Proves sharp total Gauss-Kronecker curvature inequality for convex hypersurfaces in Cartan-Hadamard manifolds with nullity index ≥ n-3 via Chern-Gauss-Bonnet, extending the isoperimetric inequality and proving the Cartan-Hadamard conjecture there.
Total absolute curvature and rigidity of surfaces in Cartan-Hadamard manifolds
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abstract
We show that closed surfaces with minimal total absolute curvature in Cartan-Hadamard 3-manifolds bound flat convex bodies. This generalizes Chern-Lashof's theorem for surfaces in Euclidean space and solves a problem posed by Gromov in 1985. Our proof is based on an isometric embedding construction via holonomy, and uses Pogorelov's theory of surfaces with bounded extrinsic curvature. Along the way, we obtain a regularity result for convex hulls and a Schur-type comparison theorem for curves in Cartan-Hadamard manifolds.
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math.DG 1years
2026 1verdicts
UNVERDICTED 1representative citing papers
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Isoperimetric and total curvature inequalities in Cartan-Hadamard manifolds with nullity
Proves sharp total Gauss-Kronecker curvature inequality for convex hypersurfaces in Cartan-Hadamard manifolds with nullity index ≥ n-3 via Chern-Gauss-Bonnet, extending the isoperimetric inequality and proving the Cartan-Hadamard conjecture there.