Ngô Quốc Anh

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 $\widehat g =\phi^{4/(n-2)}g$. Following the same strategy for the closed case, Escobar found the following invariant, still named Yamabe invariant, as follows $\displaystyle \mathcal Y(g,\partial M) = \mathop {\inf }\limits_{\phi \in {C^\infty }(M)\backslash \{ 0\} } \frac{{\int_M {\left( {\frac{{4(n - 1)}}{{n - 2}}|\nabla \phi {|^2} + \text{Scal}_g{\phi ^2}} \right)d{v_g}} + 2\int_{\partial M} {{H_g}{\phi ^2}d{s_g}} }}{{{{\left( {\int_M {|\phi {|^{2n/(n - 2)}}d{v_g}} } \right)}^{1 - 2/n}}}}.$

The original Yamabe invariant for the closed manifolds is simply the following $\displaystyle \mathcal Y(g) = \mathop {\inf }\limits_{\phi \in {C^\infty }(M)\backslash \{ 0\} } \frac{{\int_M {\left( {\frac{{4(n - 1)}}{{n - 2}}|\nabla \phi {|^2} + \text{Scal}_g{\phi ^2}} \right)d{v_g}} }}{{{{\left( {\int_M {|\phi {|^{2n/(n - 2)}}d{v_g}} } \right)}^{1 - 2/n}}}}$

which can be rewritten in terms of the conformal metric $\widehat g=\phi^{4/(n-2)}g$ as follows $\displaystyle \mathcal Y(g) = \mathop {\inf }\limits_{\widehat g}\frac{{\int_M {\text{Scal}_{\widehat g} dv_{\widehat g}} }}{{\text{vol}{{(M,\widehat g)}^{1 - 2/n}}}}.$

The purpose of this note is to obtain a short form for $\mathcal Y(g,\partial M)$ in terms of $\widehat g$ instead of the conformal factor $\phi$. To do so, let us recall that $\text{Scal}_{\widehat g}$ obeys the following rule $\displaystyle\text{Scal}_{\widehat g}=\phi^{-\frac{n+2}{n-2}} \left( -\frac{4(n-1)}{n-2}\Delta_g \phi+\text{Scal}_g \phi\right).$

Thanks to $\displaystyle d{v_{\widehat g}} = {\phi ^{\frac{{2n}}{{n - 2}}}}d{v_g}, \quad d{s_{\widehat g}} = {\phi ^{\frac{{2(n - 1)}}{{n - 2}}}}d{s_g},$

we can write $\begin{array}{lcl} \displaystyle\int_M {\text{Scal}_{\widehat g}d{v_{\widehat g}}} &=& \displaystyle\int_M {\phi \left( { - \frac{{4(n - 1)}}{{n - 2}}{\Delta _g}\phi + {\text{Sca}}{{\text{l}}_g}\phi } \right)d{v_g}} \hfill \\ &=& \displaystyle\int_M {\left( {\frac{{4(n - 1)}}{{n - 2}}|\nabla \phi {|^2} + {\text{Sca}}{{\text{l}}_g}{\phi ^2}} \right)d{v_g}} - \frac{{4(n - 1)}}{{n - 2}}\int_{\partial M} {\phi {\nabla _N}\phi d{s_g}} .\end{array}$

Therefore, the numerator of $\mathcal Y(g,\partial M)$ is nothing but $\displaystyle\int_M {\text{Scal}_{\widehat g}d{v_{\widehat g}}} + 2\left(\frac{{2(n - 1)}}{{n - 2}}\int_{\partial M} {\phi {\nabla _N}\phi d{\sigma _g}} + \int_{\partial M} {{H_g}{\phi ^2}d{s_g}} \right).$

Thanks to the conformal change rule for the mean curvature, we can further write the numerator of $\mathcal Y(g,\partial M)$ as below $\displaystyle\int_M {\text{Scal}_{\widehat g}d{v_{\widehat g}}}+ 2\int_{\partial M} {{H_{\widehat g}}{\phi ^{\frac{{2n - 2}}{{n - 2}}}}d{s_g}},$

which is equal to $\displaystyle\int_M {\text{Scal}_{\widehat g}d{v_{\widehat g}}}+ 2\int_{\partial M} {{H_{\widehat g}} d{s_{\widehat g}}}.$

Thus, the short form we obtain is the following $\displaystyle \mathcal Y(g, \partial M) = \mathop {\inf }\limits_{\widehat g}\frac{{\int_M {\text{Scal}_{\widehat g} dv_{\widehat g}} + 2\int_{\partial M} {{H_{\widehat g}} d{s_{\widehat g}}}}}{{\text{vol}{{(M,\widehat g)}^{1 - 2/n}}}}.$

To the best of my knowledge, I have not seen this type of presentation before.