Paper · Mar 18, 2024

Long time regularity of the $p$ -Gauss curvature flow with flat side

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Abstract

In this paper, we prove the long time regularity of the interface in the $p$ -Gauss curvature flow with flat side in all dimensions for $p > \frac{1}{n}$ . Here the interface is the boundary of the flat part in the flow. In dimension $2$ , this problem was solved in for $p = 1$ and in for $p \in (1/2, 1)$ . We utilize the duality method to transform the Gauss curvature flow to a singular parabolic Monge-Ampère equation, and prove the regularity of the interface by studying the asymptotic cone of the parabolic Monge-Ampère equation in the polar coordinates.

Other curvature flows Gauss curvature flow Kähler geometry

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Long time regularity of the $p$ -Gauss curvature flow with flat side

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Long time regularity of the p-Gauss curvature flow with flat side

Long time regularity of the $p$ -Gauss curvature flow with flat side