Analytic Extension of Riemannian Analytic Manifolds and Local Isometries

A Local Courant Theorem on Real Analytic Manifolds

Let M be a closed real analytic Riemannian manifold. We estimate from below the volume of a nodal domain component in an arbitrary ball, provided that this component enters the ball deeply enough. We also improve the existing estimates in the smooth case.

Neighborhoods of Analytic Varieties in Complex Manifolds

Analytic fields on compact balanced Hermitian manifolds

On a Hermitian manifold we construct a symmetric (1, 1)tensor H using the torsion and the curvature of the Chern connection. On a compact balanced Hermitian manifold we find necessary and sufficient conditions in terms of the tensor H for a harmonic 1-form to be analytic and for an analytic 1form to be harmonic. We prove that if H is positive definite then the first Betti number b1 = 0 and the ...

Baker-sprindžuk Conjectures for Complex Analytic Manifolds

The circle of problems that the present paper belongs to dates back to the 1930s, namely, to K. Mahler’s work on a classification of transcendental real and complex numbers. For a polynomial P (x) = a0 + a1x + · + anx n ∈ Z[x], let us denote by hP the height of P , that is, hP def = maxi=0,...,n |ai|. It can be easily shown using Dirichlet’s Principle that for any z ∈ C and any n ∈ N there exis...

Invariant Manifolds for Analytic Difference Equations

We use a modification of the parameterization method to study invariant manifolds for difference equations. We establish existence, regularity, smooth dependence on parameters and study several singular limits, even if the difference equations do not define a dynamical system. This method also leads to efficient algorithms that we present with their implementations. The manifolds we consider in...