On cohomogeneity one biharmonic hypersurfaces into the Euclidean space

This chapter returns to the problems which were formulated in Chapter 1, namely the Weinstein conjecture, the nonsqueezing theorem, and symplectic rigidity. These questions are all related to the existence and properties of symplectic capacities. The chapter begins by discussing some of the consequences which follow from the existence of capacities. In particular, it establishes symplectic rigidity and discusses the relation between capacities and the Hofer metric on the group of Hamiltonian symplectomorphisms. The chapter then introduces the Hofer–Zehnder capacity, and shows that its existence gives rise to a proof of the Weinstein conjecture for hypersurfaces of Euclidean space. The last section contains a proof that the Hofer–Zehnder capacity satisfies the required axioms. This proof translates the Hofer–Zehnder variational argument into the setting of (finite-dimensional) generating functions.

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ANTI-SELF-DUAL SOLUTIONS OF THE YANG-MILLS EQUATIONS IN 4n DIMENSIONS

Modern Physics Letters A ◽

10.1142/s0217732392001816 ◽

1992 ◽

Vol 07 (23) ◽

pp. 2077-2085 ◽

Cited By ~ 17

Author(s):

A. D. POPOV

Keyword(s):

Scalar Field ◽

Euclidean Space ◽

Lie Algebra ◽

Linear Equations ◽

Dual Solutions ◽

Gauge Fields ◽

System Of Equations ◽

Semisimple Lie Algebra ◽

Yang Mills ◽

Self Duality

The anti-self-duality equations for gauge fields in d = 4 and a generalization of these equations to dimension d = 4n are considered. For gauge fields with values in an arbitrary semisimple Lie algebra [Formula: see text] we introduce the ansatz which reduces the anti-self-duality equations in the Euclidean space ℝ4n to a system of equations breaking up into the well known Nahm's equations and some linear equations for scalar field φ.

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On the automorphism group of a symplectic half-flat 6-manifold

Forum Mathematicum ◽

10.1515/forum-2018-0137 ◽

2019 ◽

Vol 31 (1) ◽

pp. 265-273

Author(s):

Fabio Podestà ◽

Alberto Raffero

Keyword(s):

Automorphism Group ◽

Lie Algebra ◽

Group Action ◽

Cohomogeneity One ◽

Cohomogeneity One Action

Abstract We prove that the automorphism group of a compact 6-manifold M endowed with a symplectic half-flat {\mathrm{SU}(3)} -structure has Abelian Lie algebra with dimension bounded by {\min\{5,b_{1}(M)\}} . Moreover, we study the properties of the automorphism group action and we discuss relevant examples. In particular, we provide new complete examples on {T\mathbb{S}^{3}} which are invariant under a cohomogeneity one action of {\mathrm{SO}(4)} .

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