Convex surfaces with minimal total curvature in Cartan-Hadamard 3-space

Question

A Cartan-Hadamard 3-space $M$ is a complete simply connected 3-dimensional Riemannian manifold with nonpositive sectional curvature. A (smooth) convex surface $\Gamma\subset M$ is an embedded topological sphere with nonnegative second fundamental form $\mathrm{I\!I}$. The total (Gauss-Kronecker) curvature of $\Gamma$ is defined as $$ \mathcal{G}(\Gamma):=\int_\Gamma\det(\mathrm{I\!I}). $$ It follows quickly from Gauss' equation and Gauss-Bonnet theorem that $\mathcal{G}(\Gamma)\geq 4\pi$. Suppose that $\mathcal{G}(\Gamma)= 4\pi$. Does it follow that the compact region of $M$ bounded by $\Gamma$ is Euclidean, i.e., all its sectional curvatures are zero?

Note 1: Schroeder and Strake showed in this paper (see Theorem 2) that the answer is yes, provided that $\Gamma$ is strictly convex, i.e., the second fundamental form is positive definite. Strict convexity appears to be an essential feature of the proof.

Note 2: On page 66 of Lectures on Manifolds on Nonpositive Curvature (see Exercise (b)), Gromov poses a more general question for total absolute curvature of closed surfaces in Cartan-Hadamard $3$-spaces (the term "absolute" is not explicitly mentioned).

Answer 1

Note that curvature in sectional directions tangent to the surface, say $\Sigma$ vanish. It seems sufficient to conclude that the usual Peterson–Codazzi equations hold for the surface. It follows that there is a convex surface $\Sigma'$ in the Euclidian space that is isometric to $\Sigma$; plus they have identical second fundamental forms.

Now we want to use bow lemma (diffgeometric analog of arm lemma). The bow lemma has a version for Hadamard spaces. Applying it to plane arcs in $\Sigma'$, we get that the map $\Sigma'\to\Sigma$ is distance-noncontracting.

Applying Reshetnyak majorization theorem, to plane sections of $\Sigma'$, we get that the map $\Sigma'\to\Sigma$ is distance-preserving.

Finally, applying Kirszbraun, we get that the map $\Sigma'\to\Sigma$ extends to a distance-preserving map from the convex hull of $\Sigma'$.

Answer 2

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.