The mass of asymptotically hyperbolic Riemannian manifolds

Chruściel, Piotr T.; Herzlich, Marc

doi:10.2140/pjm.2003.212.231

Cited by 222 publications

(411 citation statements)

References 38 publications

Supporting

Mentioning

391

Contrasting

Unclassified

Order By: Relevance

“…The analogous rigidity statement for hyperbolic space was proven in [12], [18], [5], [1] by establishing appropriate versions of the positive mass theorem in this context. In [13] M. Min-Oo raised the following question:…”

Section: G) Be a Compact Orientable Riemannian 3-manifold With Nonnegmentioning

confidence: 76%

The size of isoperimetric surfaces in $3$-manifolds and a rigidity result for the upper hemisphere

Eichmair

2009

Proc. Amer. Math. Soc.

View full text Add to dashboard Cite

show abstract

Section: G) Be a Compact Orientable Riemannian 3-manifold With Nonnegmentioning

confidence: 76%

The size of isoperimetric surfaces in $3$-manifolds and a rigidity result for the upper hemisphere

Eichmair

2009

Proc. Amer. Math. Soc.

View full text Add to dashboard Cite

show abstract

“…One can prove rigidity theorems [7] for asymptotically hyperbolic ends, or existence results for asymptotically hyperbolic Einstein metrics [47]. Similar rigidity problems for asymptotically complex hyperbolic ends are the subject of [10,11,30].…”

Section: We Define ϕ : [−1 +1] → R ϕ(T) = Log(t +1)−log(1−t) ϕ(±1)mentioning

confidence: 99%

On the geometry of Riemannian manifolds with a Lie structure at infinity

Ammann

Lauter

Nistor

2004

International Journal of Mathematics and Mathematical Sciences

214

View full text Add to dashboard Cite

We study a generalization of the geodesic spray and give conditions for noncomapct manifolds with a Lie structure at infinity to have positive injectivity radius. We also prove that the geometric operators are generated by the given Lie algebra of vector fields. This is the first one in a series of papers devoted to the study of the analysis of geometric differential operators on manifolds with Lie structure at infinity.2000 Mathematics Subject Classification: 53C21, 53C27, 58J40.1. Introduction. Geometric differential operators on complete noncompact Riemannian manifolds were extensively studied due to their applications to physics, geometry, number theory, and numerical analysis. Still, their properties are not as well understood as those of differential operators on compact manifolds, one of the main reasons being that differential operators on noncompact manifolds do not enjoy some of the most useful properties enjoyed by their counterparts on compact manifolds.For example, elliptic operators on noncompact manifolds are not Fredholm in general. (We use the term "elliptic" in the sense that the principal symbol is invertible outside the zero section.) Also, one does not have a completely satisfactory pseudodifferential calculus on an arbitrary complete noncompact Riemannian manifold, which might allow us to decide whether a given geometric differential operator is bounded, Fredholm, or compact (see however [2] and the references therein).However, if one restricts oneself to certain classes of complete noncompact Riemannian manifolds, one has a chance to obtain more precise results on the analysis of the geometric differential operators on those spaces. This paper is the first in a series of papers devoted to the study of such a class of Riemannian manifolds, the class of Riemannian manifolds with a "Lie structure at infinity" (see Definition 3.1). We stress here that few results on the geometry of these manifolds have a parallel in the literature, although there are a fair number of papers devoted to the analysis on particular classes of such manifolds [12,13,15,16,17,38,39,55,57,64,67,70,74,75,76,78] A manifold M 0 with a Lie structure at infinity has, by definition, a natural compactification to a manifold with corners M = M 0 ∪∂M such that the tangent bundle T M 0 → M 0

show abstract

“…For asymptotically hyperbolic spin manifolds, the following positive mass theorem was proved in Wang [12] (see also Chrusciel-Herzlich [3]). …”

mentioning

confidence: 99%

On the Uniqueness of the ADS Spacetime

Wang

2005

Acta Math Sinica

View full text Add to dashboard Cite

The uniqueness of the ADS spacetime among all static vacuum spacetimes with the same conformal infinity is proved for dimension n ≤ 7. For dimension n > 7, the same result is established under the spin assumption. In Einstein's theory of general relativity with a negative cosmological constant Λ, a vacuum spacetime is a solution to the equation R ab = Λg ab and the lowest-energy solution is the anti-de Sitter spacetime. Moreover it was proved by Bocher-GibbonsHorowitz [1] in 3 + 1 dimensions that the anti-de Sitter spacetime is the unique static, asymptotically anti-de Sitter vacuum.To give the precise statement of their theorem, let us first recall that an (n + 1) dimensional static spacetime (N, g) has the formwhere (M, g) is a Riemannian manifold and V is a positive function on M . The vacuum Einstein equation (without loss of generality we always take the negative cosmological constant Λ to be −n)Ric (g) = −ng

show abstract

The mass of asymptotically hyperbolic Riemannian manifolds

Cited by 222 publications

References 38 publications

The size of isoperimetric surfaces in $3$-manifolds and a rigidity result for the upper hemisphere

The size of isoperimetric surfaces in $3$-manifolds and a rigidity result for the upper hemisphere

On the geometry of Riemannian manifolds with a Lie structure at infinity

On the Uniqueness of the ADS Spacetime

Contact Info

Product

Resources

About