Convexity and rigidity of hypersurfaces in Cartan-Hadamard manifolds

TL;DR

The paper proves that closed infinitesimally convex hypersurfaces $\Gamma$ in Cartan–Hadamard (i.e., $\textup{CAT}^n(k_{\le 0})$) manifolds with $K_M\equiv k$ on tangent planes bound convex $k$-flat regions, generalizing the Chern–Lashof–Sacksteder characterization to CAT$(k)$ spaces. The approach embeds $\Gamma$ into the model space $\mathcal{M}^n_k$ via Gauss–Codazzi, preserving the second fundamental form, and then uses a generalized Schur comparison in CAT spaces, via Reshetnyak majorization, to compare extrinsic distances; Kirszbraun-type extension yields a global isometry of the bounded convex bodies. The intrinsic version follows by translating the extrinsic isometry through the convex hull in the model space, and the results yield total-curvature bounds and rigidity statements that connect to Gromov’s gap theorems and classical convexity results in Euclidean and hyperbolic settings. These methods illustrate how Alexandrov geometry techniques can solve convexity/rigidity problems in nonpositively curved manifolds and potentially inform curvature-gap phenomena in broader contexts.

Abstract

We show that in Cartan-Hadamard manifolds $M^n$, $n\geq 3$, closed infinitesimally convex hypersurfaces $Γ$ bound convex flat regions, if curvature of $M^n$ vanishes on tangent planes of $Γ$. This encompasses Chern-Lashof-Sacksteder characterization of compact convex hypersurfaces in Euclidean space, and some results of Greene-Wu-Gromov on rigidity of Cartan-Hadamard manifolds. It follows that closed simply connected surfaces in $M^3$ with minimal total absolute curvature bound Euclidean convex bodies, as stated by Gromov in 1985. The proofs employ the Gauss-Codazzi equations, a generalization of Schur comparison theorem to CAT($k$) spaces, and other techniques from Alexandrov geometry outlined by Petrunin.

Theorems & Definitions (11)

• Theorem 1.1
• Theorem 1.2
• Corollary 1.3
• proof
• Proposition 2.1
• Lemma 2.2
• proof
• proof : Proof of Proposition \ref{['prop:embedding']}
• Theorem 3.1: Generalized Schur's Comparison
• Lemma 3.2: Reshetnyak's Majorization Theorem akp2019reshetnyak1968