Ngô Quốc Anh

Happy Lunar New Year, A Year of Dragon

Ngô Quốc Anh — Mon, 23 Jan 2012 04:05:44 +0000

A Hardy-Moser-Trudinger inequality: A conjecture by Wang and Ye

Ngô Quốc Anh — Sat, 31 Dec 2011 13:48:48 +0000

Let denote the standard unit disk in . The famous Moser–Trudinger inequality says that

holds. There is another important inequality in analysis, the Hardy inequality which claims that

holds. The one is usuall called the Hardy functional. One can immediately see that

for any . Recently, in a paper accepted in Advances in Mathematics journal, Wang and Ye proved that there exists a constant 0' title='C_0 >0' class='latex' /> such that the following

where is the unit ball in , and is the complement of with respect to the following norm .

Let us go back to the case . They then defined

0,\quad \forall u \in H_0^1(\Omega )\backslash \{ 0\}' title='\displaystyle {H_d}(u) = \int_\Omega {|\nabla u{|^2}dx} - \frac{1}{4}\int_\Omega {\frac{{{u^2}}}{{d{{(x,\partial \Omega )}^2}}}dx} > 0,\quad \forall u \in H_0^1(\Omega )\backslash \{ 0\}' class='latex' />

where is a regular, bounded and convex domain sitting in . They then conjectured that the following

still holds for some constant 0' title='C(\Omega)>0' class='latex' /> where denotes the completion of with the corresponding norm associated with . Apparently, the conjecture holds true for .

Conformal compactification

Ngô Quốc Anh — Tue, 15 Nov 2011 16:49:20 +0000

Start with a pseudo-Riemannian manifold , let be another pseudo-Riemannian metric on , we say that and are conformal if there exists a positive scalar function on such that (sufficient smoothness of the relevant quantities are always assumed).

Observe that two conformal metrics measure angles the same way: recall that on a pseudo-Riemannian manifold , given a point and two non-null vectors , the angle between the vectors can be defined by

(Notice that on Euclidean space, if form an angle , then .) Thus if is conformal to , they define the same angles

In fact, this inference works the other way too. If , are two pseudo-Riemannian metrics such that for any two vectors we have , then , are conformal (up to a change of sign) by the above definition (see e.g. Exercise 14, Chapter 2 from B.O’Neill, Semi-Riemannian Geometry).

So, in plain English, two metrics are conformal if they measure angles the same way.

Now, let be a pseudo-Riemannian manifold that is non-compact. A conformal compactification of is a choice of a metric such that can be isometrically embedded into a compact domain of a pseudo-Riemannian manifold (well, I am ignoring some regularity issues here). Let be the conformal factor as before. Then observe that any regular extension of to the conformal boundary must vanish on said boundary. This reflects the property of a conformal compactification that “brings infinity to a finite distance”.

The simplest example of conformal compactification is the one-point compactification of Euclidean space via the stereographic projection. In this case, the target manifold is compact itself, taken to be standard sphere. The source manifold is Euclidean space with the standard metric, and the image set is taken to be the sphere minus the north pole.

[Source]

A blowup proof of the Aubin theorem in the Yamabe problem

Ngô Quốc Anh — Tue, 08 Nov 2011 05:29:21 +0000

Yamabe’s approach was to consider first the perturbed functional

where

Set

By using a direct minimizing procedure, it can be shown that for , there exists a smooth positive function such that its -norm is equal to one, , and satisfies the equation

The direct method does not work when because the Sobolev embedding is continuous but not compact. However, if one can show that is uniformly bounded, i.e. there exists a positive constant such that in for , then there exists a sequence such that and converges to a smooth positive function which satisfies the Yamabe equation .

We discuss a blow-up argument. Suppose that no such upper bound exists. It follows that there exist sequences and such that

As is compact, we may assume that as . For a normal coordinate system centered at and with radius , let the coordinates of be , . In the local coordinates,

From the equation, we know that satisfies

The idea here is to consider the normalized function

where . We have and as . Here is defined on a ball in of radius as . Obviously,

Under the choice of , we have

where those limits are taken as . By the argument of diagonal subsequence and the property of normal coordinates, one observes that a subsequence of converges to a smooth positive function which is a nonnegative solution of the equation

where , and is the standard Laplacian on . By the strong maximum principle, 0' title='v>0' class='latex' />. It is known that

where is an invariant depending only on the conformal class of the metric . Let be the diameter of . By a change of variables we have

where denotes the open ball in with center at and radius equal to . we note that

0\quad \text{and} \frac{2s_k}{s_k-2}-n \to 0\quad\text{as}\;k\to\infty.' title='\displaystyle \frac{2s_k}{s_k-2}-n>0\quad \text{and} \frac{2s_k}{s_k-2}-n \to 0\quad\text{as}\;k\to\infty.' class='latex' />

From (2) the Fatou lemma and , we obtain

A similar argument implies

Let be a cutoff function satisfies

Defined , then

Multiplies (1) by and integration by parts, we obtain

Taking in above equation and thanks to (4) we get

If , then , and (3) implies , which is a contradiction with 0' title='v>0' class='latex' />.
If 0' title='\lambda>0' class='latex' />, . (2) (5) and the best Sobolev imbedding implies

Thus

We are led to the contradiction with

Therefore, is uniformly bounded.

[Source]

MuPad: Heart in 3D

Ngô Quốc Anh — Fri, 04 Nov 2011 16:26:49 +0000

This is not mathematics. I just found an equation so that we can draw a heart in 3D. Indeed, the following equation

will generate a heart. I have tried and the following pictures show that fact.

The MuPad code I have used is the following

plot(plot::Implicit3d((x^2+9/4*y^2+z^2-1)^3-x^2*z^3-9/80*y^2*z^3=0, x = -1.3 .. 1.3, y = -1.3 .. 1.3, z = -1.3 .. 1.3, Mesh = [21, 10, 10], AdaptiveMesh = 3), Axes = None, Scaling = Constrained)

I have also made an GIF file, you can watch it from here .

An ODE appearing in the Nirenberg problem

Ngô Quốc Anh — Tue, 01 Nov 2011 12:21:43 +0000

It is well-known that the simplest form of the Nirenberg problem is equivalent to solving the following PDE

in . Using stereographic projection, one can see that the above PDE is equivalent to

in . If we assume that the solution has finite energy in the sense that

it is well-known that the preceding PDE has unique radial solution. In terms of ODE language, our PDE can be rewritten as

The purpose of this note is to find solutions to the above ODE. Our approach consists of several steps as shown below.

Step 1. Let . We then have which implies that

Substituting this into the ODE yields

Let , then we immediately have , thus giving us .

Step 2. In order to solve the latter ODE involving , we multiply both sides by . Hence, we arrive at

and by integrating, we know that

Thus,

since is non-negative and is positive. In other words,

which yields

We again integrate to get

which obviously helps us to write down the following

Solving for gives

Hence,

or equivalently,

Step 3. Now we condition our ODE with initial condition, say, and . Then it immediately yields

that is . We now use the condition to find . We first have

which implies that

Therefore, we can write

Locally conformally flat manifolds and Weyl and Cotton tensors, 2

Ngô Quốc Anh — Fri, 07 Oct 2011 19:27:18 +0000

The purpose of this note is to prove the following result that left in the previous entry

Lemma. Provided the Weyl tensor vanishes, equation

is locally solvable if and only if the following integrability condition is satised

That is, if and only if the Cotton tensor vanishes.

Proof. It is necessary and suffcient to find a 1-form locally such that

where is a symmetric 2-tensor depending only on and . To see this, by the symmetry of the RHS, we have

which implies . Thus locally is the exterior derivative of some function . Thus solves the equation. From the equation, we have

Suppose and that the coordinates is dened in a neighborhood of . The Frobenius theorem says a necessary and suficient condition to locally solve

with for any is the following integrability condition arising from

that

More invariantly, the integrability condition arises from

and is

where we have used and

From the denition of , we obtain

Therefore, we have

which implies

Locally conformally flat manifolds and Weyl and Cotton tensors

Ngô Quốc Anh — Tue, 04 Oct 2011 12:45:03 +0000

The purpose of this note is to prove the following

Theorem. A Riemannian manifold is locally conformally flat if and only if

for , the Weyl tensor vanishes;

for , the Cotton tensor vanishes.

To this purpose, let us briefly recall some definitions

The Weyl tensor. The Weyl tensor can be defined using the following formula

where and denotes the Kulkarni–Nomizu product of two symmetric (0,2) tensors. Writing the Weyl tensor in this way means that the Weyl tensor is actually a (0,4) tensor. It can be seen that the Weyl tensor can be rewritten in this form

where the part

is called the Schouten tensor. We have the following result

Proposition. If , then

where

The Cotton tensor. The (0,3) tensor above is called the Cotton tensor. Apparently, if either the Weyl tensor or the Ricci tensor vanishes, so does the Cotton tensor.

The Weyl and Cotton tensors under the conformal changes of metric. It is well-known that these tensors are invariant under the conformal changes of metric, that is,

under the change for some smooth function (see this note).

The Riemmanian curvature tensor under the conformal changes of metric. We list here the following rule

See this note for further details.

Locally conformally flat manifolds. Roughly speaking, this is to say at each point , there exists a neighborhood of such that the conformal class of contains the flat metric in , that is to say

We are now in a position to prove the theorem.

Proof of Theorem. We first assume that is locally conformally flat, that is, . If , using the formula for we have

since the Riemmanian curvature tensor vanishes. If , we use the formula for , we obtain

since the Ricci tensor vanishes.

Conversely, if the Weyl tensor vanishes, we have

Under the conformal change, for some , we have

Since the mapping given by is injective, it suffices to show that the following equation

is locally solvable. This can be done using the following whose proof is postponed.

Lemma. Provided the Weyl tensor vanishes, equation

is locally solvable if and only if the following integrability condition is satised

That is, if and only if the Cotton tensor vanishes.

Recall that when ; the condition follows from the Weyl tensor vanishes. This concludes the proof.

Source

Concentration-Compactness principle, II

Ngô Quốc Anh — Wed, 28 Sep 2011 01:35:01 +0000

In this entry, we continue to talk about the Concentration-Compactness Principle discovered by P.L. Lions [here]. In the previous entry, we already discussed two forms of non-compactness due to unbounded domains. Here we discuss what happens when passing to the limit on those functionals along weakly convergent subsequences.

Theorem (Lions). Let be sequence in weakly convergent to and such that

converges weak* to a nonnegative measure ,

converges weak* to a nonnegative measure .

Then there exists an at most countable index set , sequence , , , , such that

and

and

where is the best Sobolev constant and are Dirac measures assigned to . If and

then is a singleton and

for some .

Apparently, the theorem does not provide any information about possible loss of mass at infinity of a weakly convergent minimizing sequence. We shall consider that case in the forthcoming topic.

The Riemannian Penrose inequality

Ngô Quốc Anh — Sun, 04 Sep 2011 17:43:35 +0000

In mathematical general relativity, the Penrose inequality, first conjectured by Sir Roger Penrose, estimates the mass of a spacetime in terms of the total area of its black holes and is a generalization of the positive mass theorem. The Riemannian Penrose inequality is the most important special case. Specifically, if is an asymptotically flat Riemannian 3-manifold with nonnegative scalar curvature and ADM mass , and is the area of the outermost minimal surface (possibly with multiple connected components), then the Riemannian Penrose inequality asserts

This is purely a geometrical fact, and it corresponds to the case of a complete three-dimensional, space-like, totally geodesic submanifold of a (3 + 1)-dimensional spacetime. Such a submanifold is often called a time-symmetric initial data set for a spacetime. The condition of having nonnegative scalar curvature is equivalent to the spacetime obeying the dominant energy condition.

This inequality was first proved by Gerhard Huisken and Tom Ilmanen in 1997 [here and here] in the case where is the area of the largest component of the outermost minimal surface. Their proof relied on the machinery of weakly defined inverse mean curvature flow, which they developed. In 1999, Hubert Bray [here] gave the first complete proof of the above inequality using a conformal flow of metrics. Both of the papers were published in 2001 in the Journal of Differential Geometry.

Source: Wiki