On locally strongly convex affine hypersurfaces with parallel cubic form. Part I

In this paper, we study locally strongly convex affine hypersurfaces of Rn+1 that have parallel cubic form with respect to the Levi-Civita connection of the affine Berwald-Blaschke metric; it is known that they are affine spheres. In dimension n?7 we give a complete classification of such hypersurfaces; in particular, we present new examples of affine spheres.     

δ-invariants and their applications to centroaffine geometry

We introduce the notion of δ-invariant for curvature-like tensor fields and establish optimal general inequalities in case the curvature-like tensor field satisfies some algebraic Gauss equation. We then study the situation when the equality case of one of the inequalities is satisfied and prove a dimension and decomposition theorem. In the second part of the paper, we apply these results to definite centroaffine hypersurfaces in Rn+1. The inequality is specified into an inequality involving the affine δ-invariants and the Tchebychev vector field. We show that if a centroaffine hypersurface satisfies the equality case of one of the inequalities, then it is a proper affine hypersphere. Furthermore, we prove that if a positive definite centroaffine hypersurface in , satisfies the equality case of one of the inequalities, it is foliated by ellipsoids. And if a negative definite centroaffine hypersurface satisfies the equality case of one of the inequalities, then it is foliated by two-sheeted hyperboloids. Some further applications of the inequalities are also provided in this article.     

On spacelike hypersurfaces with constant scalar curvature in the de Sitter space

We classify spacelike hypersurfaces of the de Sitter space with constant scalar curvature and with two principal curvatures. Moreover, we prove that if Mn is a complete spacelike hypersurface 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<(n−2)c/n. Additionally, we prove several rigidity theorems for such hypersurfaces.     

Affine Kähler hypersurfaces satisfying certain conditions on the curvature tensor

The notion of affine Kähler immersions has been recently introduced by Nomizu-Pinkall-Podestà ([N-Pi-Po]). This work is aimed at giving some results towards the classification of non degenerate affine Kähler hypersurfaces with symmetric and parallel Ricci tensor; this problem generalizes the classical results due to Nomizu-Smyth ([N-S]) in the theory of Kählerian hypersurfaces. In a second section we deal with the case of “semisymmetric” affine Kähler immersions, when the curvature tensor R satisfies R · R = 0 and the Ricci tensor is symmetric, providing a complete classification; for affine Kähler curves we prove that the conditions above are actually equivalent to saying that the immersion is isometric for a suitable Kähler metric in C².     

Geometry of curves and surfaces through the contact map

We introduce a new approach to the study of affine equidistants and centre symmetry sets via a family of maps obtained by reflexion in the midpoints of chords of a submanifold of affine space. We apply this to surfaces in R³, previously studied by Giblin and Zakalyukin, and then apply the same ideas to surfaces in R⁴, elucidating some of the connexions between their geometry and the family of reflexion maps. We also point out some connexions with symplectic topology.     

Affine minimal hypersurfaces of rotation

We give a complete list of affine minimal surfaces inA ³ with Euclidean rotational symmetry, completing the treatise given in [1] and prove that these surfaces have maximal affine surface area within the class of all affine surfaces of rotation satisfying suitable boundary conditions. Besides we show that for rotationally symmetric locally strongly convex affine minimal hypersurfaces inA ⁿ ,n4, the second variation of the affine surface area is negative definite under certain conditions on the meridian.     

Almost complex curves and Hopf hypersurfaces in the nearly Kähler 6-sphere

We characterize Hopf hypersurfaces inS ⁶ as open parts of geodesic hyperspheres or of tubes around almost complex curves ofS ⁶.     

A new variational characterization of the Wulff shape

Given a positive function F on Sn which satisfies a convexity condition, we define the rth anisotropic mean curvature function Mr for hypersurfaces in Rn+1 which is a generalization of the usual rth mean curvature function. Let be an n-dimensional closed hypersurface with , for some r with 1?r?n−1, which is a critical point for a variational problem. We show that X(M) is stable if and only if X(M) is the Wulff shape.     

Maximum properties of affine minimal hypersurfaces

It is shown that the sign of the second variation of locally strongly convex affine minimal hypersurfaces in affine space A ⁿ for n ≥ 4 can not be determined by a suitable reduction to a sum of squares as was done for n = 3 in [3]. Also we prove that strictly stable locally strongly convex affine minimal hypersurfaces are a relative weak maximum of the affine area functional, and give an affine version of the Morse-Smale index theorem [16].     

A sharp height estimate for compact spacelike hypersurfaces with constant r-mean curvature in the Lorentz-Minkowski space and application

In this paper we obtain a sharp height estimate concerning compact spacelike hypersurfaces Σn immersed in the (n+1)-dimensional Lorentz-Minkowski space Ln+1 with some nonzero constant r-mean curvature, and whose boundary is contained into a spacelike hyperplane of Ln+1. Furthermore, we apply our estimate to describe the nature of the end of a complete spacelike hypersurface of Ln+1.     

On some properties of the curvature and Ricci tensors in complex affine geometry

We present a new approach — which is more general than the previous ones — to the affine differential geometry of complex hypersurfaces inC ⁿ⁺¹. Using this general approach we study some curvature conditions for induced connections.The research supported by Alexander von Humboldt Stiftung and KBN grant no. 2 P30103004.     

Stability of spacelike hypersurfaces in foliated spacetimes

Given a generalized Robertson-Walker spacetime whose warping function verifies a certain convexity condition, we classify strongly stable spacelike hypersurfaces with constant mean curvature. More precisely, we will show that given a closed, strongly stable spacelike hypersurface of with constant mean curvature H, if the warping function ? satisfying ?^″?max{H?^′,0} along M, then Mn is either maximal or a spacelike slice Mt₀={t₀}×F, for some t₀∈I.     

Absolute gradient bounds for nonparametric hypersurfaces of constant mean curvature

We consider the problem of determining the existence of absolute apriori gradient bounds of nonparametric hypersurfaces of constant mean curvature in ann-dimensional sphereB _R, 1>R>R ₀ ⁽ⁿ⁾ , (R ₀ ⁽ⁿ⁾ being a constant depending only onn), without imposing boundary conditions or bounds of any sort.

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Sunto Consideriamo il problema di determinare stime a priori di gradienti di ipersuperfici non parametriche di curvatura media costante in una sferan-dimensionaleB _R, 1>R>R ₀ ⁽ⁿ⁾, (R ₀ ⁽ⁿ⁾ essendo una costante che dipende solo dan), senza imporre condizioni al contorno o limiti di altro tipo.

     

Complete hypersurfaces of R4 with constant mean curvature

In this paper, we completely classify complete hypersurfaces inR ⁴ with constant mean curvature and constant scalar curvature.The project was supported by NNSFC, FECC, and CPF.     

On a family of lie algebras related to homogeneous surfaces

Real affine homogeneous hypersurfaces of general position in three-dimensional complex space ?³ are studied. The general position is defined in terms of the Taylor coefficients of the surface equation and implies, first of all, that the isotropy groups of the homogeneous manifolds under consideration are discrete. It is this case that has remained unstudied after the author’s works on the holomorphic (in particular, affine) homogeneity of real hypersurfaces in three-dimensional complex manifolds. The actions of affine subgroups G ? Aff(3, ?) in the complex tangent space T ^? _p M of a homogeneous surface are considered. The situation with homogeneity can be described in terms of the dimensions of the corresponding Lie algebras. The main result of the paper eliminates “almost trivial” actions of the groups G on the spaces T _p ^? M for affine homogeneous strictly pseudoconvex surfaces of general position in ?³ that are different from quadrics.     

On the curvatures of bounded complete spacelike hypersurfaces in the Lorentz–Minkowski space

In this paper we characterize the spacelike hyperplanes in the Lorentz–Minkowski space L ^{n +1} as the only complete spacelike hypersurfaces with constant mean curvature which are bounded between two parallel spacelike hyperplanes. In the same way, we prove that the only complete spacelike hypersurfaces with constant mean curvature in L ^{n +1} which are bounded between two concentric hyperbolic spaces are the hyperbolic spaces. Finally, we obtain some a priori estimates for the higher order mean curvatures, the scalar curvature and the Ricci curvature of a complete spacelike hypersurface in L ^{n +1} which is bounded by a hyperbolic space. Our results will be an application of a maximum principle due to Omori and Yau, and of a generalization of it. Received: 5 July 1999     

An Optimal Inequality and Extremal Classes of Affine Spheres in Centroaffine Geometry

A hypersurface f : M → Rⁿ⁺¹ in an affine (n+1)-space is called centroaffine if its position vector is always transversal to f_*(TM) in Rⁿ⁺¹. In this paper, we establish a general optimal inequality for definite centroaffine hypersurfaces in Rⁿ⁺¹ involving the Tchebychev vector field. We also completely classify the hypersurfaces which verify the equality case of the inequality.     

On hypersurfaces inR n+1 with integral bounds on curvature

We show that the L ^p norm of the second fundamental form of hypersurfaces in R ⁿ⁺¹ is very coercive in the GMT setting of Gauss graphs currents, when p exceeds the dimension n. A compactness result for immersed hypersurfaces and its application to a variational problem are provided.     

Large time behavior of disturbed planar fronts in the Allen-Cahn equation

We consider the Allen-Cahn equation in Rn (with n?2) and study how a planar front behaves when arbitrarily large (but bounded) perturbation is given near the front region. We first show that the behavior of the disturbed front can be approximated by that of the mean curvature flow with a drift term for all large time up to t=+∞. Using this observation, we then show that the planar front is asymptotically stable in L^∞(Rn) under spatially ergodic perturbations, which include quasi-periodic and almost periodic ones as special cases. As a by-product of our analysis, we present a result of a rather general nature, which states that, for a large class of evolution equations, the unique ergodicity of the initial data is inherited by the solution at any later time.