Mathematical Research Letters

Volume 21 (2014)

Number 6

Rigidity of time-flat surfaces in the Minkowski spacetime

Pages: 1227 – 1240

DOI: http://dx.doi.org/10.4310/MRL.2014.v21.n6.a1

Authors

Po-Ning Chen (Department of Mathematics,Columbia University, New York, N.Y., U.S.A.)

Mu-Tao Wang (Department of Mathematics,Columbia University, New York, N.Y., U.S.A.)

Ye-Kai Wang (Department of Mathematics,Columbia University, New York, N.Y., U.S.A.; and Department of Mathematics, Michigan State University, East Lansing, Mich., U.S.A.)

Abstract

A time-flat condition on spacelike $2$-surfaces in spacetime is considered here. This condition is analogous to the constant torsion condition for curves in a three-dimensional space and has been studied in [2, 5, 6, 13, 14]. In particular, any $2$-surface in a static slice of a static spacetime is time-flat. In this paper, we address the question in the title and prove several local and global rigidity theorems for such surfaces in the Minkowski and Schwarzschild spacetimes. Higher-dimensional generalizations are also considered.

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