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On asymptotically hyperbolic anti-self-dual Einstein metrics Singer, Michael


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 infinityâ on $N$. A little more precisely, the problem is to find (Einstein) $g$ with the boundary condition $x^2g$ tends to a metric $h$ on $N$ as $x$ goes to $0$, $x$ being a boundary defining function for $N$. The freedom to rescale $x$ by an arbitrary smooth positive function means that only the conformal class of $h$ is naturally well defined. Hence the terminology `conformal infinityâ in this boundary problem. Since the pioneering work of Graham and Lee (1991) the problem has attracted attention from a number of authors. If the dimension $d$ is $4$, there is a refinement, asking that $g$ be anti-self-dual as well as Einstein (satisfying the same boundary condition). If $M$ is the ball, this is the subject of the positive frequency conjecture of LeBrun (1980s) proved by Biquard in 2002. In this talk, which is based on joint work with Joel Fine and Rafe Mazzeo, I shall explain a gauge theoretic approach to the ASDE problem which is readily applicable for general $M$ and the currently available results.

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