Curve Shortening Flow and Smooth Projective Planes

TL;DR

This paper demonstrates that curve shortening flow can deform any smooth projective plane on the sphere into the real projective plane, and shows convergence of intersecting curves to geodesics.

Contribution

It establishes a deformation process for smooth projective planes via curve shortening flow and proves convergence of intersecting curves to geodesics on RP^2.

Findings

Any smooth projective plane on S^2 can be deformed into the real projective plane.

Intersecting curves on RP^2 converge to geodesics under the flow.

The deformation preserves the smooth projective plane structure.

Abstract

In this paper, we study a family of curves on $S^2$ that defines a two-dimensional smooth projective plane. We use curve shortening flow 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.