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Conformal geometry and the Painlevé VI equation. (English) Zbl 1410.34265
Ji, Lizhen (ed.) et al., Uniformization, Riemann-Hilbert correspondence, Calabi-Yau manifolds and Picard-Fuchs equations. Based on the conference, Institute Mittag-Leffler, Stockholm, Sweden, July 13–18, 2015. Somerville, MA: International Press; Beijing: Higher Education Press. Adv. Lect. Math. (ALM) 42, 187-217 (2018).
The paper discusses the connection between the so-called mean field equations and Painlevé VI equations. Let $$\mathbb{H}$$ be the upper half-plane of the complex plane $$\mathbb{C}$$, $$\tau \in \mathbb{H}$$, $$\omega_0=0$$, $$\omega_1=1$$, $$\omega_2=\tau$$, $$\omega_3=1+\tau$$, $$\Lambda_\tau$$ be the lattice generated by $$\omega_1, \omega_2$$, $$E_\tau=\mathbb{C}/{\Lambda_\tau}$$ be a flat torus and $$F_\tau$$ be the field of meromorphic functions on $$E_\tau$$. As an example, the author considers the mean field equation of the form $\Delta u+e^u=8\pi\sum_{i=0}^3 n_i\delta(\frac{\omega_i}{2})+4\pi\delta(p)+4\pi\delta(-p)\quad\text{in }E_\tau,\tag{1}$ where $$\delta(p)$$ is the Dirac measure with the support at $$p\in E_\tau$$, and $$n_i>-1$$. For each solution $$u(z)$$ of equation (1), the expression $$u_{zz}-\frac{1}{2}u_z^2:=-2I(z)$$ is an elliptic function, which in the case of even $$u(z)$$ may be described by the expression $I(z)=\sum_{i=0}^3 n_i(n_i+1)\wp (z+\frac{\omega_i}{2})+\frac{3}{4}(\wp (z+p)+\wp (z-p))+A(\varsigma (z+p)-\varsigma (z-p))+B,$ where $$A$$ and $$B$$ are two unknown constants that are specified further.
Next, the author introduces a linear differential equation ${y}''=I(z)y\tag{2}$ (a generalized Lamé equation), which plays a key role in the task at hand. There are a number of important properties of the equation (2) due to the relationship with the equation (1). For example: for any developing function $$f$$ of the solution $$u(z)$$, the Schwarzian derivative $$\{f;z\}$$ is equal to $$-2I(z)$$; if $$\sigma$$ is a differential isomorphism over $$F_\tau$$ and $$f_1$$ is a developing function for $$\sigma(u)$$, then $$\{f_1;z\}=-2I(z)$$; if $$f_1$$ is an analytic continuation of $$f$$, then $$f_1$$ is a developing function for $$u(z)$$ and $$f_1\sim f$$ are equivalent by the fractional linear transformation from $$\mathrm{PSU}(2)$$; the singular points $$\pm p$$ of (2) are apparent; if $$\{g\in F_\tau<f>\mid g(z+\omega_1)=g(z+\omega_2)=g(z)\}\subset F_\tau$$ then the differential Galois group $$G=\mathrm{Gal}(F_\tau<f>/{F_\tau})$$ is commutative and the action of $$G$$ on the field $$F_\tau <f>$$ is determined by a pair of relations $f(z+\omega_j)=\frac{a_j f+b_j}{c_j f+d_j},\, B_j=\begin{pmatrix} a_j & b_j \\ c_j & d_j \\ \end{pmatrix}\in \mathrm{SL}(2,\mathbb{C})\;(j=1,2) \text{ and }[B_1,B_2]=E.$ The paper deals with the simplest case when $$f(z+\omega_j)=e^{2\pi i\theta_j}f(z)$$, $$\theta_j\in \mathbb{R}\;(j=1,2)$$. Under certain conditions, the (monodromy) data $$(\theta_1,\theta_2)$$ is preserved while $$\tau$$ is deforming and accordingly the group $$G$$ is saved. This is a key point that allows the author to relate equation (2) (and therefore equation (1)) to the Painlevé equation VI.
For the entire collection see [Zbl 1398.14003].
##### MSC:
 34M35 Singularities, monodromy and local behavior of solutions to ordinary differential equations in the complex domain, normal forms 34M55 Painlevé and other special ordinary differential equations in the complex domain; classification, hierarchies 34M56 Isomonodromic deformations for ordinary differential equations in the complex domain 30B50 Dirichlet series, exponential series and other series in one complex variable 53A30 Conformal differential geometry (MSC2010) 35J91 Semilinear elliptic equations with Laplacian, bi-Laplacian or poly-Laplacian 12H05 Differential algebra