I will talk on a recent joint preprint with Jean-Baptiste Casteras and Jaime Ripoll. We prove the existence of solutions to the asymptotic Plateau problem for hypersurfaces of prescribed mean curvature in Cartan-Hadamard manifolds N. More precisely, given a suitable subset L of the asymptotic boundary of N and a suitable function H on N, we are able to construct a set of locally finite perimeter whose boundary has generalized mean curvature H and asymptotic boundary L provided that N satisfies the so-called strict convexity condition and that its sectional curvatures are bounded from above by a negative constant. We also obtain a multiplicity result in low dimensions.

I will review a recently appeared joint article [Math. Z. 290 (2018), 221-250] with Jean-Baptiste Casteras and Jaime Ripoll on the asymptotic Plateau problem on Cartan-Hadamard (abbr. CH) manifolds satisfying the so-called strict convexity (abbr. SC) condition. We consider CH manifolds whose sectional curvatures are bounded from above and below by certain functions depending on the distance to a fixed point. In particular, we are able to verify the SC condition on manifolds whose curvature lower bound can go to \(-\infty\) and upper bound to \(0\) simultaneously at certain rates, or on some manifolds whose sectional curvatures go to \(-\infty\) faster than any prescribed rate. These improve previous results of Anderson, Borb\'ely, and Ripoll and Telichevsky. We then solve the asymptotic Plateau problem for locally rectifiable currents with \(\mathbb{Z}_2\)- multiplicity in a Cartan-Hadamard manifold satisfying the SC condition given any compact topologically embedded \((k-1)\)- dimensional submanifold of the sphere at infinity, \(2\le k\le n-1\), as the boundary data. These generalize previous results of Anderson, Bangert, and Lang.