By using a time slicing procedure, we represent the solution operator of a second-order parabolic pseudodifferential equation on Rn as an infinite product of zero-order pseudodifferential operators. A similar representation formula is proven for parabolic differential equations on a compact Riemannian manifold. Each operator in the multi-product is given by a simple explicit Ansatz. The proof i...

We show that solutions of the Seiberg-Witten equations lead to nontrivial estimates for the L-norm of the Weyl curvature of a smooth compact 4-manifold. These estimates are then used to derive new obstructions to the existence of Einstein metrics on smooth compact 4-manifolds with a non-zero Seiberg-Witten invariant. These results considerably refine those previously obtained [21] by using scal...

In this paper we construct a Birkhoff normal form for semiclassical magnetic Schrödinger operator with non-degenerate field, and discrete well, defined on an even-dimensional Riemannian manifold $M$. We use to get expansion of the first eigenvalues in powers $\hbar^{1/2}$, Weyl asymptotics operator.

We study the complex powers A z of an elliptic, strictly positive pseudodifferential operator A using an axiomatic method that combines the approaches of Guillemin and Seeley. In particular, we introduce a class of algebras , " extended Weyl algebras, " whose definition was inspired by Guillemin's paper [11]. An extended Weyl algebra can be thought of as an algebra of " abstract pseudodifferent...

S 1 × S 2 as a bag membrane and its Einstein-Weyl geometry Abstract In the hybrid skyrmion in which an Anti-de Sitter bag is imbedded into the skyrmion configuration a S 1 × S 2 membrane is lying on the compactified spatial infinity of the bag [H. Rosu, Nuovo Cimento B 108, 313 (1993)]. The connection between the quark degrees of freedom and the mesonic ones is made through the membrane, in a w...

We have seen that irreducible finite dimensional representations of a complex simple Lie algebra g or corresponding compact Lie group are classified and can be constructed starting from an integral dominant weight. The dominance condition depends upon a choice of positive roots (or equivalently, a choice of invariant complex structure on the flag manifold.) An obvious question is that of what h...

A functional ansatz is developed which gives certain elliptic solutions of the Witten-Dijkgraaf-Verlinde-Verlinde (or WDVV) equation. This is based on the elliptic trilogarithm function introduced by Beilinson and Levin. For this to be a solution results in a number of purely algebraic conditions on the set of vectors that appear in the ansatz, this providing an elliptic version of the idea, in...

We consider S 2 bundles P and P ′ of totally null planes of maximal dimension over a 4-dimensional manifold equipped with a Weyl or Riemannian geometry. The fibre product PP ′ of P and P ′ is found to be appropriate for the encoding of both the selfdual and the Einstein-Weyl equations for the 4-metric. This encoding is realized in terms of the properties of certain well defined geometrical obje...