Constant Higher Order Mean Curvature Research Articles

On constant higher order mean curvature hypersurfaces in H n × R ${\mathbb{H}}^{n}{\times}\mathbb{R}$

Abstract We classify hypersurfaces with rotational symmetry and positive constant r-th mean curvature in H n × R ${\mathbb{H}}^{n}{\times}\mathbb{R}$ . Specific constant higher order mean curvature hypersurfaces invariant under hyperbolic translation are also treated. Some of these invariant hypersurfaces are employed as barriers to prove a Ros–Rosenberg type theorem in H n × R ${\mathbb{H}}^{n}{\times}\mathbb{R}$ : we show that compact connected hypersurfaces of constant r-th mean curvature embedded in H n × [ 0 , ∞ ) ${\mathbb{H}}^{n}{\times}\left[0,\infty \right)$ with boundary in the slice H n × { 0 } ${\mathbb{H}}^{n}{\times}\left\{0\right\}$ are topological disks under suitable assumptions.

Rigidity of hypersurfaces with constant higher order mean curvature in space forms

Rigidity of hypersurfaces with constant higher order mean curvature in space forms

Complete Non-compact Hypersurfaces of Constant Higher Order Mean Curvature in Lorentzian Warped Product Manifolds

Complete Non-compact Hypersurfaces of Constant Higher Order Mean Curvature in Lorentzian Warped Product Manifolds

Height estimates and half-space theorems for hypersurfaces in product spaces of the type ℝ × M n.

We obtain height estimates and half-space theorems concerning a wide class of hypersurfaces immersed into a product space ℝ × M n , the so-called generalized linear Weingarten hypersurfaces, which extends that one having some constant higher order mean curvature.

Remarks on hypersurfaces with constant higher order mean curvature in Euclidean space

In this paper we consider complete oriented hypersurfaces of Euclidean space with constant higher order mean curvature and having two principal curvatures, one of them simple. As an application of the so called principal curvature theorem, a purely geometric result on the principal curvatures of the hypersurface given by Smyth and Xavier (Invent Math 90:443–450, 1987), we characterize those hypersurfaces for which the Gauss–Kronecker curvature does not change sign, extending to the general n-dimensional case a previous result for surfaces due to Klotz and Osserman.

Biharmonic hypersurfaces in 5-dimensional non-flat space forms

Abstract We prove that every biharmonic hypersurface having constant higher order mean curvature Hr for r > 2 in a space form M 5(c) is of constant mean curvature. In particular, every such biharmonic hypersurface in 𝕊5(1) has constant mean curvature. There exist no such compact proper biharmonic isoparametric hypersurfaces M in 𝕊5(1) with four distinct principal curvatures. Moreover, there exist no proper biharmonic hypersurfaces in hyperbolic space ℍ5 or in E 5 having constant higher order mean curvature Hr for r > 2.

ON THE UMBILICITY OF HYPERSURFACES IN THE HYPERBOLIC SPACE

In this paper, we establish new characterization results concerning totally umbilical hypersurfaces of the hyperbolic space$\mathbb{H}^{n+1}$, under suitable constraints on the behavior of the Lorentzian Gauss map of complete hypersurfaces having some constant higher order mean curvature. Furthermore, working with different warped product models for$\mathbb{H}^{n+1}$and supposing that certain natural inequalities involving two consecutive higher order mean curvature functions are satisfied, we study the rigidity and the nonexistence of complete hypersurfaces immersed in$\mathbb{H}^{n+1}$.

Hypersurfaces with constant higher order mean curvature in space forms

Let Mcn+1, n≥3, be a space form of constant sectional curvature c=0,1,−1 and M a complete oriented hypersurface of Mcn+1 having constant r-th mean curvature Hr for some 2≤r≤n−1 and two principal curvatures of multiplicities (n−1) and 1. We suppose further that |Hr|>0 for c=0, |Hr|>1 for c=−1 and being Hr any value for c=1. We prove that the infimum and the supremum of the squared norm of the second fundamental form of M are attained, obtain sharp bounds for them and characterize those hypersurfaces where the bounds is attained.

Hypersurfaces with constant higher order mean curvature in Euclidean space

In this paper we derive sharp estimates for the infimum and for the supremum of the squared norm of the second fundamental form of complete oriented hypersurfaces of Euclidean space with constant higher order mean curvature and having two principal curvatures, one of them simple. Besides, we characterize those hypersurfaces for which any of these bounds is attained. Our results will be an application of a purely geometric result on the principal curvatures of the hypersurface, the so called principal curvature theorem, given by Smyth and Xavier (Invent Math 90:443–450, 1987).

Weak maximum principles and geometric estimates for spacelike hypersurfaces in generalized Robertson–Walker spacetimes

In this paper we deal with complete spacelike hypersurfaces in a generalized Robertson–Walker spacetime. Using as main analytical tool a new local form of the weak maximum principle for a class of operators including the Lorentzian mean curvature operator, we obtain some mean curvature estimates and height estimates for spacelike graphs with nice Bernstein type consequences. We also give height estimates for spacelike hypersurfaces with constant higher order mean curvature, which are obtained via the local form of the weak maximum principle recently given in Alías et al. (in press) for a class of operators including those constructed from the Newton tensors of a hypersurface.

A Sharp Height Estimate for Compact Hypersurfaces with Constant k-Mean Curvature in Warped Product Spaces

AbstractIn this paper we obtain a sharp height estimate concerning compact hypersurfaces immersed into warped product spaces with some constant higher-order mean curvature and whose boundary is contained in a slice. We apply these results to draw topological conclusions at the end of the paper.

Totally Umbilical Lightlike Hypersurfaces in Robertson-Walker Spacetimes

We study the problem of lightlike hypersurface immersed into Robertson-Walker (RW) spacetimes in this paper, where the screen bundle of the hypersurface has constant higher order mean curvature. We consider the following question: under what conditions is the compact lightlike hypersurface totally umbilical? Our approach is based on the relationship between the lightlike hypersurface with its screen bundle and the Minkowski formulae for the screen bundle.

Erratum to “Spacelike hypersurfaces of constant higher order mean curvature in generalized Robertson–Walker spacetimes”. Math. Proc. Camb. Phil. Soc. (2012), 152, 365–383.

The proof of Corollary 4⋅3 in our paper [1] is not correct because there is a mistake in the expression given for ∥X* ∧ Y*∥2 on page 374. In fact, the correct expression for this term is \begin{eqnarray*} \norm{X^*\wedge Y^*}^2 & = & \norm{X^*}^2\norm{Y^*}^2-\pair{X^*,Y^*}^2\\ {} & = & 1+\pair{X,T}^2+\pair{Y,T}^2\geq 1, \end{eqnarray*} and then the inequality (4⋅9) is no longer true. Observe that all the previous reasoning before the wrong expression for ∥X* ∧ Y*∥2 is correct.

Hypersurfaces of constant higher order mean curvature in warped products

In this paper we characterize compact and complete hypersurfaces with some constant higher order mean curvature into warped product spaces. Our approach is based on the use of a new trace operator version of the Omori-Yau maximum principle which seems to be interesting in its own.

Spacelike hypersurfaces of constant higher order mean curvature in generalized Robertson–Walker spacetimes

AbstractIn this paper we analyse the problem of uniqueness for spacelike hypersurfaces with constant higher order mean curvature in generalized Robertson–Walker spacetimes. We consider first the case of compact spacelike hypersurfaces, completing some previous results given in [2]. We next extend these results to the complete noncompact case. In that case, our approach is based on the use of a generalized version of the Omori–Yau maximum principle for trace type differential operators, recently given by the authors in [3].

Conformal fields and the stability of leaves with constant higher order mean curvature

In this paper, we study hypersurfaces with constant rth mean curvature S r . We investigate the stability of such hypersurfaces in the case when they are leaves of a codimension one foliation. We also generalize recent results by Barros and Sousa, concerning conformal fields, to an arbitrary manifold. Using this we show that normal component of a Killing field is an rth Jacobi field of a hypersurface with S r + 1 constant. Finally, we study relations between rth Jacobi fields and vector fields preserving a foliation.

Capillary surfaces in a convex cone

We show that a compact embedded hypersurface $${S\subset\mathbb{R}^{n+1}}$$ with constant higher-order mean curvature in a convex piecewise smooth cone C which is perpendicular to ∂C is part of a hypersphere. Also we prove that an embedded disk type constant mean curvature surface $${S\subset\mathbb{R}^3}$$ in a convex polyhedral cone C which makes constant contact angles with ∂C is a spherical cap if C has at most five faces. This condition on the number of faces can be dropped if C is a right cone over a regular n-gon and the contact angles are the same on ∂S.

Complete spacelike hypersurfaces in conformally stationary Lorentz manifolds

We derive, for the square operator of Yau, an analogue of the Omori–Yau maximum principle for the Laplacian. We then apply it to obtain nonexistence results concerning complete noncompact spacelike hypersurfaces immersed with constant higher order mean curvature in a conformally stationary Lorentz manifold.

Uniqueness of spacelike hypersurfaces with constant higher order mean curvature in generalized Robertson–Walker spacetimes

AbstractIn this paper we study the problem of uniqueness for spacelike hypersurfaces with constant higher order mean curvature in generalized Robertson–Walker (GRW) spacetimes. In particular, we consider the following question: under what conditions must a compact spacelike hypersurface with constant higher order mean curvature in a spatially closed GRW spacetime be a spacelike slice? We prove that this happens, essentially, under the so callednull convergence condition. Our approach is based on the use of the Newton transformations (and their associated differential operators) and the Minkowski formulae for spacelike hypersurfaces.

Spacelike hypersurfaces with constant higher order mean curvature in de Sitter space

In this paper we develop Minkowski-type formulae for compact spacelike immersed hypersurfaces with boundary and having some constant higher order mean curvature in de Sitter space S 1 n + 1 . We apply them to establish a relation between the mean curvature and the geometry of the boundary, when it is a geodesic sphere contained into a horizontal hyperplane of the steady state space H n + 1 ⊂ S 1 n + 1 .