Submanifolds in de Sitter space-time satisfying \(\Delta H = \lambda H\) (Q1903123)

Let \(x:M^n\to S^{m-1}_1(1)\subset E^m_1\) be an isometric immersion from an \(n\)-dimensional Riemannian manifold \(M\) into de Sitter spacetime \(S^{m-1}_1(1)\). Theorem: 1. \(\Delta H=\lambda H\), \(\lambda<n\), if and only if \(M\) is contained in a spacelike non-totally geodesic, totally umbilical hypersurface of \(S^{m-1}_1\) as a minimal submanifold. 2. \(\Delta H=nH\), if and only if either (a) \(M\) is a minimal submanifold in \(S^{m-1}_1\) or (b) up to rigid motions of \(E^m_1\), the immersion is given by the formula \(x=(f+h,f+h,x_3,\dots,x_m)\), where \(h\) is a harmonic function, \(f\) an eigenfunction with eigenvalue \(n\), and \(y=(x_3,\dots,x_m):M\to S^{m-3}(1)\subset E^m_1\) is an isometric minimal immersion into the sphere \(S^{m-3}\). 3. Let \(M\) be isometrically immersed into \(S^{m-1}_1\) with parallel nonzero mean curvature vector. Then \(\Delta H=\lambda H\), \(\lambda>n\), if and only if \(x\) immerses \(M\) minimally in \(N_1(c):=\{x\in S^{m-1}_1:\langle c,x\rangle= \langle c,c\rangle\}\), where \(c\in E^m_1\) is a spacelike vector satisfying \(\langle c,c\rangle= 1-n/\lambda\). The author also constructs examples of spacelike submanifolds of \(S^{m-1}_1\) satisfying \(\Delta H=\lambda H\), \(\lambda>n\), but with non-parallel mean curvature vector.