An extension of Liebmann’s theorem to hypersurfaces with boundary

Key Points

• The study aims to extend Liebmann's theorem to locally convex hypersurfaces with boundaries and analyze their properties.
• Proving the extension of Liebmann's theorem for compact, connected CMC hypersurfaces
• Analyzing the conditions for these hypersurfaces relative to their boundaries
• Investigating the relationship between the geometry of hypersurfaces and their bounding submanifolds.
• Locally convex CMC hypersurfaces are constrained by closed strictly convex boundaries in a hyperplane.
• Hypersurfaces with non-zero constant mean curvature must be spherical caps when bounded by (n-1)-spheres.
• The derived geometrical conditions extend and refine the existing theorem on convex surfaces.

Abstract