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:

First, we denote N=2n/(n-2). In view of the Sobolev embedding W^{1,2}(M) \hookrightarrow L^N(M) and Sobolev trace embedding W^{1,2}(M) \hookrightarrow L^{N/2+1} (\partial M), we obtain the following inequalities

\displaystyle\|X\|_{L^N(M)} \leqslant \mathcal S_M \|X\|_{W^{1,2}(M)}

and

\displaystyle\|X\|_{L^{N/2+1}(\partial M)} \leqslant \mathcal S_{\partial M} \|X\|_{W^{1,2}(M)}

for some positive constants \mathcal S_M and \mathcal S_{\partial M} independent of X. Therefore,

\displaystyle\|X\|_{L^N(M)} + \|X\|_{L^{N/2+1}(\partial M)} \leqslant ( \mathcal S_M+ \mathcal S_{\partial M}) \|X\|_{W^{1,2}(M)} .

By simply rewriting C_g(M,\partial M) as

\displaystyle \begin{array}{lcl}C_g(M,\partial M) &=&\displaystyle\inf_X \frac{\|\mathbb L X\|_{L^2(M)}}{\|X\|_{L^N(M)} + \|X\|_{L^{N/2+1}(\partial M)}} \\&=& \displaystyle\inf_X \frac{\|\mathbb L X\|_{L^2(M)}}{\|X\|_{W^{1,2}(M)}} \frac{\|X\|_{W^{1,2}(M)}} {\|X\|_{L^N(M)} + \|X\|_{L^{N/2+1}(\partial M)}} \\&\geqslant& \displaystyle\frac{1}{\mathcal S_M+ \mathcal S_{\partial M}}\inf_X \frac{\|\mathbb L X\|_{L^2(M)}}{\|X\|_{W^{1,2}(M)}},\end{array}

it suffices to prove that

\displaystyle\inf_X \frac{\|\mathbb L X\|_{L^2(M)}}{\|X\|_{W^{1,2}(M)}} >0.

We are now able to repeat all arguments in the previous note to conclude the result. Apparently, one can easly verify that

\displaystyle C_g(M,\partial) \leqslant C_g(M).

I take this chance to explain that all arguments for proving C_g >0 work and the fact C_g>0 holds true since the numerator involves \mathbb L which is of fourth order. Thefore, one can expect that the inequalities C_g >0 are actually stronger than usual Sobolev inequalities. One main difficulty is how to connect these fourth order operators acting on X, \|\mathbb LX\|_2, and \|\nabla X\|_2.

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