Electrostatically positioned droplets are very useful for the fundamental study of solidification phenomena and the measurement of thermal physical properties. This paper descries a numerical analysis of surface deformation and surface tension driven flows in electrostatically positioned droplets in microgravity. The analysis is based on a fully coupled boundary element and finite element solution of the Maxwell equations, the Navier-Stokes equations and the energy balance equation. Results show that an applied electrostatic field results in a nonuniform electric stress distribution along the droplet surface, which, combined with surface tension, causes the droplet to deform into an ellipsoidal shape in microgravity. Laser heating induces a non-uniform temperature distribution in the droplet, which in turn produces Marangoni convection in the droplet. It is found that the viscous stress contribution to the deformation is small for a majority of cases. Also, a higher temperature gradient produces a stronger Marangoni convection in droplets with higher melting points that require more laser power. The internal recirculating flow may be reduced by more uniform laser heating. During the undercooling of the droplet, both temperature and fluid flow fields evolve in time such that the temperature gradient and the tangential velocities along the droplet surface subside in magnitude and reverse their directions.