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.

TangentSpace

Following the same strategy for the closed case, Escobar found the following invariant, still named Yamabe invariant, as follows

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