| Résume | Let M be a compact oriented d-dimensional manifold with boundary N. A natural geometric boundary value problem is to find an asymptotically hyperbolic Einstein metric g on (the interior of) M with prescribed ’conformal infinity’ on N. A little more precisely, the problem is to find (Einstein) g with the boundary condition that x?g tends to a metric h on N as x goes to 0, x being a boundary defining function for N. The prototype is the hyperbolic metric g on the ball, with conformal infinity the round metric on the boundary sphere. Since the pioneering work of Graham and Lee (1991) the problem has attracted attention from a number of authors.
In this talk, I shall explain a gauge-theoretic approach to the problem which works in dimension d=4, and explain how it can be used to obtain some new results for this boundary value problem. |
