Hilário Alencar

Hypersurfaces With Constant Mean Curvature in Spheres

Let \(M^n\)be a compact hypersurface of a sphere with constant mean curvature H. We introduce a tensor \(\phi\) related to H and to the second fundamental form, and show that if \(|\phi|^2\leq B_{H}\), where \(B_{H}\neq 0\)is a number depending only on H and n, then either \(|\phi|^2\equiv 0\) or \(|\phi|^2\equiv{B}_{H}.\) We also characterize all \(M^n\)with \(|\phi|^2\equiv{B}_{H}.\)

On the first eigenvalue of the linearized operator of ther-th mean curvature of a hypersurface

We generalize Reillys inequality for the first eigenvalue of immersed submanifolds ofIRm+1 and the total (squared) mean curvature, to hypersurfaces ofIRm+1 and the first eigenvalue of the higher order curvatures. We apply this to stability problems. We also consider hypersurfaces in hyperbolic space.

Stable hypersurfaces with constant scalar curvature

It is well known that hypersurfaces \(M^n\)with constant mean curvature in a Riemannian manifold \(\overline{M}{n+1}(c)\)of constant sectional curvature c are solutions to the variational problem of extremizing the area function for volumepreserving variations.

Integral Formulas for the r-Mean Curvature Linearized Operator of a Hypersurface

AbstractFor a normal variation of a hypersurface Mn in a space form Qcn+1 by a normal vector field fN, R. Reilly proved:

Eigenvalue estimates for a class of elliptic differential operators on compact manifolds

\frac{d}{{dt}}S_{r + 1} (t)|_{t = 0} = L_r f + (S_1 S_{r + 1} - (r + 2)S_{r + 2} )f + c(n - r)S_r f,

Stable hypersurfaces with zero scalar curvature in Euclidean space

where Lr (0 < r < n − 1) is the linearized operator of the (r + 1)-mean curvature Sr+1 of Mn given by Lr = div(Pr∇); that is, Lr = the divergence of the rth Newton transformation Pr of the second fundamental form applied to the gradient ∇, and L0 = Δ the Laplacian of Mn.From the Dirichlet integral formula for Lr

Stable hypersurfaces with constant scalar curvature.

\int {_{M^n } } (fL_r g + \left\langle {P_r \nabla f,\nabla g} \right\rangle ) = 0