Total Mean Curvature Research Articles

Isometric deformations of flat tori in the 3-sphere with nonconstant mean curvature

For an isometric immersion $f$ of a flat torus into the unit 3-sphere, we show that if the mean curvature of $f$ is not constant, then the immersion $f$ admits a nontrivial isometric deformation preserving the total mean curvature.

The total mean curvature of submanifolds in a Euclidean space.

LetM be an n-dimensional submanifold in a Euclidean spaceEn+p of dimension n+p. Denote by R the normalized scalar curvature and byH the mean curvature of M. Ōtsuki [O] introduced a kind of curvatures, λ1 ≥ λ2 ≥ · · · ≥ λp for M 2 in E2+p, and showed that they can be used to study the geometry of surfaces in higher-dimensional Euclidean space. Shiohama [S] proved that a complete oriented surface inE2+p with λα = 0 (1 ≤ α ≤ p) is a cylinder. Chen [C1] classified compact oriented surfaces in E2+p with λp ≥ 0. In higher-dimensional cases, Chen [C3] introduced the notion of αth scalar curvatures, λ1 ≥ λ2 ≥ · · · ≥ λp for M in En+p, and found a relationship between the αth scalar curvatures and the scalar curvature. When n = 2, it reduces to that introduced by Ōtsuki [O]. Chen [C3] also proved that a closed submanifold M (n ≥ 3) in En+p with ∫ Mn (λ1) n/2 dV = cn and λα = 0 (2 ≤ α ≤ p) is an n-sphere, where cn is the volume of the unit n-sphere and dV denotes the volume element of M. In this paper, we give a further description of the behavior of the αth scalar curvatures and obtain some applications of them. In Section 2, we first prove that λα ≤ 0 (2 ≤ α ≤ p) for any submanifold M in En+p. Then we prove an inequality involving the integral of λ1 for closed M in En+p with R ≥ 0. Suppose that M is closed in En+p. The total mean curvature of M is defined to be the integral ∫ Mn H dV. An interesting and outstanding problem is to find the best possible lower bound of this integral in terms of the geometric or topologic invariants of M. A special case of this problem is the famous Willmore’s conjecture. There have been many results obtained on this problem. In Section 3 we give an estimate of the total mean curvature for closed submanifolds in En+p with R ≥ 0. The main result of this paper is the following theorem.

Integrable deformations of affine surfaces via the Nizhnik-Veselov-Novikov equation

It is shown that deformations of affine conormal according to the Nizhnik-Veselov-Novikov equation generate, via the Lelieuvre's formula, integrable deformations of affine surfaces. The total affine mean curvature is preserved by these deformations. It is shown that a stationary point of these deformations for centro-affine surfaces is given by the proper affine sphere.

The Total Mean Curvature of a Complete Noncompact Surface of Nonnegative Curvature in R 3

We give a complete classification of complete noncompact oriented surfaces with nonnegative Gaussian curvature and finite total mean curvature in R3.

Total mean curvature and closed geodesics

The proofs and applications are based on a Riemannian version of Gromov’s non-squeezing theorem and classical integral geometry. Given a convex surface Σ ⊂ R and a point q in the unit sphere S we denote by UΣ(q) the perimeter of the orthogonal projection of Σ onto a plane perpendicular to q. We obtain a function UΣ on the sphere which is clearly continuous, even, and positive. Let us denote the minimum value for this function by uΣ. The analogue of the non-squeezing theorem we wish to present is the following result.

The generalized Laplace equation of capillarity I. Thermodynamic and hydrostatic considerations of the fundamental equation for interfaces

Theories of capillarity are reviewed. The limitations of the classical theory as developed by Gibbs are demonstrated. The generalized theories of capillarity initiated by Buff, and later by Murphy, Kondo, Kralchevsky, and Boruvka and Neumann are scrutinized by considering the basic requirements of formulating thermodynamic fundamental equations. The different generalized Laplace equations of different theories stem from the different setups of the fundamental equation. It is concluded that only Boruvka and Neumann's (BN) generalized theory satisfies all the requirements of thermodynamics and mathematics. Further, to test the BN theory, a hydrostatic treatment of a two phase capillary system is presented. This non-thermodynamic approach is based on the concept of virtual work as the condition for equilibrium of a capillary system, and on the concept of parallel surfaces for evaluating the stress tensor field and excess properties within the interfacial region. Following a straight-forward procedure, it is shown that the hydrostatic results for surface tension γ and two bending moments, C 1, and C 2, agree with the results of the BN generalized thermodynamic theory of capillarity. This agreement indicates that the form of the BN fundamental equation for surfaces with the extensive geometric curvatures ( J and K total mean and total Gaussian curvatures, respectively) is the proper expression required to generalize the theory of capillarity.

The sufficient condition for a convex body to enclose another in 𝑅⁴

We follow Hadwiger and Ren’s ideas to estimate the kinematic measure of a convex body D 1 {D_1} with C 2 {C^2} -boundary ∂ D 1 \partial {D_1} moving inside another convex body D 0 {D_0} with the same kind of boundary ∂ D 0 \partial {D_0} under the isometry group G in R 4 {\mathbb {R}^4} . By using Chern and Yen’s kinematic fundamental formula, C-S. Chen’s kinematic formula for the total square mean curvature ∫ ∂ D 0 ∩ g ∂ D 1 H 2 d v {\smallint _{\partial {D_0} \cap g\partial {D_1}}}{H^2}dv , and some well-known results about the curvatures of the 2-dimensional intersection submanifold ∂ D 0 ∩ g ∂ D 1 \partial {D_0} \cap g\partial {D_1} , we obtain a sufficient condition to guarantee that one convex body can enclose another in R 4 {\mathbb {R}^4} .

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

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

Conformal degeneracy and conformal diffusion of vesicles.

The shape of vesicles with genus g=2, i.e., with two holes or two handles, is studied in the framework of curvature models. These vesicles exhibit a new phase which also persists for higher genus g>2. In this phase, the ground state of the vesicle is conformally degenerate even when the volume, the area, and the total mean curvature of the vesicle are kept constant. It is predicted that such vesicles undergo a new type of diffusive motion, termed conformal diffusion, which should be observable in experiments as pronounced shape fluctuations.

Quasilocal energy and conserved charges derived from the gravitational action

The quasilocal energy of gravitational and matter fields in a spatially bounded region is obtained by employing a Hamilton-Jacobi analysis of the action functional. First, a surface stress-energy-momentum tensor is defined by the functional derivative of the action with respect to the three-metric on $^{3}$B, the history of the system's boundary. Energy surface density, momentum surface density, and spatial stress are defined by projecting the surface stress tensor normally and tangentially to a family of spacelike two-surfaces that foliate $^{3}$B. The integral of the energy surface density over such a two-surface B is the quasilocal energy associated with a spacelike three-surface \ensuremath{\Sigma} whose orthogonal intersection with $^{3}$B is the boundary B. The resulting expression for quasilocal energy is given in terms of the total mean curvature of the spatial boundary B as a surface embedded in \ensuremath{\Sigma}. The quasilocal energy is also the value of the Hamiltonian that generates unit magnitude proper-time translations on $^{3}$B in the timelike direction orthogonal to B. Conserved charges such as angular momentum are defined using the surface stress tensor and Killing vector fields on $^{3}$B. For spacetimes that are asymptotically flat in spacelike directions, the quasilocal energy and angular momentum defined here agree with the results of Arnowitt, Deser, and Misner in the limit that the boundary tends to spatial infinity. For spherically symmetric spacetimes, it is shown that the quasilocal energy has the correct Newtonian limit, and includes a negative contribution due to gravitational binding.

Membrane interactions and total mean curvature

The dynamical properties of a time-like hypersurface in Minkowski spacetime are described in terms of a membrane model in which electromagnetic self-stresses are balanced with stresses derived from the shape tensor of the immersion. Radial modes are investigated and in the harmonic approximation a tantalising result emerges for the energy of a harmonic quantum. The particular model is put into the perspective of a class of Willmore-type immersions.

Conservation laws in spacetimes with boundary

Symmetries and conservation laws and their dependence on the boundary conditions are analysed. It is shown that, for spacetimes with boundary, the gravitational energy is proportional to the total mean curvature of the boundary of any Cauchy surface implications for positivity of energy are discussed.

ON TOTAL MEAN CURVATURES

ON TOTAL MEAN CURVATURES

TOTAL MEAN CURVATURE AND SUBMANIFOLDS OF FINITE TYPE (World Scientific Series in Pure Mathematics-Volume 1) By Bang -Yen Chen : pp. 352. £15.70. (Published by World Scientific Publishing, distributed by John Wiley & Sons Ltd, 1984.)

TOTAL MEAN CURVATURE AND SUBMANIFOLDS OF FINITE TYPE (World Scientific Series in Pure Mathematics-Volume 1) <i>By</i> B<scp>ang</scp> -Y<scp>en</scp> C<scp>hen</scp> : pp. 352. £15.70. (Published by World Scientific Publishing, distributed by John Wiley &amp; Sons Ltd, 1984.)

Lipschitzian Mappings and Total Mean Curvature of Polyhedral Surfaces. I

For a smooth closed surface $C$ in ${E^3}$ the classical total mean curvature is defined by $M(C) = \frac {1} {2}\int ({\kappa _1} + {\kappa _2})\;d\sigma (p)$, where ${\kappa _1},{\kappa _2}$ are the principal curvatures at $p$ on $C$. If $C$ is a polyhedral surface, there is a well known discrete version given by $M(C) = \frac {1} {2}\Sigma {l_i}(\pi - {\alpha _i})$, where ${l_i}$ represents edge length and ${\alpha _i}$ the corresponding dihedral angle along the edge. In this article formulas involving differentials of total mean curvature (closely related to the differential formula of L. Schláfli) are applied to several questions concerning Lipschitizian mappings of polyhedral surfaces. For example, the simplest formula $\Sigma {l_i} d{\alpha _i} = 0$ may be used to show that the remarkable flexible polyhedral spheres of R. Connelly must flex with constant total mean curvature. Related differential formulas are instrumental in showing that if $f: {E^2} \to {E^2}$ is a distance-increasing function and $K \subset {E^2}$, then $\operatorname {Per}(\operatorname {conv}\;K) \leqslant \operatorname {Per}(\operatorname {conv}\;f[K])$. This article (part I) is mainly concerned with problems in ${E^n}$. In the sequel (part II) related questions in ${S^n}$ and ${H^n}$, as well as ${E^n}$, will be considered.

Lipschitzian mappings and total mean curvature of polyhedral surfaces. I

For a smooth closed surface C in E3 the classical total mean curvature is defined by M(C) = ?/(«i + k2) do(p), where kx, k2 are the principal curvatures at p on C. If C is a polyhedral surface, there is a well known discrete version given by M(C) = IE/,(w - a,), where 1i represents edge length and a, the corresponding dihedral angle along the edge. In this article formulas involving differentials of total mean curvature (closely related to the differential formula of L. Schlafli) are applied to several questions concerning Lipschitizian mappings of polyhedral surfaces. For example, the simplest formula T.I, da, = 0 may be used to show that the remarkable flexible polyhedral spheres of R. Connelly must flex with constant total mean curvature. Related differential formulas are instrumental in showing that if/: E2 -> E2 is a distance-increasing function and KciE2, then Per(convK~)< Per(conv/(A:)). This article (part I) is mainly concerned with problems in E. In the sequel (part II) related questions in S and H, as well as E, will be considered.

Total Mean Curvature of Immersed Surfaces in E m

Total Mean Curvature of Immersed Surfaces in E m

Total mean curvature of immersed surfaces in 𝐸^{𝑚}

Total mean curvature and value-distribution of mean curvature for certain pseudo-umbilical surfaces are studied.

Divergence properties of curvatures of a vector field and a family of surfaces

A representation in divergent form is given for the total curvature and mean curvature of a vector field.

Die symmetrische Kugelzone als extremaler Rotationskörper

For a convex body in three dimensions letV be the volume,F the surface area, andM the total mean curvature.