Classification Theorems of Complete Space-Like Lagrangian \(\xi \)-Surfaces in the Pseudo-Euclidean Space \({\mathbb R}^4_2\)

Abstract

\(\xi \)-submanifolds and \(\xi \)-translators are, respectively, the natural generalizations of self-shrinkers and translators of the mean curvature flow and, in the case of codimension one, they are previously known as \(\lambda \)-hypersurfaces and \(\lambda \)-translators, respectively. In this paper, we study the complete Lagrangian space-like \(\xi \)-surfaces and \(\xi \)-translators in \({\mathbb R}^4_2\), the pseudo-Euclidean 4-spaces of signature 2 endowed with the canonical complex structure. As the result, we first obtain a classification theorem for all complete Lagrangian space-like \(\xi \)-surfaces in \({\mathbb R}^4_2\) of constant square norm of the second fundamental form. Then the main idea of the proof also allows us to obtain a similar classification theorem for \(\xi \)-translators in \({\mathbb R}^4_2\) by a Bernstein-type theorem for space-like translators in a general pseudo-Euclidean space \({\mathbb R}^{m+p}_p\), which is of independent significance.

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Correspondence to Xingxiao Li.

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Research supported by National Natural Science Foundation of China (No. 11671121, No. 11871197 and No 11971153).

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Li, X., Qiao, R. & Liu, Y. Classification Theorems of Complete Space-Like Lagrangian \(\xi \)-Surfaces in the Pseudo-Euclidean Space \({\mathbb R}^4_2\). Bull Braz Math Soc, New Series (2021). https://doi.org/10.1007/s00574-020-00235-4

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Keywords

  • Mean curvature
  • Second fundamental form
  • Lagrangian space-like \(\xi \)-submanifolds
  • Classification

Mathematics Subject Classification

  • 53C40
  • 53C44