Translation Minimal Surfaces In Sol₃

In this article we deal with one of Thurston geometries, the Sol3 geometry. In the space Sol3, a translation surface is defined by φ(s,t)=β(t)*α(s) where α and β are graphs of functions contained in coordinate planes. Sol3 is also a solvable lie group with group operator*.Our work is focusing on the translation minimal surfaces, that is, surfaces whose mean curvature H vanishes on the surface. As the curves of α(s) and β(t) are in different coordinate planes, we can get three kinds of translation minimal surfaces, namely, M1:φ(s,t)=(t+e-g(t)s,eg(t)f(s),g(t)),M2:φ(s,t)=(e-g(t)s,t+eg(t)f(s),g(t)) and M3:φ(s,t)=(e-g(t)s,t,f(s)+g(t)).Each of surfaces are satisfied second-order nonlinear ordinary differential equations, some examples of minimal surfaces M1 and M2 can get by solving equations. With respect to α(s)=(s,0,f(s)) and β(t)=(0,t,9(t)),the equation for surface M3 is very elaborate. So we solve it through adding the initial condition. In our calculations, when doing Taylor series expansion at 0 point launched to cube of variate, the solution is given by formal power series. Finally, we made a discussion about the existence and uniqueness of solution for initial value problem.