The following paper develops the outline described by Anton Petrunin to solve Gromov's problem on total absolute curvature for simply connected surfaces:

Convexity and rigidity of hypersurfaces in Cartan-Hadamard manifolds

The main result of the paper generalizes Chern-Lashof characterization of convex hypersurfaces in Euclidean space, and rigidity theorem of Greene-Wu-Gromov for Cartan-Hadamard manifolds.

The following paper develops the outline described by Anton Petrunin to solve Gromov's problem on total absolute curvature for simply connected surfaces:

Convexity and rigidity of hypersurfaces in Cartan-Hadamard manifolds

The main result of the paper generalizes Chern-Lashof characterization of convex hypersurfaces in Euclidean space, and rigidity theorem of Greene-Wu-Gromov for Cartan-Hadamard manifolds.

I have just finished a paper which develops the outline described by Anton Petrunin to solve Gromov's problem on total absolute curvature for simply connected surfaces:

Convexity and rigidity of hypersurfaces in Cartan-Hadamard manifolds

The main result of the paper generalizes Chern-Lashof characterization of convex hypersurfaces in Euclidean space, and rigidity theorem of Greene-Wu-Gromov for Cartan-Hadamard manifolds.

The following paper develops the outline described by Anton Petrunin to solve Gromov's problem on total absolute curvature for simply connected surfaces:

Convexity and rigidity of hypersurfaces in Cartan-Hadamard manifolds

The main result of the paper generalizes Chern-Lashof characterization of convex hypersurfaces in Euclidean space, and rigidity theorem of Greene-Wu-Gromov for Cartan-Hadamard manifolds.