# 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