On almost-fuchsian submanifolds of Hadamard spaces and the asymptotic Plateau problem

We consider minimal submanifolds of negatively curved spaces with small curvature. We show that in a Hadamard space with negatively pinched curvature <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>-</mml:mo> <mml:mi>C</mml:mi> <mml:mo>≤</mml:mo> <mml:mi>K</mml:mi> <mml:mo>≤</mml:mo> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> , complete minimal submanifolds with second fundamental form less than <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mn>1</mml:mn> </mml:math> everywhere bound a class of spheres at infinity for which the asymptotic Plateau problem is uniquely solvable.