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An Arzela-Ascoli theorem for immersed submanifolds

Graham Smith (2007)

Annales de la faculté des sciences de Toulouse Mathématiques

The classical Arzela-Ascoli theorem is a compactness result for families of functions depending on bounds on the derivatives of the functions, and is of invaluable use in many fields of mathematics. In this paper, inspired by a result of Corlette, we prove an analogous compactness result for families of immersed submanifolds which depends only on bounds on the derivatives of the second fundamental forms of these submanifolds. We then show how the result of Corlette may be obtained as an immediate...

An index inequality for embedded pseudoholomorphic curves in symplectizations

Michael Hutchings (2002)

Journal of the European Mathematical Society

Let Σ be a surface with a symplectic form, let φ be a symplectomorphism of Σ , and let Y be the mapping torus of φ . We show that the dimensions of moduli spaces of embedded pseudoholomorphic curves in ℝ × 𝕐 , with cylindrical ends asymptotic to periodic orbits of φ or multiple covers thereof, are bounded from above by an additive relative index. We deduce some compactness results for these moduli spaces. This paper establishes some of the foundations for a program with Michael Thaddeus, to understand...

An inequality for symplectic fillings of the link of a hypersurface K3 singularity

Hiroshi Ohta, Kaoru Ono (2009)

Banach Center Publications

Some relations between normal complex surface singularities and symplectic fillings of the links of the singularities are discussed. For a certain class of singularities of general type, which are called hypersurface K3 singularities in this paper, an inequality for numerical invariants of any minimal symplectic fillings of the links of the singularities is derived. This inequality can be regarded as a symplectic/contact analog of the 11/8-conjecture in 4-dimensional topology.

An introduction to gerbes on orbifolds

Ernesto Lupercio, Bernardo Uribe (2004)

Annales mathématiques Blaise Pascal

This paper is a gentle introduction to some recent results involving the theory of gerbes over orbifolds for topologists, geometers and physicists. We introduce gerbes on manifolds, orbifolds, the Dixmier-Douady class, Beilinson-Deligne orbifold cohomology, Cheeger-Simons orbifold cohomology and string connections.

An o-minimal structure which does not admit C ∞ cellular decomposition

Olivier Le Gal, Jean-Philippe Rolin (2009)

Annales de l’institut Fourier

We present an example of an o-minimal structure which does not admit C ∞ cellular decomposition. To this end, we construct a function H whose germ at the origin admits a C k representative for each integer k , but no C ∞ representative. A number theoretic condition on the coefficients of the Taylor series of H then insures the quasianalyticity of some differential algebras 𝒜 n ( H ) induced by H . The o-minimality of the structure generated by H is deduced from this quasianalyticity property.

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