[Paper Review] Constant higher order mean curvature hypersurfaces in Riemannian spaces

Luis J. Alı́as, Jorge H. de Lira+1 more|ArXiv.org|Nov 20, 2003

Geometric Analysis and Curvature Flows12 references3 citations

TL;DR

This paper investigates compact hypersurfaces with constant higher-order mean curvature in Riemannian manifolds, extending symmetry results to non-Euclidean spaces. By analyzing boundary geometry and using flux formulas and reflection principles, it proves that such hypersurfaces in hyperbolic space and the sphere with spherical boundaries must be spherical caps, generalizing prior results in Euclidean space.

ABSTRACT

It is still an open question whether a compact embedded hypersurface in the Euclidean space R^{n+1} with constant mean curvature and spherical boundary is necessarily a hyperplanar ball or a spherical cap, even in the simplest case of surfaces in R^3. In a recent paper the first and third authors have shown that this is true for the case of hypersurfaces in R^{n+1} with constant scalar curvature, and more generally, hypersurfaces with constant higher order r-mean curvature, when r>1. In this paper we deal with some aspects of the classical problem above, by considering it in a more general context. Specifically, our starting general ambient space is an orientable Riemannian manifold, where we will consider a general geometric configuration consisting of an immersed hypersurface with boundary on an oriented hypersurface P. For such a geometric configuration, we study the relationship between the geometry of the hypersurface along its boundary and the geometry of its boundary as a hypersurface of P, as well as the geometry of P. Our approach allows us to derive, among others, interesting results for the case where the ambient space has constant curvature. In particular, we are able to extend the previous symmetry results to the case of hypersurfaces with constant higher order r-mean curvature in the hyperbolic space and in the sphere.

Motivation & Objective

• To resolve the open problem of whether compact embedded hypersurfaces with constant mean curvature and spherical boundary in Euclidean space are necessarily spherical caps or flat discs.
• To generalize symmetry results for constant r-mean curvature hypersurfaces beyond Euclidean space to include hyperbolic space and the sphere.
• To establish a geometric relationship between the hypersurface's boundary, its boundary on a supporting hypersurface P, and the ambient manifold’s curvature.
• To prove that compact hypersurfaces with constant higher-order mean curvature and spherical boundary in space forms (Rⁿ⁺¹, Hⁿ⁺¹, Sⁿ⁺¹) must be spherical caps.
• To develop a reflection principle and use flux formulas to rule out non-symmetric configurations, thereby proving uniqueness under symmetry assumptions.

Proposed method

• Analyzes the geometry of an immersed hypersurface M in a Riemannian manifold M̄, with boundary on a hypersurface P, using Newton transformations and higher-order mean curvatures.
• Applies a flux formula to relate the integral of the r-th Newton transformation over the boundary to the geometry of M and P.
• Employs a reflection principle via one-parameter families of totally geodesic spheres or hypersurfaces to detect symmetry.
• Uses transversality and homology arguments to rule out components of M ∩ ext(D) that could break symmetry, leading to contradiction if non-symmetric.
• Constructs a continuous family of spheres R(t) and uses continuity of the reflection process to show that symmetry must be preserved.
• Applies the maximum principle and convexity of the boundary Σ to show that any symmetry sphere must coincide with the original, proving uniqueness.

Experimental results

Research questions

• RQ1 Are compact embedded hypersurfaces with constant r-mean curvature and spherical boundary in Rⁿ⁺¹ necessarily spherical caps or flat discs?
• RQ2 Can the symmetry results for constant scalar and higher-order mean curvature hypersurfaces in Euclidean space be extended to hyperbolic space and the sphere?
• RQ3 What geometric conditions ensure that a hypersurface with boundary on a totally geodesic hypersurface P inherits symmetry from its boundary?
• RQ4 Under what conditions does the reflection process through a family of spheres preserve the hypersurface’s structure and lead to a contradiction if symmetry is broken?
• RQ5 How does the flux formula relate the geometry of M, its boundary ∂M, and the ambient manifold’s curvature in the context of constant r-mean curvature?

Key findings

• In the sphere Sⁿ⁺¹ and hyperbolic space Hⁿ⁺¹, compact embedded hypersurfaces with constant r-mean curvature and spherical boundary are necessarily spherical caps.
• The paper proves that if a hypersurface M in a space form has boundary Σ that is a round (n−1)-sphere and M is compact with constant r-mean curvature, then M must be symmetric under reflection across any hyperplane through the center of Σ.
• The authors establish that M ∩ P = Σ and M lies entirely in the closure of one side of P, ruling out self-intersections or components extending beyond the boundary.
• Using a continuous family of reflection spheres T(t), the paper shows that if the reflection process reaches a symmetry sphere, then M must be symmetric, contradicting the absence of boundary points in certain regions.
• The flux formula and homology arguments are used to show that no component of M ∩ ext(D) can be homologous to zero unless it leads to a contradiction, thereby enforcing symmetry.
• The result generalizes Alexandrov’s theorem to the case of non-empty boundary, showing that spherical caps are the only compact, embedded, constant r-mean curvature hypersurfaces with spherical boundary in space forms.