Ngô Quốc Anh

January 31, 2014

2014 Vietnamese New Year, aka Tết

Filed under: Linh Tinh — Ngô Quốc Anh @ 1:55

tet

Vietnamese New Year, more commonly known by its shortened name Tết or Tết Nguyên Đán, is the most important and popular holiday and festival in Vietnam. It is the Vietnamese New Year marking the arrival of spring based on the Chinese calendar, a lunisolar calendar. For those who do not know about Tết, please read an article in wikipedia for details.

At the first moment of the new year, I wish you a good health and prosperity all year round and thank you for your interest in my blog.

January 17, 2014

Short form of the Yamabe invariant on compact manifolds with boundary

Filed under: Uncategorized — Tags: — Ngô Quốc Anh @ 7:10

Suppose that (M,g) is a compact Riemannian manifold with boundary \partial M. Let N be an outer unit normal vector field to the boundary \partial M.

Using notation introduced in a previous note, the unnormalized mean curvature H_g computed using the trace of the associated second fundamental form \mathrm{I\!I} obeys the following conformal change rule

\displaystyle {H_{\widehat g}} = {\phi ^{ - 2/(n - 2)}} \bigg( {H_g} + \frac{{2(n - 1)}}{{n - 2}}\frac{{{\nabla _N}\phi }}{\phi } \bigg).

where the conformal metric \widehat g in terms of the background metric g is defined to be \widehat g =\phi^{4/(n-2)}g.

TangentSpace

Following the same strategy for the closed case, Escobar found the following invariant, still named Yamabe invariant, as follows

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January 8, 2014

Conformal metric having strictly negative scalar curvature in a given region

Filed under: PDEs, Riemannian geometry — Ngô Quốc Anh @ 0:39

Let (M,g) be a closed (compact without boundary) Riemannian manifold of dimension 3 with a C^2 metric g of arbitrary Yamabe class. Suppose \Omega \subset M be an open subset of M with regular boundary \partial\Omega and with M\backslash (\Omega \cup \partial \Omega ) non-empty and open in M.NegativeScalarCurvatureInOmega

In this note, we mention a very interesting result basically due to O’Murchadha-York from here and Isenberg from here. The result says that there exists a conformal metric \widehat g \in [g] such that

\displaystyle \text{Scal}_{\widehat g} < -\xi<0

in \Omega for some constant \xi>0 to be specify later. The novelty of this result is that although the metric g may be of positive Yamabe class which tells us that it is impossible to construct a conformal metric which is everywhere negative, it is possible to make it negative in a proper subset of M. A proof for this result goes as follows:

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January 4, 2014

A Picone type identity for bi-Laplacian

Filed under: PDEs — Tags: — Ngô Quốc Anh @ 22:22

Simultaneously, I have recently found the following identity in the same fashion of the Picone identity for \Delta. It says that

\displaystyle \left( \Delta u -\frac uv \Delta v\right)^2-\frac {2\Delta v}{v}\left| \nabla u - \frac uv \nabla v\right|^2 = |\Delta u|^2 -\Delta \left( \frac {u^2}v\right)\Delta v

for any function v \ne 0. It is worth noticing that the original Picone identity says that

\displaystyle \left| \nabla u - \frac{u}{v}\nabla v\right|^2= \left|\nabla u\right|^2 - \nabla \left( \frac{u^2}{v} \right) \cdot \nabla v \geqslant 0

for any function v \ne 0. It turns out that a few days ago, this identity appeared in a recent notes by Dwivedi  and Tyagi, see Lemma 2.1 from here. The extra term

\displaystyle\frac {2\Delta v}{v}\left| \nabla u - \frac uv \nabla v\right|^2

naturally appears since it only involves up to third order derivatives. However, to compare this term with 0, we only need to assume that \Delta v has a fixed sign. To see how this identity could be useful, let us consider the following equation

\displaystyle (-\Delta)^2 u +hu= f u ^\frac {n+4}{n-4}, \quad u>0, h<0

naturally arises from prescribing Q-curvature in Riemannian manifolds of dimension n \geqslant 5.

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