Ngô Quốc Anh

March 29, 2014

A new Rayleigh-type quotient for the conformal Killing operator on manifolds with boundary

Filed under: Riemannian geometry — Tags: — Ngô Quốc Anh @ 15:22

In the previous note, I showed a Rayleigh-type quotient for the conformal Killing operator \mathbb L on manifolds (M,g) with boundary \partial M, i.e. the following result holds:

Whenever M admits no non-zero conformal Killing vector fields, the following holds

\displaystyle C_g(M)=\inf \frac{{{{\left( {\int_M {|\mathbb LX|^2 d{v_g}} } \right)}^{1/2}}}}{{{{\left( {\int_M {|X|^{2n/(n - 2)}d{v_g}} } \right)}^{(n - 2)/(2n)}}}} > 0

where the infimum is taken over all smooth vector fields X on M with X \not\equiv 0.

Today, I am going to prove a slightly stronger version of the above inequality, namely, when some terms on the boundary \partial M take part in. Precisely, we shall prove

Whenever M admits no non-zero conformal Killing vector fields, the following holds

\displaystyle C_g(M,\partial M)=\inf \frac{{{{\left( {\int_M {|\mathbb LX|^2 d{v_g}} } \right)}^{1/2}}}}{{{{\left( {\int_M {|X|^\frac{2n}{n - 2}d{v_g}} } \right)}^\frac{n - 2}{2n}}} + \left( \int_{\partial M}|X|^\frac{2(n-1)}{n-2}ds_g\right)^\frac{n-2}{2(n-1)}} > 0

where the infimum is taken over all smooth vector fields X on M with X \not\equiv 0.

However, a proof for this new inequality remains the same. To do so, we first make use of some Sobolev embeddings as follows:

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October 20, 2013

Rayleigh-Type Quotient For The Conformal Killing Operator on manifolds with boundary

Filed under: Uncategorized — Tags: — Ngô Quốc Anh @ 22:50

Following the previous note, today we discuss a similar Rayleigh-type quotient for the conformal Killing operator \mathbb L on manifolds (M,g) with boundary. We also prove that

whenever M admits no non-zero conformal Killing vector fields, the following holds

\displaystyle\inf \frac{{{{\left( {\int_M {|\mathbb LX|^2 d{v_g}} } \right)}^{1/2}}}}{{{{\left( {\int_M {|X|^{2n/(n - 2)}d{v_g}} } \right)}^{(n - 2)/(2n)}}}} > 0

where the infimum is taken over all smooth vector fields X on M with X \not\equiv 0.

Since the Bochner-type formula for the conformal Killing operator on manifolds without boundary, i.e.

\displaystyle \frac{1}{2}\int_M |\mathbb L X|^2 dv_g= \int_M |\nabla X|^2 dv_g + \left( 1-\frac{2}{n}\right)\int_M |{\rm div}X|^2 dv_g - \int_M {\rm Ric}(X,X)dv_g,

is no longer available, we use a new approach in order to estimate \int_M |\mathbb L X|^2 dv_g from below. To this purpose, we make use of a Riemannian version for the Korn inequality recently proved by S. Dain [here].

First, in view of Corollary 1.2 in Dain’s paper, the following inequality holds

\displaystyle \int_M |\nabla X|^2 dv_g \leqslant C \left( \int_M |X|^2 dv_g + \int_M |\mathbb LX|^2 dv_g \right)

for some positive constant C independent of X. This helps us to conclude that

\displaystyle C\int_M |\mathbb LX|^2 dv_g \geqslant \|X\|_{H^1}^2 - (C+1)\|X\|_{L^2}^2

as in Dahl et al’ paper. Therefore, we can argue by contradiction by assuming that there exists a sequence of vector fields \{X_k\}_k \in H^1(M) such that

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October 13, 2013

Rayleigh-type quotient for the conformal Killing operator

Filed under: Riemannian geometry — Tags: — Ngô Quốc Anh @ 10:05

Suppose (M,g) is a compact Riemannian manifold without boundary of dimension n \geqslant 3. We further assume that M admits no conformal Killing vector fields.

In this entry, we discuss a beautiful result due to Dahl-Gicquaud-Humbert recently published in Duke Math. J. They proved that

\displaystyle\inf \frac{{{{\left( {\int_M {|\mathbb LX|^2 d{v_g}} } \right)}^{1/2}}}}{{{{\left( {\int_M {|X|^{2n/(n - 2)}d{v_g}} } \right)}^{(n - 2)/(2n)}}}} > 0

where the infimum is taken over all smooth vector fields X on M with X \not\equiv 0.

Their proof goes as follows: First by the compactness of M, {\rm Ric} \leqslant \lambda g for some constant \lambda. We now use the the Bochner-type formula for the conformal Killing operator on manifolds without boundary, i.e.,

\displaystyle \frac{1}{2}\int_M |\mathbb L X|^2 dv_g= \int_M |\nabla X|^2 dv_g + \left( 1-\frac{2}{n}\right)\int_M |{\rm div}X|^2 dv_g - \int_M {\rm Ric}(X,X)dv_g ,

which now yields

\displaystyle \frac{1}{2}\int_M |\mathbb L X|^2 dv_g \geqslant \int_M |\nabla X|^2 dv_g - \lambda\int_M |X|^2 dv_g,

thanks to 1-\frac{2}{n}>0. Using the standard norm for H^1(M), we rewrite the preceeding inequality as follows

\displaystyle \frac{1}{2}\int_M |\mathbb L X|^2 dv_g \geqslant \|X\|_{H^1}^2 - (\lambda+1)\|X\|_{L^2}^2.

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October 12, 2013

Bochner-type formula for the conformal Killing operator on manifolds with boundary

Filed under: Uncategorized — Tags: , — Ngô Quốc Anh @ 8:39

Given a Riemannian manifold (M,g) without boundary, in the previous note, we derived a Bochner-type formula for the conformal Killing operator \mathbb L. Precisely, we obtained

\displaystyle \frac{1}{2}\int_M |\mathbb L X|^2 dv_g= \int_M |\nabla X|^2 dv_g + \left( 1-\frac{2}{n}\right)\int_M |{\rm div}X|^2 dv_g - \int_M {\rm Ric}(X,X)dv_g

or equivalenlty,

\displaystyle \frac{1}{2}\int_M |\mathbb LX |^2 dv_g=- \int_M {({g^{ij}}{\nabla _i}{\nabla _j}{X^h}){X_h} - {\rm Ric}(X,X)d{v_g}}+\left( 1-\frac{2}{n}\right)\int_M |{\rm div}X|^2 dv_g.

Today, we try to derive a similar formula for the operator \mathbb L assuming the manifold has boundary \partial M.  Our starting point again is the Bochner formula for vector fields mentioned here, i.e.

\displaystyle\frac{1}{2}\Delta (|X|^2) = |\nabla X|^2 + {\rm div}({\mathbb L_X}g)(X) - {\nabla _X}{\rm div}X - {\rm Ric}(X,X).

Using this and the formula for \Delta (|X|^2) that we derived here, we arrive at

\displaystyle\frac{1}{2}{\Delta _g}(|X{|^2}) = ({g^{ij}}{\nabla _i}{\nabla _j}{X^h}){X_h} + |\nabla X{|^2}

which now yields

\displaystyle -{\rm div}({\mathbb L_X}g)(X)=-({g^{ij}}{\nabla _i}{\nabla _j}{X^h}){X_h}- {\nabla _X}{\rm div}X - {\rm Ric}(X,X).

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October 11, 2013

Bochner-type formula for the conformal Killing operator on manifolds without boundary

Filed under: Uncategorized — Tags: , — Ngô Quốc Anh @ 9:59

Assuming the Riemannian manifold (M,g) is compact without boundary. In the previous post, we showed that

\displaystyle \frac{1}{2}\int_M |\mathbb L_X g|^2 dv_g= \int_M |\nabla X|^2 dv_g +\int_M |{\rm div}X|^2 dv_g - \int_M {\rm Ric}(X,X)dv_g .

Today, we use the above formula to derive a Bochner-type formula for the conformal Killing operator \mathbb L given by

\displaystyle \mathbb L X = \mathbb L_Xg - \frac{2}{n}\text{div}(X)g.

Clearly, for any vector field X,

\begin{array}{lcl} |\mathbb{L}X{|^2} &=& \displaystyle \Big|{\mathbb{L}_X}g - \frac{2}{n}( \text{div}X)g \Big|^2 \hfill \\ &=&\displaystyle {\left( {{\mathbb{L}_X}g - \frac{2}{n}( \text{div}X)g} \right)_{im}}{\left( {{\mathbb{L}_X}g - \frac{2}{n}( \text{div}X)g} \right)_{jn}}{g^{ij}}{g^{mn}} \hfill \\ &=& \displaystyle |{\mathbb{L}_X}g{|^2} + \frac{4}{{{n^2}}}{( \text{div}X)^2}\underbrace {{g_{im}}{g_{jn}}{g^{ij}}{g^{mn}}}_{\delta _n^i\delta _i^n = n} - \frac{2}{n}( \text{div}X)\left[ {{g_{im}}{\mathbb{L}_X}{g_{jn}} + {g_{jn}}{\mathbb{L}_X}{g_{im}}} \right]{g^{ij}}{g^{mn}} \hfill \\ &=& \displaystyle |{\mathbb{L}_X}g{|^2} + \frac{4}{n}{( \text{div}X)^2} - \frac{2}{n}( \text{div}X)[({\mathbb{L}_X}{g_{jn}})\underbrace {{g_{im}}{g^{ij}}{g^{mn}}}_{\delta _m^j{g^{mn}} = {g^{jn}}} + ({\mathbb{L}_X}{g_{im}})\underbrace {{g_{jn}}{g^{ij}}{g^{mn}}}_{\delta _n^i{g^{mn}} = {g^{im}}}] \hfill \\ &=&\displaystyle |{\mathbb{L}_X}g{|^2} + \frac{4}{n}{( \text{div}X)^2} - \frac{2}{n}( \text{div}X)[({\nabla _j}{X_n}+\nabla_n X_j){g^{jn}} + ({\nabla _i}{X_m}+\nabla_m X_i){g^{im}}] \hfill \\ &=& \displaystyle |{\mathbb{L}_X}g{|^2} + \frac{4}{n}{( \text{div}X)^2} - \frac{2}{n}( \text{div}X)[\underbrace {2{\nabla ^n}{X_n} + 2{\nabla ^m}{X_m}}_{4 \text{div}X}] \hfill \\ &=& \displaystyle |{\mathbb{L}_X}g{|^2} - \frac{4}{n}{( \text{div}X)^2}. \end{array}

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January 17, 2013

Conformal Killing Operator

Filed under: Riemannian geometry — Tags: — Ngô Quốc Anh @ 16:14

In this notes, I want to summarize some properties of the so-called Conformal Killing Operator relative to the metric g, say \mathbb L_g.

1. First, we start with its definition. Roughly speaking, the conformal killing operator is a generalization of the Killing operator relative to the metric g. It maps any vector field on M to some tensor of type (2,0). More precisely, in components, we have

\displaystyle (\mathbb L_g v)^{ij} = \nabla^iv^j + \nabla^jv^i - \frac{2}{n}\nabla_kv^k g^{ij},

where v is a vector field on M.

Immediately, one can check that \mathbb L_gv is traceless as can be seen in the following

\displaystyle {g_{ij}}{({\mathbb{L}_g}v)^{ij}} = {g_{ij}}{\nabla ^i}{v^j} + {g_{ij}}{\nabla ^j}{v^i} - 2{\nabla _k}{v^k} = 0.

2. We now define the so-called Conformal Vector Laplacian associated to the metric g. Basically, it is given by

\displaystyle (\Delta_{g,\text{conf}}v)^i= \nabla_j(\mathbb L_g v)^{ij}.

In components, we have

\displaystyle\begin{array}{lcl} {({\Delta _{g,{\text{conf}}}}v)^i} &=&\displaystyle {\nabla _j}{\nabla ^i}{v^j} + {\nabla _j}{\nabla ^j}{v^i} - \frac{2}{n}{\nabla ^i}{\nabla _k}{v^k} \hfill \\ &=&\displaystyle {\nabla ^i}{\nabla _j}{v^j} + R_j^i{v^j} + {\nabla _j}{\nabla ^j}{v^i} - \frac{2}{n}{\nabla ^i}{\nabla _k}{v^k} \hfill \\ &=&\displaystyle \frac{{n - 2}}{n}{\nabla ^i}{\nabla _j}{v^j} + R_j^i{v^j} + {\nabla _j}{\nabla ^j}{v^i}. \hfill \\ \end{array}

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