The conformal rigidity of certain subdomains on a Riemann surface

Abstract

The differential dP2 is said to have a pole of order k at a point p if h has a pole of order k there. The point p is often referred to as a pole of d 2. If d 2 is non-negative on the contours F, then dD2 is called a quadratic differential of V, every quadratic differential on a compact surface being a quadratic differential of the surface. A curve along which dD2 is positive is called a trajectory [3] of dD2. It is known [7 ] that if Vi is a subregion of V with the property that there are no mappings of Vi into V which are arbitrarily close to the identity mapping of V1 into V, then V1 must be a dense subregion of V whose boundary relative to V consists of slits along which some quadratic differential d 2 of V is non-negative. The principal result of this paper is the proof of a strong form of the converse of this fact, a proof that under the proper conditions there are not only no conformal mappings near the identity but no conformal mappings which are homotopic to the identity. More precisely we have the following theorem.