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HALF-SPACE THEOREMS FOR MINIMAL SURFACES IN Nil(3) AND Sol(3)

Abstract : We prove some half-space theorems for minimal surfaces in the Heisenberg group Nil(3) and the Lie group Sol(3) endowed with their standard left-invariant Riemannian metrics. If S is a properly immersed minimal surface in Nil(3) that lies on one side of some entire minimal graph G, then S is the image of G by a vertical translation. If S is a properly immersed minimal surface in Sol(3) that lies on one side of a special plane epsilon(t) (see the discussion just before Theorem 1.5 for the definition of a special plane in Sol(3)), then S is the special plane epsilon(u) for some u is an element of R.
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https://hal-upec-upem.archives-ouvertes.fr/hal-00692989
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Submitted on : Tuesday, May 1, 2012 - 7:15:54 PM
Last modification on : Thursday, March 19, 2020 - 12:26:02 PM

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  • HAL Id : hal-00692989, version 1

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Benoit Daniel, William H. Meeks, Harold Rosenberg. HALF-SPACE THEOREMS FOR MINIMAL SURFACES IN Nil(3) AND Sol(3). Journal of Differential Geometry, 2011, 88 (1), pp.41--59. ⟨hal-00692989⟩

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