# Top PDF The Seiberg-Witten equations on 3-manifolds with boundary ### The Seiberg-Witten equations on 3-manifolds with boundary

xii List of Notations YM , £ energy functional on 4-manifolds CSV Chern-Simons-Dirac functional on 3-manifolds F Dirac functional on Riemann surfaces CIV, g Clifford module on vector spa[r] ### Boundary value problems for thend-order Seiberg-Witten equations

Let X be a compact smooth 4-manifold with nonempty boundary. In our context, the Seiberg-Witten equations are the 2nd-order Euler-Lagrange equation of the functional defined in Definition 2.3. When the boundary is empty, their variational aspects were first studied in  and the topological ones in . Thus, the main aim here is to obtain the existence of a solution to the nonhomogeneous equations whenever ∂X = ∅ . The nonemptiness of the boundary inflicts boundary conditions on the problem. Classically, this sort of problem is classified according to its boundary conditions in Dirichlet problem (Ᏸ) or Neumann problem (ᏺ). ### Quasiconvexity of bands in hyperbolic 3 manifolds

homotopy equivalence to Σ. We will assume (for now) that the cusps of M are in bijective correspondence with the boundary components of Σ (that is, the action of Γ is “strictly type preserving”). Let η > 0 be some constant less than the 3-dimensional Margulis constant. Let Ψ(M, η ) be the closed non-cuspidal part of M , that is M minus union of open η-Margulis cusps. By tameness [Bon], Ψ(M, η) is homeomorphic to Σ × R . The proof of the Ending Lamination Conjecture [Mi, BrCM] has lead to a reasonably good understanding of such manifolds in terms of model spaces. An important feature of their geometry are “bands” — subsets homeomorphic to a subsurface of Σ times an interval, where the boundary curves of the subsurface are represented by short geodesics in M (see for example, [Mi, BrCM, Mj1, Mj2, Bow1, Bow4]). Subsets of this sort are termed “blocks” in [Mj1, Mj2], though in this paper, we use the terminology from [Bow1] to avoid a clash with the term “block” as used in [Mi]. Bands are related to the “scaffolds” featuring in [BrCM]. The aim of this paper is to show that a lift of any such band (together with the associated Margulis tubes and cusps) is uniformly ### Closed essential surfaces in hyperbolizable acylindrical 3 manifolds

finitely generated fundamental group, using a standard trick. Let Γ be a finitely generated torsion- free Kleinian group which has infinite co-volume and which contains no Z ⊕ Z subgroups. By the Core Theorem of Scott , there exists a compact submanifold M of H 3 /Γ whose inclusion is a homotopy equivalence. Since Γ has infinite co-volume, the boundary of M is non-empty; since Γ contains no Z ⊕ Z subgroups, M cannot contain an incompressible torus, and so by Thurston’s uniformization theorem (see for example Morgan ), there exists a convex co-compact Kleinian group Φ uniformizing M . If it happens that M is acylindrical, Theorem 4.2 implies that there exists a closed immersed essential surface S in M , and hence that there exists a closed essential surface S in M Γ . Note that this argument works in the presence of parabolics, though the fundamental ### On Hyperbolic 3 Manifolds Obtained by Dehn Surgery on Links

Lastly, for positive integers m > 1, n ≥ 3, and k ≥ 1, it was proved that a family of closed 3-manifold M2m 1, n, k as the identification space of certain polyhedron P2m 1, n, k whose finitely many boundary faces are glued together in pair and which is another method to construct 3-manifolds, is the n/d-fold strongly cyclic covering of the 3-sphere branched over the link L m,d , where gcdn, k d 8. Since our link L m,d is hyperbolic link for m > 1, ### CONTACT STRUCTURES ON OPEN 3-MANIFOLDS

Recently, there has been much work towards the classification of tight contact structures on compact 3-manifolds up to isotopy (relative to the boundary). In particular, Honda and Giroux provided several classification theorems for solid tori, toric annuli, torus bundles over the circle, and circle bundles over surfaces [Gi1, Gi2, Gi3, Ho2, Ho3]. In compar- ison, tight contact structures on open 3-manifolds have been virtually unstudied. Two main results dealing with open contact manifolds are due to Eliashberg. In [El1], Eliash- berg shows that R 3 has a unique tight contact structure. It is immediate from his proof that S 2 × [0, ∞ ) has a unique tight contact structure with a fixed characteristic foliation on S 2 × 0. Therefore, the classification of tight contact structures on open manifolds ### Sphere systems in 3 manifolds and arc graphs

As already mentioned, all the components of M f g \ Σ f 1 are three-holed 3- spheres. Each 2-piece of Σ f 2 is properly embedded in one of these three-holed spheres. Since, by definition of standard form, spheres intersect minimally in M f g , then, by Lemma 2.1.10, a sphere in Σ f 2 intersects each sphere in Σ f 1 at most once. Therefore no two-piece of Σ f 2 can intersect a sphere in Σ f 1 in more than one circle, and consequently no 2-piece can have more than three boundary components. Moreover there can be no bigons, where by a bigon I mean a 2-piece of Σ f 1 and a 2-piece of Σ f 2 whose union bounds a ball in M f g . Therefore a 2-piece of Σ f 2 embedded in a connected component C of M f g \ Σ f 1 can be of the following three types: ### The relation between Maxwell, Dirac, and the Seiberg Witten equations

1. Introduction. In [1, 2, 3, 4, 5], using standard covariant spinor ﬁelds, Campolattaro proposed that Maxwell equations are equivalent to a nonlinear Dirac-like equation. The subject has been further developed in [35, 39] using the Cliﬀord bundle formalism, which is discussed together with some of its applications in a series of papers, for example, [11, 12, 13, 17, 18, 19, 22, 26, 28, 29, 30, 35, 39, 40]. The crucial point in proving the Maxwell-Dirac equiva- lence (MDE) starts once we observe that to any given representative of a Dirac- Hestenes spinor ﬁeld (for more information, see Section 2, and for details, see [17, 22, 26, 30]) ψ ∈ sec[ 0 (M) + 2 (M)+ 4 (M)] ⊂ sec Ꮿ (M, g), there is an associated electromagnetic ﬁeld F ∈ sec 2 (M) ⊂ sec Ꮿ (M, g) (F 2 ≠ 0) through ### 4-Manifolds With Inequivalent Symplectic Forms and 3-Manifolds With Inequivalent Fibrations

The SeibergWitten polynomial. A central feature of the fiber-sum X = X(P, L) is that its SeibergWitten polynomial is directly computable. Assume that X is simply-connected and b + 2 (X) > 1. Then the SeibergWitten invariant of X can be regarded as a map ### The Kapustin Witten Equations with Singular Boundary Conditions

arguments, cf. [2, 3], that the domain of J as an unbounded operator on L 2 ( S + 2 ) is compactly contained in L 2 . This implies the discreteness of the spectrum. Another proof which provides more accurate information uses that J is itself an incomplete uniformly degenerate operator, as analyzed thoroughly in . The main theorem in that paper produces a particular degenerate pseudodifferential operator G which invers J on L 2 . It is also shown there that G : L 2 (S + 2 ) → ψ 2 H 0 2 (S + 2 ) (where H 0 2 is the scale-invariant Sobolev space associated to the vector fields ψ∂ ψ , ψ∂ θ ). The compactness of ψ 2 H 0 2 (S + 2 ) , → L 2 (S + 2 ) follows from the L 2 Arzela-Ascoli theorem. There is an accompanying regularity theorem: if ( J − λ ) w = f where (for simplicity) f is smooth and vanishes to all orders at ψ = 0 and λ ∈ R (or more generally can be any bounded polyhomogeneous function), then w is polyhomogeneous with an expansion of the form ### Toric duality as Seiberg duality and brane diamonds

On the other hand we give an example in the other direction, namely two Picard- Lefschetz dual theories which are not Seiberg duals. Consider the case given in Figure 14, this is a phase of the theory for the complex cone over dP3 as given in . This is PL dual to any of the 4 four phases in Figure 12 in the previous section by construction with (p, q)-webs. Note that the total rank remains 6 under PL even though the number of nodes changed. However Seiberg duality on any of the allowed node on any of the 4 phases cannot change the number of nodes. Therefore, this example in Figure 14 is not Seiberg dual to the other 4. ### EFFECTIVE FIELD THEORY ON MANIFOLDS WITH BOUNDARY Benjamin I. Albert A DISSERTATION in Mathematics

where R ≥ 1. In the next section, the covering lemma that was proved by Costello in  is strengthened and proved. Much more detail about the nature of the sets in the cover is given. Other preliminary concepts needed for the renormalization procedure like local functionals and the form of their Feynman weights are then discussed. ### Global continuation of periodic solutions for retarded functional differential equations on manifolds

Let X be a metric ANR and consider a locally compact (continuous) X -valued map k deﬁned on a subset D(k) of X . Given an open subset U of X contained in D(k), if the set of ﬁxed points of k in U is compact, the pair (k, U) is called admissible. We point out that such a condition is clearly satisﬁed if U ⊆ D(k), k(U) is compact and k(p) = p for all p in the boundary of U. To any admissible pair (k, U), one can associate an integer ind X (k, U) - the ﬁxed point index of k in U - which satisﬁes properties analogous to those of the classical Leray-Schauder degree []. The reader can see, for instance, [, –] for a comprehensive presentation of the index theory for ANRs. As regards the connection with the homology theory, we refer to standard algebraic topology textbooks (e.g., [, ]). ### Boundary value problems for fractional differential equations with nonlocal boundary conditions

To the best of our knowledge, we can see the fact that, although the fractional diﬀerential equation boundary value problems have been studied by some authors, very little is known in the literature on the boundary value problems with integral boundary conditions. In order to enrich the theoretical knowledge of the above, in this paper, we investigate two classes of fractional diﬀerential equation boundary value problems with integral boundary conditions. ### The Inertial Manifolds for a Class of Higher Order Coupled Kirchhoff Type Equations

DOI: 10.4236/jamp.2018.65091 1064 Journal of Applied Mathematics and Physics er-order coupled Kirchhoff-type equations. In the process of research, we take advantage of Hadamard’s graph to get the equivalent form of the original equa- tions and then use spectral gap condition. Based on some of the work above, we prove the existence of the inertial manifolds of the system. For this problem, we will study the exponential attractors, blow-up, random attractors and so on. ### Harnack inequality for parabolic Lichnerowicz equations on complete noncompact Riemannian manifolds

Theorem . (Song and Zhao []) Let M be a compact Riemannian manifold without boundary, Ric(M) ≥ . Let c(t) ∈ C  (, ∞). Assume that u(x, t) is any positive solution of (.) on M with A(x) ≡ A, B(x) ≡ B, and h(x) ≡ h. Denote ϕ = ln u, suppose that A ≤ , B ≥ , c(t) ≥ , c (t) ≥ . If |∇ ϕ |  –  ### Biharmonic submanifolds in 3 dimensional (κ,μ) manifolds

in Sasakian space forms. Inoguchi  classified proper biharmonic Legendre curves and Hopf cylinders (automatically anti-invariant surfaces) in Sasakian 3-space forms. Sasa- hara  classified proper biharmonic Legendre surfaces in Sasakian 5-space forms. Also, in , the authors studied anti-invariant submanifolds in Sasakian 5-space forms. ### CONHARMONIC CURVATURE TENSOR OF -PARA SASAKIAN 3- MANIFOLDS

Abstract: The present paper deals with the study of 3-dimensional ( ) -para Sasakian manifolds satisfying certain curvature conditions on the conharmonic curvature tensor C . We characterize 3- dimensional (  ) -para Sasakian manifold satisfying  -conharmonically flat, C ( X , Y )  S  0 and locally  -recurrent condition, where S is Ricci tensor. ### A focal boundary value problem for difference equations

Comparison of eigenvalues for focal point problems for nth order difference equations, Differential and Integral Equations 3 1990, 363-380.... Sturmian theory for a class of nonselfadjoi[r] ### A singular boundary value problem for neutral equations

The aim of the present paper is to obtain conditions for the existence and uniqueness of an absolutely continuous solution of a singular boundary value problem for neutral equations, usi[r]