Constant Mean Curvature Hypersurfaces Condensing to Geodesic Segments and Rays in Riemannian Manifolds

TL;DR

This paper constructs special constant mean curvature hypersurfaces in symmetric Riemannian manifolds by gluing small spheres along geodesics, revealing how ambient scalar curvature enables their existence.

Contribution

It introduces a novel gluing method to build constant mean curvature hypersurfaces condensing to geodesic segments in curved spaces, unlike in Euclidean space.

Findings

Existence of compact constant mean curvature hypersurfaces in symmetric manifolds.

Scalar curvature gradient acts as a 'friction term' facilitating the construction.

Surfaces cannot exist in Euclidean space, highlighting curvature's role.

Abstract

We construct examples of compact and one-ended constant mean curvature surfaces with large mean curvature in Riemannian manifolds with axial symmetry by gluing together small spheres positioned end-to-end along a geodesic. Such surfaces cannot exist in Euclidean space, but we show that the gradient of the ambient scalar curvature acts as a `friction term' which permits the usual analytic gluing construction to be carried out.