Results 1  10
of
35
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 ..."
Abstract

Cited by 38 (14 self)
 Add to MetaCart
(Show Context)
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.
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 ..."
Abstract

Cited by 18 (11 self)
 Add to MetaCart
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.
Some new entire solutions of semilinear elliptic . . .
"... We prove existence of a new type of positive solutions of the semilinear equation −∆u+u = up on Rn, where 1 < p < n+2n−2. These solutions are bounded, but do not tend to zero at infinity. Indeed, they decay to zero away from three halflines with a common origin, and their asymptotic profile ..."
Abstract

Cited by 17 (1 self)
 Add to MetaCart
We prove existence of a new type of positive solutions of the semilinear equation −∆u+u = up on Rn, where 1 < p < n+2n−2. These solutions are bounded, but do not tend to zero at infinity. Indeed, they decay to zero away from three halflines with a common origin, and their asymptotic profile is periodic along these halflines.
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 ..."
Abstract

Cited by 14 (3 self)
 Add to MetaCart
(Show Context)
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.
On the nondegeneracy of constant mean curvature surfaces
"... We prove that many complete, noncompact, constant mean curvature (CMC) surfaces f: Σ → R 3 are nondegenerate; that is, the Jacobi operator ∆f + Af  2 has no L 2 kernel. In fact, if Σ has genus zero and f(Σ) is contained in a halfspace, then generically the dimension of the L 2 kernel is at most t ..."
Abstract

Cited by 10 (4 self)
 Add to MetaCart
We prove that many complete, noncompact, constant mean curvature (CMC) surfaces f: Σ → R 3 are nondegenerate; that is, the Jacobi operator ∆f + Af  2 has no L 2 kernel. In fact, if Σ has genus zero and f(Σ) is contained in a halfspace, then generically the dimension of the L 2 kernel is at most the number of noncylindrical ends of f(Σ), minus three. Our main tool is a conjugation operation on Jacobi fields which linearizes the conjugate cousin construction. Consequences include partial regularity for CMC moduli space, a larger class of CMC surfaces to use in gluing constructions, and a surprising characterization of CMC surfaces via rolling spheres. 1
Complex antiselfdual instantons and Cayley submanifolds
"... Let M be a CalabiYau manifold of complex dimension 4 with symplectic form ω and complex volume form θ. A connection A on a vector bundle ..."
Abstract

Cited by 10 (0 self)
 Add to MetaCart
(Show Context)
Let M be a CalabiYau manifold of complex dimension 4 with symplectic form ω and complex volume form θ. A connection A on a vector bundle