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25
Existence of nontrivial, vacuum, asymptotically simple spacetimes
 2002) L 71  L 79. Erratum Class. Quantum Grav
"... We construct nontrivial vacuum spacetimes with a global I +. The construction proceeds by proving extension results across compact boundaries for initial data sets, adapting the gluing arguments of Corvino and Schoen. Another application of the extension results is existence of initial data which ..."
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Cited by 38 (2 self)
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We construct nontrivial vacuum spacetimes with a global I +. The construction proceeds by proving extension results across compact boundaries for initial data sets, adapting the gluing arguments of Corvino and Schoen. Another application of the extension results is existence of initial data which are exactly Schwarzschild both near infinity and near each of the connected component of the apparent horizon. Finally the construction allows one to add EinsteinRosen bridges to timesymmetric initial data sets at points satisfying a local parity condition, with the perturbation of the metric localized in an arbitrarily small neighbourhood of the bridge. 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
SMOOTH YAMABE INVARIANT AND SURGERY
, 2008
"... We prove a surgery formula for the smooth Yamabe invariant σ(M) of a compact manifold M. Assume that N is obtained from M by surgery of codimension at least 3. We prove the existence of a positive constant Λn, depending only on the dimension n of M, such that σ(N) ≥ min{σ(M), Λn}. ..."
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Cited by 9 (1 self)
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We prove a surgery formula for the smooth Yamabe invariant σ(M) of a compact manifold M. Assume that N is obtained from M by surgery of codimension at least 3. We prove the existence of a positive constant Λn, depending only on the dimension n of M, such that σ(N) ≥ min{σ(M), Λn}.
On Optimal 4Dimensional Metrics
, 2007
"... We completely determine, up to homeomorphism, which simply connected compact oriented 4manifolds admit scalarflat, antiselfdual Riemannian metrics. The key new ingredient is a proof that the connected sum CP2#CP2#CP2#CP2#CP2 of five reverseoriented complex projective planes admits such metrics. ..."
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Cited by 9 (1 self)
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We completely determine, up to homeomorphism, which simply connected compact oriented 4manifolds admit scalarflat, antiselfdual Riemannian metrics. The key new ingredient is a proof that the connected sum CP2#CP2#CP2#CP2#CP2 of five reverseoriented complex projective planes admits such metrics. 1
Curvature functionals, optimal metrics, and the differential topology of 4manifolds, Different faces of geometry
 MR MR2102997 (2005h:53055
, 2004
"... Abstract. This paper investigates the question of which smooth compact 4manifolds admit Riemannian metrics that minimize the L 2norm of the curvature tensor. Metrics with this property are called optimal; Einstein metrics and scalarflat antiselfdual metrics provide us with two interesting class ..."
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Cited by 8 (0 self)
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Abstract. This paper investigates the question of which smooth compact 4manifolds admit Riemannian metrics that minimize the L 2norm of the curvature tensor. Metrics with this property are called optimal; Einstein metrics and scalarflat antiselfdual metrics provide us with two interesting classes of examples. Using twistor methods, optimal metrics of the second type are constructed on the connected sums kCP2 for k> 5. However, related constructions also show that large classes of simply connected 4manifolds do not admit any optimal metrics at all. Interestingly, the difference between existence and nonexistence turns out to delicately depend on one’s choice of smooth structure; there are smooth 4manifolds which carry optimal metrics, but which are homeomorphic to infinitely many distinct smooth 4manifolds on which no optimal metric exists. 1.
Surgery and equivariant Yamabe invariant
, 2006
"... We consider the equivariant Yamabe problem, i.e. the Yamabe problem on the space of Ginvariant metrics for a compact Lie group G. The GYamabe invariant is analogously defined as the supremum of the constant scalar curvatures of unit volume Ginvariant metrics minimizing the total scalar curvature ..."
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Cited by 6 (0 self)
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We consider the equivariant Yamabe problem, i.e. the Yamabe problem on the space of Ginvariant metrics for a compact Lie group G. The GYamabe invariant is analogously defined as the supremum of the constant scalar curvatures of unit volume Ginvariant metrics minimizing the total scalar curvature functional in their Ginvariant conformal subclasses. We prove a formula about how the GYamabe invariant changes under the surgery of codimension 3 or more, and compute some GYamabe invariants.
THE SPINORIAL τINVARIANT AND 0DIMENSIONAL SURGERY.
, 2008
"... Abstract. Let M be a compact manifold with a metric g and with a fixed spin structure χ. Let λ + 1 (g) be the first nonnegative eigenvalue of the Dirac operator on (M, g, χ). We set τ(M, χ): = sup inf λ + 1 (g) where the infimum runs over all metrics g of volume 1 in a conformal class [g0] on M and ..."
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Abstract. Let M be a compact manifold with a metric g and with a fixed spin structure χ. Let λ + 1 (g) be the first nonnegative eigenvalue of the Dirac operator on (M, g, χ). We set τ(M, χ): = sup inf λ + 1 (g) where the infimum runs over all metrics g of volume 1 in a conformal class [g0] on M and where the supremum runs over all conformal classes [g0] on M. Let (M # , χ # ) be obtained from (M, χ) by 0dimensional surgery. We prove that τ(M # , χ #)≥τ(M, χ).
Generalized connected sum construction for constant scalar curvature metrics
"... In this paper we construct constant scalar curvature metrics on the generalized connected sum M = M1 ♯K M2 of two compact Riemannian manifolds (M1, g1) and (M2, g2) along a common Riemannian submanifold (K, gK), in the case where the codimension of K is ≥ 3 and the manifolds M1 and M2 carry the same ..."
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In this paper we construct constant scalar curvature metrics on the generalized connected sum M = M1 ♯K M2 of two compact Riemannian manifolds (M1, g1) and (M2, g2) along a common Riemannian submanifold (K, gK), in the case where the codimension of K is ≥ 3 and the manifolds M1 and M2 carry the same nonzero constant scalar curvature S. In particular the structure of the metrics we build is investigated and described. 1 Introduction and statement of the result Connected sum of solutions of nonlinear problems has revealed to be a very powerful tool in understanding solutions of many geometric problems (minimal and constant mean curvature surfaces [7], [8], constant scalar curvature metrics [4], [9], [6], and recently even Einstein metrics [1]). However, generalized connected
Connected sum construction for σkYamabe metrics
"... In this paper we produce families of Riemannian metrics with positive constant σkcurvature equal to ..."
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In this paper we produce families of Riemannian metrics with positive constant σkcurvature equal to