Index theory on conformally compact manifolds
نویسنده
چکیده
I will discuss index theory in the context of Poincare-Einstein manifolds of AdS/CFT fame. Complications arise because Dirac operators are not Fredholm and the Atiyah-Singer integrands are not integrable. In some cases, renormalization allows us to circumvent these difficulties.
منابع مشابه
On Positive Solutions to Semi-linear Conformally Invariant Equations on Locally Conformally Flat Manifolds
In this paper we study the existence and compactness of positive solutions to a family of conformally invariant equations on closed locally conformally flat manifolds. The family of conformally covariant operators Pα were introduced via the scattering theory for Poincaré metrics associated with a conformal manifold (Mn, [g]). We prove that, on a closed and locally conformally flat manifold with...
متن کاملA Renormalized Index Theorem for Some Complete Asymptotically Regular Metrics: the Gauss-bonnet Theorem
The Gauss-Bonnet Theorem is studied for edge metrics as a renormalized index theorem. These metrics include the Poincaré-Einstein metrics of the AdS/CFT correspondence. Renormalization is used to make sense of the curvature integral and the dimensions of the L-cohomology spaces as well as to carry out the heat equation proof of the index theorem. For conformally compact metrics even mod x, the ...
متن کاملOn the Topology of Conformally Compact Einstein 4-manifolds
In this paper we study the topology of conformally compact Einstein 4-manifolds. When the conformal infinity has positive Yamabe invariant and the renormalized volume is also positive we show that the conformally compact Einstein 4-manifold will have at most finite fundamental group. Under the further assumption that the renormalized volume is relatively large, we conclude that the conformally ...
متن کاملTopics in Conformally Compact Einstein Metrics
Conformal compactifications of Einstein metrics were introduced by Penrose [38], as a means to study the behavior of gravitational fields at infinity, i.e. the asymptotic behavior of solutions to the vacuum Einstein equations at null infinity. This has remained a very active area of research, cf. [27], [19] for recent surveys. In the context of Riemannian metrics, the modern study of conformall...
متن کاملOn the Rigidity for Conformally Compact Einstein Manifolds
In this paper we prove that a conformally compact Einstein manifold with the round sphere as its conformal infinity has to be the hyperbolic space. We do not assume the manifolds to be spin, but our approach relies on the positive mass theorem for asymptotic flat manifolds. The proof is based on understanding of positive eigenfunctions and compactifications obtained by positive eigenfunctions. ...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2006