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Mohammad Ghomi
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Convex surfaces with minimal total curvature in Cartan-Hadamard space

A Cartan-Hadamard space $M$ is a complete simply connected 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?

Mohammad Ghomi
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