Zejun Hu

Hypersurfaces of the hyperbolic space with constant scalar curvature

We classify hypersurfaces of the hyperbolic space ℍn+1(c) with constant scalar curvature and with two distinct principal curvatures. Moreover, we prove that if Mn is a complete hypersurfaces with constant scalar curvature n(n − 1) R and with two distinct principal curvatures such that the multiplicity of one of the principal curvatures is n− 1, then R ≥ c. Additionally, we prove two rigidity theorems for such hypersurfaces.

WILLMORE SUBMANIFOLDS IN A RIEMANNIAN MANIFOLD

where S = ∑ α,i,j(h α ij) 2 and H are respectively the norm square of the second fundamental form and the mean curvature of the immersion x, dv is the volume element of M . In this survey paper, we calculate the Euler-Lagrangian equation of W (x) for an n-dimensional submanifold in an (n + p)-dimensional Riemannian manifold Nn+p and give many applications as well as many examples of Willmore submanifolds.

Scalar Curvature, Killing Vector Fields and Harmonic One-Forms on Compact Riemannian Manifolds

It is well known that no non-trivial Killing vector field exists on a compact Riemannian manifold of negative Ricci curvature; analogously, no non-trivial harmonic one-form exists on a compact manifold of positive Ricci curvature. One can consider the following, more general, problem. By reducing the assumption on the Ricci curvature to one on the scalar curvature, such vanishing theorems cannot hold in general. This raises the question: “What information can we obtain from the existence of non-trivial Killing vector fields (or, respectively, harmonic one-forms)?” This paper gives answers to this problem; the results obtained are optimal.

Classification of the Locally Strongly Convex Centroaffine Hypersurfaces with Parallel Cubic Form

In this paper, we study locally strongly convex centroaffine hypersurfaces with parallel cubic form with respect to the Levi–Civita connection of the centroaffine metric. As the main result, we obtain a complete classification of such centroaffine hypersurfaces. The result of this paper is a centroaffine version of the complete classification of locally strongly convex equiaffine hypersurfaces with parallel cubic form due to Hu et al. (J Differ Geom 87:239–307, 2011).