Let @math be a symplectic manifold, @math a coisotropic submanifold, and @math a compact oriented (real) surface. I define a natural Maslov index for each continuous map @math that sends every connected component of @math to some isotropic leaf of @math . This index is real valued and generalizes the usual Lagrangian Maslov index. The idea is to use the linear holonomy of the isotropic foliation of @math to compensate for the loss of boundary data in the case codimension @math . The definition is based on the Salamon-Zehnder (mean) Maslov index of a path of linear symplectic automorphisms. I prove a lower bound on the number of leafwise fixed points of a Hamiltonian diffeomorphism, if @math is geometrically bounded and @math is closed, regular (i.e. "fibering"), and monotone. As an application, we obtain a presymplectic non-embedding result. I also prove a coisotropic version of the Audin conjecture.
In his recent paper @cite_1 (Theorem 1.5) V. L. Ginzburg proved an upper bound on the minimal Maslov number of a closed, stable, displaceable coisotropic submanifold.
@cite_1
1893146140 2008724760 2053519960 2015059094
This is the second in a series of papers in which we derive a @math -expansion for the two-dimensional non-local Ginzburg-Landau energy with Coulomb repulsion known as the Ohta-Kawasaki model in connection with diblock copolymer systems. In this model, two phases appear, which interact via a nonlocal Coulomb type energy. Here we focus on the sharp interface version of this energy in the regime where one of the phases has very small volume fraction, thus creating small "droplets" of the minority phase in a "sea" of the majority phase. In our previous paper, we computed the @math -limit of the leading order energy, which yields the averaged behavior for almost minimizers, namely that the density of droplets should be uniform. Here we go to the next order and derive a next order @math -limit energy, which is exactly the Coulombian renormalized energy obtained by Sandier and Serfaty as a limiting interaction energy for vortices in the magnetic Ginzburg-Landau model. The derivation is based on the abstract scheme of Sandier-Serfaty that serves to obtain lower bounds for 2-scale energies and express them through some probabilities on patterns via the multiparameter ergodic theorem. Without thus appealing to the Euler-Lagrange equation, we establish for all configurations which have "almost minimal energy" the asymptotic roundness and radius of the droplets, and the fact that they asymptotically shrink to points whose arrangement minimizes the renormalized energy in some averaged sense. Via a kind of @math -equivalence, the obtained results also yield an expansion of the minimal energy for the original Ohta-Kawasaki energy. This leads to expecting to see triangular lattices of droplets as energy minimizers. In this paper we study a fundamental open problem in the area of probabilistic checkable proofs: What is the smallest s such that NP ⊆ naPCP1,s[O(log n),3]? In the language of hardness of approximation, this problem is equivalent to determining the smallest s such that getting an s-approximation for satisfiable 3-bit constraint satisfaction problems ("3-CSPs") is NP-hard. The previous best upper bound and lower bound for s are 20 27+µ by Khot and Saket [KS06], and 5 8 (assuming NP subseteq BPP) by Zwick [Zwi98]. In this paper we close the gap assuming Khot's d-to-1 Conjecture. Formally, we prove that if Khot's d-to-1 Conjecture holds for any finite constant integer d, then NP naPCP1,5 8+ µ[O(log n),3] for any constant µ > 0. Our conditional result also solves Hastad's open question [Has01] on determining the inapproximability of satisfiable Max-NTW ("Not Two") instances and confirms Zwick's conjecture [Zwi98] that the 5 8-approximation algorithm for satisfiable 3-CSPs is optimal. Abstract Let M be a compact submanifold with boundary of a Euclidean space or a Sphere. In this paper, we derive an upper bound for the first non-zero eigenvalue p 1 of Steklov problem on M in terms of the r -th mean curvatures of its boundary ∂ M . The upper bound obtained is sharp. Let (L ) be a closed orientable Lagrangian submanifold of a closed symplectic six-manifold ((X , ) ). We assume that the first homology group (H_1 (L ; A) ) with coefficients in a commutative ring (A ) injects into the group (H_1 (X ; A) ) and that (X ) contains no Maslov zero pseudo-holomorphic disc with boundary on (L ). Then, we prove that for every generic choice of a tame almost-complex structure (J ) on (X ), every relative homology class (d H_2 (X , L ; Z ) ) and adequate number of incidence conditions in (L ) or (X ), the weighted number of (J )-holomorphic discs with boundary on (L ), homologous to (d ), and either irreducible or reducible disconnected, which satisfy the conditions, does not depend on the generic choice of (J ), provided that at least one incidence condition lies in (L ). These numbers thus define open Gromov–Witten invariants in dimension six, taking values in the ring (A ).