth function on . Then

**Proof:** Take an orthonormal frame field such that is tangent to . Notice

where . It follows from Remark 2.3 that

where is the outward unit normal to . Changing the coordinates to and , we can get

It is easy to see

and

Therefore the proposition is established.

Remark: Robert Reilly, On the hessian of a function and the curvature of its graph

]]>The principle curvatures are taken from the eigenvalues of the Jacobian matrix . One can define its mean curvature using the notation of above

Now let us consider general matrix ,

where is the generalized Kronecker delta.

Then we define the Newton tensor

For any vector field on , is a matrix, where and , denote , we have

Since for any

because is skew-symmetric in . Then

Applying the formula (Check [1] Propsition 1.2) and , we get

It follows from the result of Reilly, Remark 2.3(a), that

Suppose is a vector field normal to , then

where is the outer ward unit normal to .

**Remark:** [1] R.C. Reilly, On the Hessian of a function and the curvatures of its graph., Michigan Math. J. 20 (1974) 373â€“383. doi:10.1307/mmj/1029001155.

[2] N. Trudinger, Apriori bounds for graphs with prescribed curvature.

We probably have to assume is a symmetric matrix in order to use the formula of Reilly. Not sure about this.

]]>This formula has more general forms.

]]>What is the unit normal to this radial graph?

Suppose is a smooth local frame on . Let be the covariant derivative on . Tangent space of consists of which are

In order to get the unit normal, we need some simplification. Let us assume are orthonormal basis of the tangent space of and . Then

Then we obtain an orthonormal basis of the tangent of

We are able to get the normal by projecting to this subspace

After normalization, the (outer)unit normal can be written

**Remark: **Guan, Bo and Spruck, Joel. *Boundary-value Problems on \mathbb{S}^n for Surfaces of Constant Gauss Curvature.*

Choose a local orthonormal frame such that , and in the neighborhood of some point . We want to change (1) to some expression on or . To do that, we need to apply both sides of (1) to , getting

where we have used . Similarly

Now let us calculate the Laplacian of second fundamental form

Then

Combining all the above estimates to (1), we get

where we write . Using this equation and the other one on , one can derive that if is mean convex then it is actually convex.

]]>Denote for short. If we pull the metric of back to , denote as , then

where and . Then one can use the local coordinate to calculate

also one can see from another definition of Laplacian

By using the expression of stated above, we can calculate

It follows from the definition of tangential derivative on , see, that

then

where is the mean curvature of the . Combining all the above calculations,

]]>

Denote and is the blow down map. On (near A), takes the form

where is the boundary defining function for ff and is boundary defining function for rb. Similarly on (near B), takes the form

where is a bdf for lb and is a bdf for ff. One can verify that is a diffeomorphism . Let . Then is a polyhomogeneous conormal function on . Its index can be denoted correspond to lb, ff and rb.

Suppose is the projection to coordinate. Consider , is actually a fibration. Then the push forward map maps to a polyhomogeneous function on .

In order to make is integrable, let us assume support and . What is the index for on ?

For the first integral, letting

For the second integral, letting

Since the Taylor series

Consequently

Combining all the above analysis, the index for is

From another point of view, the vanishing order of on each boundary hypersurface of are , and . maps lb and ff to the boundary of . Therefore the index of is contained in

**Remark:** Daniel Grieser, Basics of Calculus.

where . Then one can see , . Under this stereographic projection, a strip will be equivalent to some lens domain on the sphere.

Let us pull the standard metric of to the . For the following statement, we will always omit and . Calculation shows,

therefore

here we used the fact that . Suppose is the connection on equipped with the standard metric. We want to calculate . To that end, it is better to use the coordinates in

Since we know

where means the tangential part to and is the unit normal to . Using the connection in , we get

One can verify from the above equalities that

Similarly

]]>

What is the critical point of this area functional? To see that,

From some basic calculation in minimal surface(see colding minicozzi’s book)

Stokes’ theorem implies

Combining these above fact, we get

where is the unit normal of . Therefore the critical point of will satisfy

This is so called translating soliton.

Now consider the second variation at a translation soliton,

While

Suppose , where , then

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