Möbius geometry of hypersurfaces with constant mean curvature and scalar curvature

Let be a hypersurface in the (m+1)-dimensional unit sphere S^m+1 without umbilics. Four basic invariants of x under the Möbius transformation group in S^m+1 are a Riemannian metric g called Möbius metric, a 1-form called Möbius form, a symmetric (0,2) tensor A called Blaschke tensor and symmetric (0,2) tensor B called Möbius second fundamental form. In this paper, we prove the following classification theorem: let be a hypersurface, which satisfies (i) 0, (ii) A+g+B0 for some functions and , then and must be constant, and x is Möbius equivalent to either (i) a hypersurface with constant mean curvature and scalar curvature in S^m+1; or (ii) the pre-image of a stereographic projection of a hypersurface with constant mean curvature and scalar curvature in the Euclidean space R^m+1; or (iii) the image of the standard conformal map of a hypersurface with constant mean curvature and scalar curvature in the (m+1)-dimensional hyperbolic space H^m+1. This result shows that one can use Möbius differential geometry to unify the three different classes of hypersurface with constant mean curvature and scalar curvature in S^m+1, R^m+1 and H^m+1.Partially supported the Alexander Humboldt Stiftung and Zhongdian grant of NSFC.Partially supported by RFDP, Qiushi Award, 973 Project and Jiechu grant of NSFC.Mathematics Subject Classification (2000):Primary 53A30; Secondary 53B25     

Rotationally invariant hypersurfaces with constant mean curvature in the Heisenberg group ℍ n

In this article we study sets in the (2n + 1)-dimensional Heisenberg group ℍ ⁿ which are critical points, under a volume constraint, of the sub-Riemannian perimeter associated to the distribution of horizontal vector fields in ℍ ⁿ .We define a notion of mean curvature for hypersurfaces and we show that the boundary of a stationary set is a constant mean curvature (CMC) hypersurface. Our definition coincides with previous ones. Our main result describes which are the CMC hypersurfaces of revolution in ℍ ⁿ .The fact that such a hypersurface is invariant under a compact group of rotations allows us to reduce the CMC partial differential equation to a system of ordinary differential equations. The analysis of the solutions leads us to establish a counterpart in the Heisenberg group of the Delaunay classification of constant mean curvature hypersurfaces of revolution in the Euclidean space. Hence, we classify the rotationally invariant isoperimetric sets in ℍ ⁿ .     

Stable constant mean curvature hypersurfaces in ℝ n+1 and ℍ n+1(−1)

In this paper, we show that all complete stable hypersurfaces in ⁿ⁺¹(or ⁿ⁺¹ (-1)) (n = 3, 4, 5) with constant mean curvature H > 0 (or H > 1, respectively) and finite L ² norm of traceless second fundamental form are compact geodesic spheres. Keywords: stable hypersurface, constant mean curvature, isometric immersion, Bernstein theorem.*Supported by PolyU grant G-T575.**Partially supported by CNPq of Brazil.     

Asymptotic behavior of Cauchy hypersurfaces in constant curvature space–times

We present some new examples of families of cubic hypersurfaces in \(\mathbb {P}^5 (\mathbb {C})\) containing a plane whose associated quadric bundle does not have a rational section.     

Some remarks on compact constant mean curvature hypersurfaces in a halfspace of ℍ n+1

We consider embedded compact hypersurfacesM in a halfspace of hyperbolic space with boundaryM in the boundary geodesic hyperplaneP of the halfspace and with non-zero constant mean curvature. We prove the following. Let {M _n } be a sequence of such hypersurfaces withM _n contained in a disk of radiusr _n centered at a point P such thatr _n 0 and that eachM _n is a large. H-hypersurface,H > 1. Then there exists a subsequence of {M _n } converging to the sphere of mean curvatureH tangent toP at. In the case of smallH-hypersurfaces orH 1, if we add a condition on the curvature of the boundary, there exists a subsequence of {M _n } which are graphs. The convergence is smooth on compact subset of ³ .     

On the Gauss mapping of hypersurfaces with constant scalar curvature in ℍ n+1

In this paper, we deal with complete hypersurfaces immersed in the hyperbolic space with constant scalar curvature. By supposing suitable restrictions on the Gauss mapping of such hypersurfaces we obtain some rigidity results. Our approach is based on the use of a generalized maximum principle, which can be seen as a sort of extension to complete (noncompact) Riemannian manifolds of the classical Hopf’s maximum principle.     

Willmore hypersurfaces with constant Möbius curvature in R n+1

For an immersed hypersurface ${f : M^n \rightarrow R^{n+1}}$ without umbilical points, one can define the Möbius metric g on f which is invariant under the Möbius transformation group. The volume functional of g is a generalization of the well-known Willmore functional, whose critical points are called Willmore hypersurfaces. In this paper, we prove that if a n-dimensional Willmore hypersurfaces ${(n \geq 3)}$ has constant sectional curvature c with respect to g, then c = 0, n = 3, and this Willmore hypersurface is Möbius equivalent to the cone over the Clifford torus in ${S^{3} \subset R^{4}}$ . Moreover, we extend our previous classification of hypersurfaces with constant Möbius curvature of dimension ${n \ge 4}$ to n = 3, showing that they are cones over the homogeneous torus ${S^1(r) \times S^1(\sqrt{1 - r^2}) \subset S^3}$ , or cylinders, cones, rotational hypersurfaces over certain spirals in the space form R ², S ², H ², respectively.     

Classification of hypersurfaces with constant M?bius curvature in {\mathbb{S}^{m+1}}

Let ${x: M^{m} \rightarrow \mathbb{S}^{m+1}}$ be an m-dimensional umbilic-free hypersurface in an (m?+?1)-dimensional unit sphere ${\mathbb{S}^{m+1}}$ , with standard metric I?= dx · dx. Let II be the second fundamental form of isometric immersion x. Define the positive function ${\rho=\sqrt{\frac{m}{m-1}}\|II-\frac{1}{m}tr(II)I\|}$ . Then positive definite (0,2) tensor ${\mathbf{g}=\rho^{2}I}$ is invariant under conformal transformations of ${\mathbb{S}^{m+1}}$ and is called M?bius metric. The curvature induced by the metric g is called M?bius curvature. The purpose of this paper is to classify the hypersurfaces with constant M?bius curvature.     

Ricci flow of warped product metrics with positive isotropic curvature on S p+1 × S 1

We study the asymptotic behaviour of the ODE associated to the evolution of curvature operator in the Ricci flow of a doubly warped product metric on S ^p?+?1 ×S ¹ with positive isotropic curvature.     

Bright solitons of the variants of the Novikov–Veselov equation with constant and variable coefficients

In this paper, the existence of the bright soliton solution of four variants of the Novikov–Veselov equation with constant and time varying coefficients will be studied. We analyze the solitary wave solutions of the Novikov–Veselov equation in the cases of constant coefficients, time-dependent coefficients and damping term, generalized form, and in 1 + N dimensions with variable coefficients and forcing term. We use the solitary wave ansatz method to derive these solutions. The physical parameters in the soliton solutions are obtained as functions of the dependent coefficients. Parametric conditions for the existence of the exact solutions are given. The solitary wave ansatz method presents a wider applicability for handling nonlinear wave equations.     

Spectral Geometry of Closed Minimal Real Hypersurfaces in CP~(n+1) and QP~(n+1)

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On the mean curvature of spacelike surfaces in certain three-dimensional Robertson–Walker spacetimes and Calabi–Bernstein’s type problems

Several uniqueness results for the spacelike slices in certain Robertson–Walker spacetimes are proved under boundedness assumptions either on the mean curvature function of the spacelike surface or on the restriction of the time coordinate on the surface when the mean curvature is a constant. In the nonparametric case, a uniqueness result and a nonexistence one are proved for bounded entire solutions of some constant mean curvature spacelike differential equations.     

Classification of hypersurfaces with parallel Mbius second fundamental form in S~(n+1)

Let M~n(n≥2) be an immersed umbilic-free hypersurface in the (n+1)-dimensional unit sphere S~(n+1). Then M~n is associated with a so-called Mobius metric g,and a Mobius second fundamental form B which are invariants of M~n under the Mobiustransformation group of S~(n+1). In this paper, we classify all umbilic-free hypersurfaces withparallel Mobius second fundamental form.     

Generalized normal form and the formal equivalence of systems with zero approximation (x 2 3 , — x 1 3 )

We consider formal systems of differential equations of the form

(S − 1,S) Perishable systems with stochastic leadtimes

This article discusses a one-to-one ordering perishable system, in which reorders are processed in the order of their arrival and the processing times are arbitrarily distributed, and as such, the leadtimes are not independent. The Markov renewal techniques are employed to obtain the various operating characteristics for the case of Poisson demand and exponential lifetimes. The problem of minimizing the steady state expected cost rate is also discussed, and in the special case of exponential processing times, the optimal stock level is derived explicitly.     

Stability and bifurcation analysis of a mathematical model for tumor–immune interaction with piecewise constant arguments of delay

In this paper, we propose and analyze a Lotka–Volterra competition like model which consists of system of differential equations with piecewise constant arguments of delay to study of interaction between tumor cells and Cytotoxic T lymphocytes (CTLs). In order to get local and global behaviors of the system, we use Schur–Cohn criterion and constructed a Lyapunov function. Some algebraic conditions which satisfy local and global stability of the system are obtained. In addition, we investigate the possible bifurcation types for the system and observe that the system may undergo Neimark–Sacker bifurcation. Moreover, it is predicted a threshold value above which there is uncontrollable tumor growth, and below periodic solutions that leading to tumor dormant state occur.