We prove that every plane passing through the origin divides an embedded compact free boundary minimal surface of the euclidean
3
3
-ball in exactly two connected surfaces. We also show that if a region in the ball has mean convex boundary and contains a nullhomologous diameter, then this region is a closed halfball. Moreover, we prove the regularity at the corners of currents minimizing a partially free boundary problem by following ideas by Grüter and Simon. Our first result gives evidence to a conjecture by Fraser and Li.
In this paper we prove existence of complete minimal surfaces in some metric semidirect products. These surfaces are similar to the doubly and singly periodic Scherk minimal surfaces in R 3 . In particular, we obtain these surfaces in the Heisenberg space with its canonical metric, and in Sol 3 with a one-parameter family of non-isometric metrics.Mathematics Subject Classification (2010): 53C42.
In this paper we develop the theory of properly immersed minimal surfaces in the quotient space H 2 × R /G, where G is a subgroup of isometries generated by a vertical translation and a horizontal isometry in H 2 without fixed points. The horizontal isometry can be either a parabolic translation along horocycles in H 2 or a hyperbolic translation along a geodesic in H 2 . In fact, we prove that if a properly immersed minimal surface in H 2 × R /G has finite total curvature then its total curvature is a multiple of 2π, and moreover, we understand the geometry of the ends. These theorems hold true more generally for properly immersed minimal surfaces in M × S 1 , where M is a hyperbolic surface with finite topology whose ends are isometric to one of the ends of the above spaces H 2 × R /G.
We prove an Alexandrov type theorem for a quotient space of H 2 × R . More precisely we classify the compact embedded surfaces with constant mean curvature in the quotient of H 2 × R by a subgroup of isometries generated by a horizontal translation along horocycles of H 2 and a vertical translation. Moreover, we construct some examples of periodic minimal surfaces in H 2 × R and we prove a multi-valued Rado theorem for small perturbations of the helicoid in H 2 × R.
We prove a half-space theorem for an ideal Scherk graph Σ ⊂ M × R over a polygonal domain D ⊂ M, where M is a Hadamard surface whose curvature is bounded above by a negative constant. More precisely, we show that a properly immersed minimal surface contained in D × R and disjoint from Σ is a translate of Σ.
We prove a version of the Jenkins-Serrin theorem for the existence of CMC graphs over bounded domains with infinite boundary data in Sol 3 . Moreover, we construct examples of admissible domains where the results may be applied.
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