Consider the incompressible viscoelasticity, a coupling system of the Navier-Stokes equations for the fluid motion with a transport equation for the deformation tensor. Under a natural force balance law on the free boundary with the surface tension, its well-posedness theory has been established for any fixed viscosity coefficient ε > 0. In this talk, we solve the corresponding elastic case on a short time interval, i.e. ε = 0. Our method is the vanishing viscosity limit by establishing a uniform a priori estimates with respect to the viscosity. We point out that based on a crucial new observation on the inherent structure of the elastic term on the free boundary, the framework here is established all in standard Sobolev spaces, but not the co-normal ones. This is a joint work with Prof. Zhen Lei and Prof. Fanghua Lin.