Nevanlinna and algebraic hyperbolicity

Motivated by the notion of the algebraic hyperbolicity, we introduce the notion of Nevanlinna hyperbolicity for a pair [Formula: see text], where [Formula: see text] is a projective variety and [Formula: see text] is an effective Cartier divisor on [Formula: see text]. This notion links and unifies the Nevanlinna theory, the complex hyperbolicity (Brody and Kobayashi hyperbolicity), the big Picard-type extension theorem (more generally the Borel hyperbolicity). It also implies the algebraic hyperbolicity. The key is to use the Nevanlinna theory on parabolic Riemann surfaces recently developed by Păun and Sibony [Value distribution theory for parabolic Riemann surfaces, preprint (2014), arXiv:1403.6596 ].

Entire solutions of certain type of non-linear differential equations

Utilizing Nevanlinna’s value distribution theory of meromorphic functions, we study transcendental entire solutions of the following type nonlinear differential equations in the complex plane$$\begin{array}{} \displaystyle f^{n}(z)+P(z,f,f',\ldots,f^{(t)})=P_{1}\text{e}^{\alpha_{1}z}+P_{2}\text{e}^{\alpha_{2}z}+P_{3}\text{e}^{\alpha_{3}z}, \end{array}$$where Pj and αi are nonzero constants for j = 1, 2, 3, such that |α1| > |α2| > |α3| and P(z, f, f′, …, f(t) is an algebraic differential polynomial in f(z) of degree no greater than n – 1.

Painlevé σ-equations S1, S2, S4 and their value distribution

Local solutions of Painlev? equations P1, P2, P4, as well as of the equations S1, S2, S4 satisfied by their Hamiltonians, can be extended to functions meromorphic in C. This way they become a point of interest for value distribution theory. Distribution of values of solutions of P1, P2 and P4 is already well described. In the paper we discuss mostly S1, S2 and S4 in this context. In particular, we pay attention to deficient, asymptotic and ramified values of solutions of these equations.