  In this paper, we study the boundary effect on the surface (or frame) tension
of elastic membrane surface models. The frame tension generally depends only on
the projected area of the boundary over which the surface spans. However, from
a spin model analogy, the frame tension is expected to be dependent also on the
boundary shape at the continuous transition point. We confirm this expectation
using the following fixed-connectivity and tethered surface models: the surface
model of Helfrich and Polyakov and a surface model with deficit angle term. We
also discuss the reason why this expectation is worthwhile to study.
