  The 2-D Peskin problem describes a 1-D closed elastic string immersed and
moving in a 2-D Stokes flow that is induced by its own elastic force. The
geometric shape of the string and its internal stretching configuration evolve
in a coupled way, and they combined govern the dynamics of the system. In this
paper, we show that certain geometric quantities of the moving string satisfy
extremum principles and decay estimates. As a result, we can prove that the 2-D
Peskin problem admits a unique global solution when the initial data satisfies
a medium-size geometric condition on the string shape, while no assumption on
the size of stretching is needed.
