Paper · Oct 23, 2024

Lagrangian Mean Curvature Flow in Pseudo-Euclidean Space II

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Abstract

In this paper, we consider the mean curvature flow of entire Lagrangian graphs with initial data in the pseudo-Euclidean space, which is related to the special Lagrangian parabolic equation. We show that the parabolic equation has a smooth solution $u (x, t)$ for three corresponding nonlinear equations between the Monge-Amp $e ˋ$ re type equation( $τ = 0$ ) and the special Lagrangian parabolic equation( $τ = \frac{π}{2}$ ). Furthermore, we get the bound of $D^{l} u$ , $l = {3, 4, 5, \dots}$ for $τ = \frac{π}{4}$ and the decay estimates of the higher order derivatives when $0 < τ < \frac{π}{4}$ and $\frac{π}{4} < τ < \frac{π}{2}$ . We also prove that $u (x, t)$ converges to smooth self-expanding solutions of.

Mean curvature flow Lagrangian mean curvature flow Self-shrinkers & translators

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Mean curvature flow Wider flows A hypersurface moves in the direction of its mean curvature vector, which decreases area as fast as possible. Start with: Flow by mean curvature of convex surfaces into spheres (Huisken, 1984) Lagrangian mean curvature flow Wider flows In a Calabi–Yau manifold, mean curvature flow keeps Lagrangian submanifolds Lagrangian. Its hoped-for limits are special Lagrangians, the volume… Start with: A canonical way to deform a Lagrangian submanifold (Smoczyk, 1996) Self-shrinkers & translators Wider flows Self-similar solutions of mean curvature flow: shrinkers move by scaling, expanders by scaling outward, translators by sliding. Start with: Asymptotic behavior for singularities of the mean curvature flow (Huisken, 1990)

In the guidebook

6A.10
Curve Shortening and the First Geometric FlowsBook 6A · The Heat Equation and Its Relatives
Shrinking circles, grim reapers, neckpinches and Huisken’s monotonicity, seen in the plane.

Lagrangian Mean Curvature Flow in Pseudo-Euclidean Space II

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