Mass in Kähler Geometry
نویسندگان
چکیده
We prove a simple, explicit formula for the mass of any asymptotically locally Euclidean (ALE) Kähler manifold, assuming only the sort of weak fall-off conditions required for the mass to actually be well-defined. For ALE scalar-flat Kähler manifolds, the mass turns out to be a topological invariant, depending only on the underlying smooth manifold, the first Chern class of the complex structure, and the Kähler class of the metric. When the metric is actually AE (asymptotically Euclidean), our formula not only implies a positive mass theorem for Kähler metrics, but also yields a Penrose-type inequality for the mass. A complete connected non-compact Riemannian manifold (M, g) of dimension n ≥ 3 is said to be asymptotically Euclidean (or AE ) if there is a compact subset K ⊂ M such that M − K consists of finitely many components, each of which is diffeomorphic to the complement of a closed ball D ⊂ R, in a manner such that g becomes the standard Euclidean metric plus terms that fall off sufficiently rapidly at infinity. More generally, a Riemannian n-manifold (M, g) is said to be asymptotically locally Euclidean (or ALE ) if the complement of a compact set K consists of finitely many components, each of which is diffeomorphic to a quotient (R −D)/Γj , where Γj ⊂ O(n) is a finite subgroup which acts freely on the unit sphere, in such a way that g again becomes the Euclidean metric plus error terms that fall off sufficiently rapidly at infinity. The components of M −K are called the ends of M ; their fundamental groups are the afore-mentioned groups Γj, which may in principle be different for different ends of the manifold. Research funded in part by NSF grant DMS-1514709. Research funded in part by NSF grant DMS-1510094.
منابع مشابه
Integrable Systems in Gauge Theory, Kähler Geometry and Super KP Hierarchy – Symmetries and Algebraic Point of View
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