Curve shortening flow and smooth projective planes

In this paper, we study a family of curves on $\mathbb{S}^2$ 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 $\mathbb{RP}^2$ which intersect transversally at exactly one point converge to two different geodesics under the flow.

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Received 16 August 2013

Accepted 3 June 2017

Published 12 December 2019