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 $\|\nabla X\|_2$.