An area bound for surfaces in Riemannian manifolds

TL;DR

The paper proves that in a compact Riemannian manifold $M$ without complete totally geodesic surfaces, the area of any complete surface immersed into $M$ is bounded by a multiple of the extrinsic curvature energy $E(f)=\frac{1}{2}\int_F|A|^2$. The authors develop a multi-layered framework combining $L^1$-almost geodesics, integral-geometry bounds via Liouville measure, and Bol–Fiala-type area estimates, together with a Gromov–Hausdorff convergence analysis and a density argument for good points to produce totally geodesic leaves in the ambient manifold when energy concentrates. A key outcome is the global bound ${\rm vol}_2^{f}(F) \le C E(f)$ under the nonexistence of complete totally geodesic surfaces, together with local area and topology controls for intrinsic metric balls. The work also connects to the existence and structure of totally geodesic immersions through a detailed GH-limit and foliation-like analysis, culminating in a Reeb-style stability argument that governs the asymptotic geometry of immersed surfaces.

Abstract

Let $M$ be a compact Riemannian manifold not containing any totally geodesic surface. Our main result shows that then the area of any complete surface immersed into $M$ is bounded by a multiple of its extrinsic curvature energy, i.e. by a multiple of the integral of the squared norm of its second fundamental form.

Theorems & Definitions (81)

• Theorem 1.1: Area bound
• Theorem 1.2
• Theorem 1.3
• Corollary 1.4
• Lemma 2.1
• Lemma 2.2
• Proposition 2.3
• Remark 2.4
• Lemma 3.1
• Lemma 3.2