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].

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].

Reviewer: Mykola Grygorenko (Kyïv)

##### 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 |

##### Keywords:

mean field equations; developing function; generalized Lamé equation; singular points; differential Galois group; monodromy data; Painlevé equation VI
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\textit{C.-S. Lin}, in: 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. 187--217 (2018; Zbl 1410.34265)