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Lagrangian Mean Curvature Flow in Pseudo-Euclidean Space II

Shanshan Li, Jiaru Lv, Rongli Huang

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 \eqref{11} has a smooth solution $u(x,t)$ for three corresponding nonlinear equations between the Monge-Amp$\grave{e}$re type equation($τ=0$) and the special Lagrangian parabolic equation($τ=\fracπ{2}$). Furthermore, we get the bound of $D^lu$, $l=\{3,4,5,\cdots\}$ for $τ=\fracπ{4}$ and the decay estimates of the higher order derivatives when $0<τ<\fracπ{4}$ and $\fracπ{4}<τ<\fracπ{2}$. We also prove that $u(x,t)$ converges to smooth self-expanding solutions of \eqref{12}.

Lagrangian Mean Curvature Flow in Pseudo-Euclidean Space II

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 \eqref{11} has a smooth solution for three corresponding nonlinear equations between the Monge-Ampre type equation() and the special Lagrangian parabolic equation(). Furthermore, we get the bound of , for and the decay estimates of the higher order derivatives when and . We also prove that converges to smooth self-expanding solutions of \eqref{12}.

Paper Structure

This paper contains 5 sections, 20 theorems, 142 equations.

Table of Contents

  1. Introduction
  2. lagrangian self-expanding solution in $(\mathbb{R}^{2n},g_\tau)$
  3. proof of Theorem\ref{['1.2']} and \ref{['1.3']}
  4. proof of Theorem \ref{['1.21']} and \ref{['1.312']}
  5. proof of Theorem \ref{['1.112']} and Theorem \ref{['1.31']}

Key Result

Theorem 1.2

Let $u_{0} : \mathbb{R}^{n}\to\mathbb{R}$ be a $C^{2}$ function satisfying condition $B$ and consider the equation Then there exists a unique solution of 121 such that where $u(\cdot,t)$ satisfies condition $B$. More generally, for $l=\{3,4,5,\cdots\}$ and $\varepsilon_{0}>0$, there holds where $C$ depends only on $n$, $\zeta$, $\varrho$, $\frac{1}{\varepsilon_{0}}.$

Theorems & Definitions (36)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Proposition 2.1
  • proof
  • Definition 2.2
  • ...and 26 more