Lagrangian Mean Curvature Flow in Pseudo-Euclidean Space II
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866909360367599616 |
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| author | Li, Shanshan Lv, Jiaru Huang, Rongli |
| author_facet | Li, Shanshan Lv, Jiaru Huang, Rongli |
| contents | 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}. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_17794 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Lagrangian Mean Curvature Flow in Pseudo-Euclidean Space II Li, Shanshan Lv, Jiaru Huang, Rongli Differential Geometry 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}. |
| title | Lagrangian Mean Curvature Flow in Pseudo-Euclidean Space II |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2410.17794 |