| In this dissertation, we study Gamma a family of curves on S 2 that defines a two-dimensional smooth projective plane. We use curve shortening flow (CSF) to prove that any two-dimensional smooth projective plane can be smoothly deformed through a family of smooth projective planes into one which is isomorphic to the real projective plane. In addition, as a consequence of our main result, we show that any two smooth embedded curves on RP 2 which intersect transversally at exactly one point converge to two different geodesics under the flow.;We first prove a regularity result: Gamma converges in the C infinity topology as t → infinity under CSF. We prove that if a smooth curve on S 2 is area-bisecting then under the flow its curvature function converges to zero uniformly and exponentially in the Cinfinity norm. This turns out to be a crucial estimate in proving the regularity result which leads to the proposition: CSF gives rise to a smooth homotopy when applied to an arbitrary two-dimensional smooth projective plane.;Then we show that the smooth projective structure is preserved under the smooth homotopy at any time. We study the solution to the linearized curve shortening equation and prove that it cannot vanish as t → infinity. This leads to our main result that the smooth homotopy is in fact a smooth homotopy of smooth projective planes. |
