A half-space property for hypersurfaces in the hyperbolic space
Annales Polonici Mathematici
MSC: Primary 53C42; Secondary 53C20
DOI: 10.4064/ap230629-31-1
Published online: 3 April 2024
Abstract
Through the study of geometry of the hyperspheres (also known as equidistant spheres) of the hyperbolic space $\mathbb H^{n+1}$, we establish a nonexistence result for complete noncompact hypersurfaces immersed into $\mathbb H^{n+1}$ and a characterization of complete totally geodesic hypersurfaces of $\mathbb H^{n +1}$; namely, we characterize those complete hyperspheres of $\mathbb H^{n +1}$ with the following geometric property: any geodesic contained in a complete hypersurface is also a geodesic of $\mathbb H^{n+1}$. Our approach is based on a suitable maximum principle at infinity for complete Riemannian manifolds.
