Workshops## DPW method for indefinite proper affine spheres and finite gap solutions

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Seiichi Udagawa

2012-03-17

14:00:00 - 14:50:00

308 , Mathematics Research Center Building (ori. New Math. Bldg.)

This is a joint work with Jun-ichi INOGUCHI. We consider indefinite proper affine spheres in affine 3- space A^3. Given some kind of potential, we may produce a Blaschke immersion of an indefinite proper affine sphere by the DPW method. In our case, we consider the twisted loop group $\lambda G_{\tau}$ with the order 6 outer automorphism $\tau$, where $G=SL(3, R)$. To obtain the above DPW method for affine spheres, we need to decompose the twisted loop group on a big cell and we need some involution $\sigma$. To obtain the solutions of affine spheres explicitly, we need to solve the Tzitzeica equation : $\partial_{x}\partial_{t}u=e^{ u}-e^{-2u}$, where $h=e^{u}dxdt$ is an indefinite Blaschke metric on affine sphere. We may give the solutions in terms of not only elliptic functions but also Riemann theta functions, which belongs to the class of affine spheres of finite type. This is already known in the mathematical physics literature [CS]. However, they do not give the geometric object explicitly. We may give the geometric objects for expressing the solutions explicitly, and consequently we may give the Blaschke immersions of affine spheres of finite type in terms of Riemann theta functions and the explicit geometric objects. [CS] I. Yu. Cherdantsev and R. A. Sharipov, "Finite-gap solutions of the Bullough-Dodd-Zhiber-Shabat equation", Teor i Mathem. Fizika 82(1990), 155-160. English translation: Theoret. Math. Phys. 82(1990), no.1, 108-111.For material related to this talk, click here.