Abstract
In this paper, we prove that the global existence of solutions to timelike minimal surface equations having arbitrary co-dimension with slow decay initial data in two space dimensions and three space dimensions, provided that the initial value is suitably small.MSC:35L70.
Highlights
The theory of minimal surfaces has a long history, originating with the papers of Lagrange ( ) and the famous Plateau problem; we refer to the classical papers by Calabi [ ] and by Cheng and Yau [ ]
Timelike minimal submanifolds may be viewed as simple but nontrivial examples of D-branes, which play an important role in string theory, and the system under consideration here has natural generalizations motivated by string theory
Huang and Kong [ ] studied the motion of a relativistic torus in the Minkowski space R +n (n ≥ ). They derived the equations for the motion of relativistic torus in the Minkowski space R +n (n ≥ )
Summary
The theory of minimal surfaces has a long history, originating with the papers of Lagrange ( ) and the famous Plateau problem; we refer to the classical papers by Calabi [ ] and by Cheng and Yau [ ]. This kind of equation describes the three dimensional timelike extremal submanifolds in the Minkowski space R +n They showed that these equations can be reduced to a ( + ) dimensional quasilinear symmetric hyperbolic system and the system possesses some interesting properties, such as nonstrict hyperbolicity, constant multiplicity of eigenvalues, linear degeneracy of all characteristic fields, and the strong null condition (see [ ] and [ ]). We show the global existence of solutions to timelike minimal surface in two space dimensions and three space dimensions, provided that the initial value is suitably small. The global existence of solutions to timelike minimal surface equations with slow decay initial value in two space dimensions and three space dimensions will be proved in Section and Section , respectively.
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