Mean Curvature Vector Research Articles

On mean curvatures in submanifolds geometry

By using moving frame theory, first we introduce 2p-th mean curvatures and (2p + 1)-th mean curvature vector fields for a submanifold. We then give an integral expression of them that characterizes them as mean values of symmetric functions of principle curvatures. Next we apply it to derive directly the celebrated Weyl-Gray tube formula in terms of integrals of the 2p-th mean curvatures and some Minkowski-type integral formulas.

Conformal and semi-conformal biharmonic maps

We show that a conformal mapping between Riemannian manifolds of the same dimension n ≥ 3 is biharmonic if and only if the gradient of its dilation satisfies a certain second-order elliptic partial differential equation. On an Einstein manifold solutions can be generated from isoparametric functions. We characterise those semi-conformal submersions that are biharmonic in terms of their dilation and the fibre mean curvature vector field.

Analysis aspects of Willmore surfaces

A new formulation for the Euler–Lagrange equation of the Willmore functional for immersed surfaces in ℝ m is given as a nonlinear elliptic equation in divergence form, with non-linearities comprising only Jacobians. Letting $\vec{H}$ be the mean curvature vector of the surface, our new formulation reads ${\mathcal{L}}\vec{H}=0$ , where $\mathcal{L}$ is a well-defined locally invertible self-adjoint elliptic operator. Several consequences are studied. In particular, the long standing open problem asking for a meaning to the Willmore Euler–Lagrange equation for immersions having only L 2-bounded second fundamental form is now solved. The regularity of weak Willmore immersions with L 2-bounded second fundamental form is also established. Its proof relies on the discovery of conservation laws which are preserved under weak convergence. A weak compactness result for Willmore surfaces with energy less than 8π (the Li–Yau condition ensuring the surface is embedded) is proved, via a point removability result established for Wilmore surfaces in ℝ m , thereby extending to arbitrary codimension the main result in [KS3]. Finally, from this point-removability result, the strong compactness of Willmore tori below the energy level 8π is proved both in dimension 3 (this had already been settled in [KS3]) and in dimension 4.

Minimality in CR geometry and the CR Yamabe problem on CR manifolds with boundary

We study the minimality of an isometric immersion of a Riemannian manifold into a strictly pseudoconvex CR manifold M endowed with the Webster metric hence consider a version of the CR Yamabe problem for CR manifolds with boundary. This occurs as the Yamabe problem for the Fefferman metric (a Lorentzian metric associated to a choice of contact structure θ on M , [20]) on the total space of the canonical circle bundle S 1 →C(M) → π M (a manifold with boundary ∂C(M)= π −1 (∂M) and is shown to be a nonlinear subelliptic problem of variational origin. For any real surface N={φ=0}⊂ H 1 we show that the mean curvature vector of N⇀ H 1 is expressed by H=− 1 2 ∑ j=1 2 X j (|Xφ | −1 X j φ)ξ provided that N is tangent to the characteristic direction T of ( H 1 , θ 0 ) , thus demonstrating the relationship between the classical theory of submanifolds in Riemannian manifolds (cf. e.g.[7]) and the newer investigations in [1], [6], [8] and [16]. Given an isometric immersion Ψ:N→ H n of a Riemannian manifold into the Heisenberg group we show that ΔΨ=2J T ⊥ hence start a Weierstrass representation theory for minimal surfaces in H n .

Generalizations of the Liouville theorem

The purpose of this paper is to generalize the Liouville theorem for functions which are defined on the complete Riemannian manifolds. Then, we apply it to the isometric immersions between complete Riemannian manifolds in order to obtain an estimate for the size of the image of immersions in terms of the supremum of the length of their mean curvature vector in a quite general setting. The proofs are based on the Calabi's generalization of maximum principle for functions which are not necessarily differentiable.

Classification of marginally trapped Lorentzian flat surfaces in [formula omitted] and its application to biharmonic surfaces

A surface in a semi-Riemannian manifold is called marginally trapped if its mean curvature vector field is light-like at each point. In this article, we classify marginally trapped Lorentzian flat surfaces in the pseudo-Euclidean space E 2 4 . As an application, we obtain the complete classification of biharmonic Lorentzian surfaces in E 2 4 with light-like mean curvature vector.

Hypersurfaces of $E^4_8$ with proper mean curvature vector

Submanifolds satisfying Δ H → = λ H → are named by B. Y. Chen submanifolds with proper mean curvature vector. We prove that a hypersurface of the pseudo-Euclidean space E s 4 with Δ H → = λ H → and diagonalizable shape operator, has constant mean curvature.

Spatial and Lorentzian surfaces in Robertson-Walker space times

Let L14(f,c)=(I×fS,gfc) be a Robertson-Walker space time which does not contain any open subset of constant curvature. In this paper, we provide a general study of nondegenerate surfaces in L14(f,c). First, we prove the nonexistence of marginally trapped surfaces with positive relative nullity. Then, we classify totally geodesic submanifolds. Finally, we classify the family of surfaces with parallel second fundamental form and the family of totally umbilical surfaces with parallel mean curvature vector.

On isometric immersions with flat normal connection of the hyperbolic space L n into Euclidean space E n+m

We prove that the hyperbolic space Ln cannot be immersed in an Euclidean space En+m with a flat normal connection provided the module of the mean curvature vector is bounded.

Dirichlet sigma models and mean curvature flow

The mean curvature flow describes the parabolic deformation of embedded branes in Riemannian geometry driven by their extrinsic mean curvature vector, which is typically associated to surface tension forces. It is the gradient flow of the area functional, and, as such, it is naturally identified with the boundary renormalization group equation of Dirichlet sigma models away from conformality, to lowest order in perturbation theory. D-branes appear as fixed points of this flow having conformally invariant boundary conditions. Simple running solutions include the paper-clip and the hair-pin (or grim-reaper) models on the plane, as well as scaling solutions associated to rational (p, q) closed curves and the decay of two intersecting lines. Stability analysis is performed in several cases while searching for transitions among different brane configurations. The combination of Ricci with the mean curvature flow is examined in detail together with several explicit examples of deforming curves on curved backgrounds. Some general aspects of the mean curvature flow in higher dimensional ambient spaces are also discussed and obtain consistent truncations to lower dimensional systems. Selected physical applications are mentioned in the text, including tachyon condensation in open string theory and the resistive diffusion of force-free fields in magneto-hydrodynamics.

Hyper-Lagrangian submanifolds of Hyperkähler manifolds and mean curvature flow

In this article, we define a new class of middle dimensional submanifolds of a Hyperkähler manifold which contains the class of complex Lagrangian submanifolds, and show that this larger class is invariant under the mean curvature flow. Along the flow, the complex phase map satisfies the generalized harmonic map heat equation. It is also related to the mean curvature vector via a first order differential equation. Moreover, we proved a result on nonexistence of Type I singularity.

NEW RELATIONSHIPS INVOLVING THE MEAN CURVATURE OF SLANT SUBMANIFOLDS IN S-SPACE-FORMS

Relationships between the Ricci curvature and the squared mean curvature and between the shape operator associated with the mean curvature vector and the sectional curvature function for slant submanifolds of an S-space-form are proved, particularizing them to invariant and anti-invariant submanifolds tangent to the structure vector fields.

Mean-curvature flow and quasilocal mass for 2-surfaces in Hamiltonian general relativity

A family of quasilocal mass definitions that includes as special cases the Hawking mass and the Brown-York “rest mass” energy is derived for spacelike 2-surfaces in space-time. The definitions involve an integral of powers of the norm of the space-time mean-curvature vector of the 2-surface, whose properties are connected with apparent horizons. In particular, for any spacelike 2-surface, the direction of mean curvature is orthogonal (dual in the normal space) to a unique normal direction in which the 2-surface has vanishing expansion in space-time. The quasilocal mass definitions are obtained by an analysis of boundary terms arising in the gravitational ADM Hamiltonian on hypersurfaces with a spacelike 2-surface boundary, using a geometric time flow chosen proportional to the dualized mean-curvature vector field at the boundary surface. A similar analysis is made choosing a geometric rotational flow given in terms of the twist covector of the dual pair of mean-curvature vector fields, which leads to a family of quasilocal angular momentum definitions involving the squared norm of the twist. The large sphere limit of these definitions is shown to yield the ADM mass and angular momentum in asymptotically flat space-times, while at apparent horizons a quasilocal version of the Gibbons-Penrose inequality is derived. Finally, some results concerning positivity are proved for the quasilocal masses, motivated by consideration of spacelike mean-curvature flow of 2-surfaces in space-time.

Null 2-type submanifolds of the Euclidean space E5 with non-parallel mean curvature vector

Let M be a 3-dimensional submanifold of the Euclidean space E5 such that M is not of 1-type. We show that if M is flat and of null 2-type with constant mean curvature and non-parallel mean curvature vector then the normal bundle is flat. We also prove that M is an open portion of a 3-dimensional helical cylinder if and only if M is flat and of null 2-type with constant mean curvature and non-parallel mean curvature vector.

Constraint-induced mean curvature dependence of Cartesian momentum operators

The Hermitian Cartesian quantum momentum operator p for an embedded surface M in R3 is proved to be a constant factor −iℏ times the mean curvature vector field Hn added to the usual differential term. With use of this form of momentum operators, the operator-ordering ambiguity exists in the construction of the correct kinetic energy operator and three different operator-orderings lead to the same result.

R-Minimal submanifolds in space forms

Let x: M → R n+p (c) be an n-dimensional compact, possibly with bound- ary, submanifold in an (n + p)-dimensional space form R n+p (c). Assume that r is even and r ∈{ 0,1, ... ,n − 1}, in this paper we introduce rth mean curvature function Sr and (r + 1)-th mean curvature vector field Sr+1 .W e callM to be an r-minimal submanifold if Sr+1 ≡ 0o nM, we note that the concept of 0-minimal submani- fold is the concept of minimal submanifold. In this paper, we define a functional Jr(x) = � M Fr(S0,S2, ... ,Sr)dv of x: M → R n+p (c), by calculation of the first varia- tional formula of Jr we show that x is a critical point of Jr if and only if x is r-minimal. Besides, we give many examples of r-minimal submanifolds in space forms. We cal- culate the second variational formula of Jr and prove that there exists no compact without boundary stable r-minimal submanifold with Sr > 0 in the unit sphere S n+p . When r = 0, noting S0 = 1, our result reduces to Simons' result: there exists no compact without boundary stable minimal submanifold in the unit sphere S n+p .

Biharmonic Lorentz hypersurfaces in E14

A submanifold Mrn of pseudo-Euclidean space Es4 is said to have harmonic mean curvature vector if ?H = 0, where H denotes the mean curvature vector field and ? the Laplacian of the induced pseudo-Riemannian metric. We prove that every nondegenerate Lorentz hypersurface M13 of E14 with harmonic mean curvature vector is minimal.

Submanifolds of positive Ricci curvature in a Euclidean space

In this paper we study the role of constant vector fields on a Euclidean space R n+p in shaping the geometry of its compact submanifolds. For an n-dimensional compact submanifold M of the Euclidean space R n+p with mean curvature vector field H and a constant vector field $${\xi }$$ on R n+p, the smooth function $${\varphi =\left\langle H,\xi \right\rangle }$$ is used to obtain a characterization of sphere among compact submanifolds of positive Ricci curvature (cf. main Theorem).

On surfaces whose twistor lifts are harmonic sections

We study surfaces whose twistor lifts are harmonic sections, and characterize these surfaces in terms of their second fundamental forms. As a corollary, under certain assumptions for the curvature tensor, we prove that the twistor lift is a harmonic section if and only if the mean curvature vector field is a holomorphic section of the normal bundle. For surfaces in four-dimensional Euclidean space, a lower bound for the vertical energy of the twistor lifts is given. Moreover, under a certain assumption involving the mean curvature vector field, we characterize a surface in four-dimensional Euclidean space in such a way that the twistor lift is a harmonic section, and its vertical energy density is constant.

Classification of marginally trapped Lagrangian surfaces in Lorentzian complex space forms

A Lagrangian surface in a Lorentzian Kähler surface is called marginally trapped if its mean curvature vector is lightlike at each point. In this paper we classify marginally trapped Lagrangian surfaces in Lorentzian complex space forms. Our main results state that there exist three families of marginally trapped Lagrangian surfaces in C12, nine families in CP12, and nine families in CH12. Conversely, all marginally trapped Lagrangian surfaces in Lorentzian complex space forms are obtained from these 21 families.