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Gluing and Wormholes for the Einstein Constraint Equations
 COMMUNICATIONS IN MATHEMATICAL PHYSICS
, 2002
"... We establish a general gluing theorem for constant mean curvature solutions of the vacuum Einstein constraint equations. This allows one to take connected sums of solutions or to glue a handle (wormhole) onto any given solution. Away from this handle region, the initial data sets we produce can be m ..."
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Cited by 38 (14 self)
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We establish a general gluing theorem for constant mean curvature solutions of the vacuum Einstein constraint equations. This allows one to take connected sums of solutions or to glue a handle (wormhole) onto any given solution. Away from this handle region, the initial data sets we produce can be made as close as desired to the original initial data sets. These constructions can be made either when the initial manifold is compact or asymptotically Euclidean or asymptotically hyperbolic, with suitable corresponding conditions on the extrinsic curvature. In the compact setting a mild nondegeneracy condition is required. In the final section of the paper, we list a number ways this construction may be used to produce new types of vacuum spacetimes.
Refined asymptotics for constant scalar curvature metrics with isolated singularities
, 1998
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Constant scalar curvature metrics with isolated singularities
 Duke Math. Journal
, 1999
"... We extend the results and methods of [6] to prove the existence of constant positive scalar curvature metrics g which are complete and conformal to the standard metric on SN \ Λ, where Λ is a disjoint union of submanifolds of dimensions between 0 and (N − 2)/2. The existence of solutions with isolat ..."
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Cited by 37 (8 self)
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We extend the results and methods of [6] to prove the existence of constant positive scalar curvature metrics g which are complete and conformal to the standard metric on SN \ Λ, where Λ is a disjoint union of submanifolds of dimensions between 0 and (N − 2)/2. The existence of solutions with isolated singularities occupies the majority of the paper; their existence was previously established by Schoen [12], but the proof we give here, based on the techniques of [6], is more direct, and provides more information about their geometry. When Λ is discrete we also establish that these solutions are smooth points in the moduli spaces of all such solutions introduced and studied in [7] and [8] 1 Introduction and statement of the results In this paper we construct solutions of the Yamabe problem on the sphere (SN, g0) with its standard metric which are singular at a specified closed set Λ ⊂ SN. More specifically, we seek a new metric g conformal to g0 and complete on Λ ⊂ SN, and with constant positive scalar curvature R. The problem may be translated into a differential equation as follows. Since g is
Connected sums of constant mean curvature surfaces in Euclidean 3space
 J. Reine Angew. Math
, 2001
"... Amongst the recent developments in the study of embedded complete minimal and constant mean curvature surfaces in R 3 is the realization that these objects are far more robust and flexible than ..."
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Cited by 35 (13 self)
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Amongst the recent developments in the study of embedded complete minimal and constant mean curvature surfaces in R 3 is the realization that these objects are far more robust and flexible than
Constant scalar curvature metrics on connected sums
 Int. J. Math. Math. Sci
"... In 1960 Yamabe [7] proposed the problem of finding a metric of constant scalar curvature conformal to any compact Riemannian manifold of dimension at least ..."
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Cited by 25 (0 self)
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In 1960 Yamabe [7] proposed the problem of finding a metric of constant scalar curvature conformal to any compact Riemannian manifold of dimension at least
The Toda system and multipleend solutions of autonomous planar elliptic problems
, 2009
"... We construct a new class of positive solutions for the classical elliptic problem ∆u − u + u p = 0, p> 2, in R 2. We show that these solutions are of the form u(x, z) ∼ Pk j=1 w(x − fj(z)), where w is the unique even, positive, asymptotically vanishing solution of w ′ ′ − w + wp = 0 in R. Fun ..."
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Cited by 18 (11 self)
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We construct a new class of positive solutions for the classical elliptic problem ∆u − u + u p = 0, p> 2, in R 2. We show that these solutions are of the form u(x, z) ∼ Pk j=1 w(x − fj(z)), where w is the unique even, positive, asymptotically vanishing solution of w ′ ′ − w + wp = 0 in R. Functions fj(z), representing the multiple ends of u(x, z), solve the Toda system c 2 f ′′ j = ef j−1−f j − e f j −f j+1 in R, j = 1,..., k, are asymptotically linear, and satisfy f0 ≡ − ∞ < f1 ≪ · · · ≪ fk < fk+1 ≡ +∞. The solutions of the elliptic problem we construct have their counterparts in the theory of constant mean curvature surfaces. An analogy can also be made between their construction and the gluing of constant scalar curvature Fowler singular metrics in the sphere.
SINGULAR YAMABE METRICS AND INITIAL DATA WITH EXACTLY KOTTLER–SCHWARZSCHILD–DE SITTER ENDS II. GENERIC METRICS
"... Abstract. We present a gluing construction which adds, via a localized deformation, exactly Delaunay ends to generic metrics with constant positive scalar curvature. This provides timesymmetric initial data sets for the vacuum Einstein equations with positive cosmological constant with exactly Kott ..."
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Cited by 14 (3 self)
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Abstract. We present a gluing construction which adds, via a localized deformation, exactly Delaunay ends to generic metrics with constant positive scalar curvature. This provides timesymmetric initial data sets for the vacuum Einstein equations with positive cosmological constant with exactly Kottler–Schwarzschild–de Sitter ends, extending the results in [5]. 1.
Maskit combinations of PoincaréEinstein metrics
"... We establish a boundary connected sum theorem for asymptotically hyperbolic Einstein metrics, and also show that if the two metrics have scalar positive conformal infinities, then the same is true for this boundary join. This construction is also extended to spaces with a finite number of interior c ..."
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Cited by 11 (1 self)
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We establish a boundary connected sum theorem for asymptotically hyperbolic Einstein metrics, and also show that if the two metrics have scalar positive conformal infinities, then the same is true for this boundary join. This construction is also extended to spaces with a finite number of interior conic singularities, and as a result we show that any 3manifold which is a finite connected sum of quotients of S 3 and S 2 ×S 1 bounds such a space (with conic singularities); putatively, any 3manifold admitting a metric of positive scalar curvature is of this form. 1
An end to end gluing construction for metrics of constant positive scalar curvature
 Indiana Univ. Math. J
"... The goal of this paper to to describe a general process by which one can glue together metrics of constant positive scalar curvature on punctured spheres along their ends to obtain new metrics ..."
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Cited by 5 (0 self)
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The goal of this paper to to describe a general process by which one can glue together metrics of constant positive scalar curvature on punctured spheres along their ends to obtain new metrics