Harmonic Hadamard Manifolds and Gauss Hypergeometric Differential Equations

Appell hypergeometric partial differential equations

Mathematicians often speak of the unity of their subject. Such a unity from the 19th century origins in the history of linear ordinary differential equations, especially those closely connected with elliptic and modular functions. It is concerned with the impact of group theoretical and geometrical ideas upon the problem of understanding the nature of the solutions to a differential equation (s...

Notes on differential equations and hypergeometric functions

Harmonic Maps with Prescribed Singularities into Hadamard Manifolds

Let M a Riemannian manifold of dimension m ≥ 3, let Σ be a closed smooth submanifold of M of co-dimension at least 2, and let H be a Hadamard manifold with pinched sectional curvatures. We prove the existence and uniqueness of harmonic maps φ : M \ Σ → H with prescribed singularities along Σ. When M = R, and H = H C , the complex hyperbolic space, this result has applications to the problem of ...

Arithmetic Properties of Mirror Maps Associated with Gauss Hypergeometric Equations

— We draw up the list of Gauss hypergeometric differential equations having maximal unipotent monodromy at 0 whose associated mirror map has, up to a simple rescaling, integral Taylor coefficients at 0. We also prove that these equations are characterized by much weaker integrality properties (of p-adic integrality for infinitely many primes p in suitable arithmetic progressions). It turns out ...

Degenerate Gauss hypergeometric functions

Throughout the paper, we denote this equation by E(a, b, c). Since Kummer we know that in general there are 24 hypergeometric series which express solutions of E(a, b, c). The subject of this paper is Gauss hypergeometric functions when some or all of the numbers a, b, c, c − a, c − b, a − b, c − a − b are integers. We refer to these functions and to corresponding hypergeometric equations as de...