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

Benoit Daniel (1) , William H. Meeks , Harold Rosenberg

1 LAMA - Laboratoire d'Analyse et de Mathématiques Appliquées

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.

Domaines: Physique mathématique [math-ph]

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⟩