Explicit Solutions to the mean field equations on hyperelliptic curves of genus two

Dispersionless Hirota equations and the genus 3 hyperelliptic divisor

Equations of dispersionless Hirota type have been thoroughly investigated in the mathematical physics and differential geometry literature. It is known that the parameter space of integrable Hirota type equations in 3D is 21-dimensional and the action of the natural equivalence group Sp(6, R) on the parameter space has an open orbit. However the structure of the `master-equation' corresponding to this orbit remained elusive. Here we prove that the master-equation is specified by the vanishing of any genus 3 theta constant with even characteristic. The rich geometry of integrable Hirota type eq…

Hyperelliptic Loop Solitons with Genus g: Investigations of a Quantized Elastica

In the previous work (J. Geom. Phys. {\bf{39}} (2001) 50-61), the closed loop solitons in a plane, \it i.e., loops whose curvatures obey the modified Korteweg-de Vries equations, were investigated for the case related to algebraic curves with genera one and two. This article is a generalization of the previous article to those of hyperelliptic curves with general genera. It was proved that the tangential angle of loop soliton is expressed by the Weierstrass hyperelliptic al function for a given hyperelliptic curve $y^2 = f(x)$ with genus $g$.

亏格为三的聚焦规范修正KdV方程超椭圆解的闭合实平面曲线

Closed real plane curves of hyperelliptic solutions of focusing gauged modified KdV equation of genus three

本研究将欧拉弹性线推广，针对亏格为三的超椭圆曲线，通过调节模参数与初始条件，构造出满足聚焦规范修正KdV方程、超越欧拉8字形弹性线的闭合实平面曲线。

Filling area conjecture and ovalless real hyperelliptic surfaces

We prove the filling area conjecture in the hyperelliptic case. In particular, we establish the conjecture for all genus 1 fillings of the circle, extending P. Pu's result in genus 0. We translate the problem into a question about closed ovalless real surfaces. The conjecture then results from a combination of two ingredients. On the one hand, we exploit integral geometric comparison with orbifold metrics of constant positive curvature on real surfaces of even positive genus. Here the singular points are Weierstrass points. On the other hand, we exploit an analysis of the combinatorics on unio…