Stable constant mean curvature hypersurfaces in the real projective space
Let \(\psi: M= M^n\to \mathbb{R}\mathbb{P}^{n+1}\) be a compact two-sided hypersurface immersed into the real projective space \(\mathbb{R}\mathbb{P}^{n+1}\). The strong index \(\text{Ind}(M)/\text{weak}\) index \(\text{Ind}_T(M)\) of the hypersurface \(\psi\) is the maximum dimension of a subspace \(V\) of \(C^\infty(M)/\) of \(C^\infty_T(M):= \{u\in C^\infty(M): \int_M u\,dv= 0\}\) on which the quadratic form \(Q(u):= -\int_M u\,J(u)\,dv\) acting on the space \(C^\infty(M)\) of smooth functions on \(M\) is negative definite, where \(J = \Delta+|A|^2+ n\) (with \(\Delta\) = Laplacian operator on \(M\), \(A\) = second fundamental form of \(\psi\)) stands for the Jacobi or stability operator of \(\psi\). The hypersurface \(\psi\) is called strongly stable/weakly stable if and only if \(\text{Ind}(M)= 0/\text{Ind}_T(M)= 0\), respectively. In this paper, the authors demonstrate that no constant mean curvature hypersurface is strongly stable, and -- with regard to weak stability -- they prove, under the additional hypothesis of constant scalar curvature, the following Theorem 1: The only compact two-sided hypersurfaces with constant mean curvature \(H\) which are weakly stable in \(\mathbb{R}\mathbb{P}^{n+1}\) and have constant scalar curvature are (i) the twofold covering of a totally geodesic projective space; (ii) the geodesic spheres in \(\mathbb{R}\mathbb{P}^{n+1}\); and (iii) the quotient to \(\mathbb{R}\mathbb{P}^{n+1}\) of the hypersurface \(\mathbb{S}^k(r)\times \mathbb{S}^{n-k}(\sqrt{1- r^2})\hookrightarrow\mathbb{S}^{n+ 1}\) obtained as the product of two spheres of dimensions \(k\) and \(n-k\), with \(k= 1,\dots, n- 1\), and radii \(r\) and \(\sqrt{1-r^2}\), respectively, with \(\sqrt{k/(n+ 2)}\leq r\leq \sqrt{(k+2)/(n+ 2)}\).
scientific article; zbMATH DE number 5118888