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Ellipticity of the symplectic twistor complex

Svatopluk Krýsl (2011)

Archivum Mathematicum

For a Fedosov manifold (symplectic manifold equipped with a symplectic torsion-free affine connection) admitting a metaplectic structure, we shall investigate two sequences of first order differential operators acting on sections of certain infinite rank vector bundles defined over this manifold. The differential operators are symplectic analogues of the twistor operators known from Riemannian or Lorentzian spin geometry. It is known that the mentioned sequences form complexes if the symplectic...

Embedding of Hilbert manifolds with smooth boundary into semispaces of Hilbert spaces

J. Margalef-Roig, Enrique Outerelo-Domínguez (1994)

Archivum Mathematicum

In this paper we prove the existence of a closed neat embedding of a Hausdorff paracompact Hilbert manifold with smooth boundary into H × [ 0 , + ∞ ) , where H is a Hilbert space, such that the normal space in each point of a certain neighbourhood of the boundary is contained in H × { 0 } . Then, we give a neccesary and sufficient condition that a Hausdorff paracompact topological space could admit a differentiable structure of class ∞ with smooth boundary.

Energy and Morse index of solutions of Yamabe type problems on thin annuli

Mohammed Ben Ayed, Khalil El Mehdi, Mohameden Ould Ahmedou, Filomena Pacella (2005)

Journal of the European Mathematical Society

We consider the Yamabe type family of problems ( P ε ) : − Δ u ε = u ε ( n + 2 ) / ( n − 2 ) , u ε > 0 in A ε , u ε = 0 on ∂ A ε , where A ε is an annulus-shaped domain of ℝ n , n ≥ 3 , which becomes thinner as ε → 0 . We show that for every solution u ε , the energy ∫ A ε | ∇ u | 2 as well as the Morse index tend to infinity as ε → 0 . This is proved through a fine blow up analysis of appropriate scalings of solutions whose limiting profiles are regular, as well as of singular solutions of some elliptic problem on ℝ n , a half-space or an infinite strip. Our argument also involves a Liouville type theorem...

Energy gaps for exponential Yang-Mills fields

Zhen Rong Zhou (2018)

Archivum Mathematicum

In this paper, some inequalities of Simons type for exponential Yang-Mills fields over compact Riemannian manifolds are established, and the energy gaps are obtained.

Energy machineries on a manifold; application to the construction of new energy representations of Gauge groups.

Jean-Yves Marion (1990)

Publicacions Matemàtiques

The introduction of the concepts of energy machinery and energy structure on a manifold makes it possible a large class of energy representations of gauge groups including, as a very particular case, the ones known up to now. By using an adaptation of methods initiated by I. M. Gelfand, we provide a sufficient condition for the irreducibility of these representations.

Energy of measures on compact Riemannian manifolds

Kathryn E. Hare, Maria Roginskaya (2003)

Studia Mathematica

We investigate the energy of measures (both positive and signed) on compact Riemannian manifolds. A formula is given relating the energy integral of a positive measure with the projections of the measure onto the eigenspaces of the Laplacian. This formula is analogous to the classical formula comparing the energy of a measure in Euclidean space with a weighted L² norm of its Fourier transform. We show that the boundedness of a modified energy integral for signed measures gives bounds on the Hausdorff...

Energy quantization and mean value inequalities for nonlinear boundary value problems

Katrin Wehrheim (2005)

Journal of the European Mathematical Society

We give a unified statement and proof of a class of well known mean value inequalities for nonnegative functions with a nonlinear bound on the Laplacian. We generalize these to domains with boundary, requiring a (possibly nonlinear) bound on the normal derivative at the boundary. These inequalities give rise to an energy quantization principle for sequences of solutions of boundary value problems that have bounded energy and whose energy densities satisfy nonlinear bounds on the Laplacian and normal...

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