Reduction of variables for Willmore-Chen submanifolds in seven spheres

We obtain a reduction of variables criterion for 4-dimensional Willmore-Chen submanifolds associated with the generalized Kaluza-Klein conformal structures on the 7-sphere. This argument connects the variational problem of Willmore-Chen with a variational problem for closed curves into 4-spheres. It involves an elastic energy functional with potential. The method is based on the extrinsic conformal invariance of the Willmore-Chen variational problem, and the principle of symmetric criticality. It also uses several techniques from the theory of pseudo-Riemannian submersions. Furthermore, we give some applications, in particular, a result of existence for constant mean curvature Willmore-Chen submanifolds which is essentially supported on the nice geometry of closed helices in the standard 3-sphere. Partially supported by a DGICYT Grant No. PB97-0784.     

An exotic sphere with positive curvature almost everywhere

In this article we show that there is an exotic sphere with positive sectional curvature almost everywhere. In 1974 Gromoll and Meyer found a metric of nonnegative sectional on an exotic 7-sphere. They showed that the metric has positive curvature at a point and asserted, without proof, that the metric has positive sectional curvature almost everywhere [4]. We will show here that this assertion is wrong. In fact, the Gromoll-Meyer sphere has zero curvatures on an open set of points. Never the less, its metric can be perturbed to one that has positive curvature almost everywhere.     

On two-dimensional immersions that are stable for parametric functionals of constant mean curvature type

We consider two-dimensional immersions in Euclidean 3-space that are stable for parametric functionals of constant mean curvature type. We develop analytical and geometric concepts to give a perturbation result to estimate the principle curvatures of such mappings via uniformization.     

Complete stable CMC surfaces with empty singular set in Sasakian sub-Riemannian 3-manifolds

For constant mean curvature surfaces of class C ² immersed inside Sasakian sub-Riemannian 3-manifolds we obtain a formula for the second derivative of the area which involves horizontal analytical terms, the Webster scalar curvature of the ambient manifold, and the extrinsic shape of the surface. Then we prove classification results for complete surfaces with empty singular set which are stable, i.e., second order minima of the area under a volume constraint, inside the 3-dimensional sub-Riemannian space forms. In the first Heisenberg group we show that such a surface is a vertical plane. In the sub-Riemannian hyperbolic 3-space we give an upper bound for the mean curvature of such surfaces, and we characterize the horocylinders as the unique ones with squared mean curvature 1. Finally we deduce that any complete surface with empty singular set in the sub-Riemannian 3-sphere is unstable.     

Prescribed Mean Curvature Hypersurfaces in Hn+1(-1) with Convex Planar Boundary, I

We study immersed prescribed mean curvature compact hypersurfaces with boundary in Hⁿ⁺¹(-1). When the boundary is a convex planar smooth manifold with all principal curvatures greater than 1, we solve a nonparametric Dirichlet problem and use this, together with a general flux formula, to prove a parametric uniqueness result, in the class of all immersed compact hypersurfaces with the same boundary. We specialize this result to a constant mean curvature, obtaining a characterization of totally umbilic hypersurface caps.     

An estimate for the scalar curvature of constant mean curvature hypersurfaces in space forms

In this paper we derive a sharp estimate for the supremum of the scalar curvature (or, equivalently, the infimum of the squared norm of the second fundamental form) of a constant mean curvature hypersurface with two principal curvatures immersed into a Riemannian space form of constant curvature. Our results will be an application of the generalized Omori-Yau maximum principle, following the approach by Pigola et al. (Memoirs Am Math Soc 822, 2005).     

On Biconservative Lorentz Hypersurface with Non-diagonalizable Shape Operator

In this paper, we obtain some properties of biconservative Lorentz hypersurface \(M_{1}^{n}\) in \(E_{1}^{n+1}\) having shape operator with complex eigenvalues. We prove that every biconservative Lorentz hypersurface \(M_{1}^{n}\) in \(E_{1}^{n+1}\) whose shape operator has complex eigenvalues with at most five distinct principal curvatures has constant mean curvature. In addition, we investigate such a type of hypersurface with constant length of second fundamental form having six distinct principal curvatures.     

Typical curvature behaviour of bodies of constant width

It is known that an n-dimensional convex body, which is typical in the sense of Baire category, shows a simple, but highly non-intuitive curvature behaviour: at almost all of its boundary points, in the sense of measure, all curvatures are zero, but there is also a dense and uncountable set of boundary points at which all curvatures are infinite. The purpose of this paper is to find a counterpart to this phenomenon for typical convex bodies of given constant width. Such bodies cannot have zero curvatures. A main result says that for a typical n-dimensional convex body of constant width 1 (without loss of generality), at almost all boundary points, in the sense of measure, all curvatures are equal to 1. (In contrast, note that a ball of width 1 has radius 1/2, hence all its curvatures are equal to 2.) Since the property of constant width is linear with respect to Minkowski addition, the proof requires recourse to a linear curvature notion, which is provided by the tangential radii of curvature.     

Eigenvalue estimate on complete noncompact Riemannian manifolds and applications

We obtain some sharp estimates on the first eigenvalues of complete noncompact Riemannian manifolds under assumptions of volume growth. Using these estimates we study hypersurfaces with constant mean curvature and give some estimates on the mean curvatures.

     

Stability problems in a theorem of F. Schur

Schur's theorem states that an isotropic Riemannian manifold of dimension greater than two has constant curvature. It is natural to guess that compact almost isotropic Riemannian manifolds of dimension greater than two are close to spaces of almost constant curvature. We take the curvature anisotropy as the discrepancy of the sectional curvatures at a point. The main result of this paper is that Riemannian manifolds in Cheeger's class ℜ(n,d,V,A) withL ₁-small integral anisotropy haveL _p-small change of the sectional curvature over the manifold. We also estimate the deviation of the metric tensor from that of constant curvature in theW _p ² -norm, and prove that compact almost isotropic spaces inherit the differential structure of a space form. These stability results are based on the generalization of Schur' theorem to metric spaces.     

Kenmotsu–Bryant Type Representation Formulas for Constant Mean Curvature Surfaces in ” 3(-c2) and S 3 1(c2)

Let c be a positive constant and H a constant satisfying |H| > c. Our primary object of this paper is to give representation formulas for branched CMC H (constant mean curvature H) surfaces in the hyperbolic 3-space ³(-c²) of constant curvature c², and for spacelike CMC H surfaces in the de Sitter 3-space S ³ ₁(c²) of constant curvature c². These formulas imply, for example, that every CMC H surface in ³(-c²) can be represented locally by a harmonic map to the unit 2-sphere S².     

Convergence of phase interfaces in the van der Waals-Cahn-Hilliard theory

We study the general asymptotic behavior of critical points, including those of non-minimal energy type, of the functional for the van der Waals-Cahn-Hilliard theory of phase transitions. We prove that the interface is close to a hypersurface with mean curvature zero when no Lagrange multiplier is present, and with locally constant mean curvature in general. The energy density of the limiting measure has integer multiplicity almost everywhere modulo division by a surface energy constant. Received March 16, 1999 / Accepted June 11, 1999     

Minimal 2-spheres in a complex projective space

In this paper conformal minimal 2-spheres immersed in a complex projective space are studied by applying Lie theory and moving frames. We give differential equations of Kähler angle and square length of the second fundamental form. By applying these differential equations we give characteristics of conformal minimal 2-spheres of constant Kähler angle and obtain pinching theorems for curvature. We also discuss conformal minimal 2-spheres of constant normal curvature and prove that there does not exist any linearly full minimal 2-sphere immersed in a complex projective space CPn (n>2) with non-positive constant normal curvature. We also prove that a linearly full minimal 2-sphere immersed in a complex projective space CPn (n>2) with constant normal curvature and constant Kähler angle is of constant curvature.     

Helicoidal surfaces under the cubic screw motion in Minkowski 3-space

In this paper, we construct helicoidal surfaces under the cubic screw motion with prescribed mean or Gauss curvature in Minkowski 3-space . We also find explicitly the relation between the mean curvature and Gauss curvature of them. Furthermore, we discuss helicoidal surfaces under the cubic screw motion with H²=K and prove that these surfaces have equal constant principal curvatures.     

Global estimates and energy identities for elliptic systems with antisymmetric potentials

We derive global estimates in critical scale invariant norms for solutions of elliptic systems with antisymmetric potentials and almost holomorphic Hopf differential in two dimensions. Moreover, we obtain new energy identities in such norms for sequences of solutions of these systems. The results apply to harmonic maps into general target manifolds and surfaces with prescribed mean curvature. In particular, the results confirm a conjecture of Rivière in the two-dimensional setting.     

Discrete quasi-Einstein metrics and combinatorial curvature flows in 3-dimension

Motivated by the definition of combinatorial scalar curvature given by Cooper and Rivin, we introduce a new combinatorial scalar curvature. Then we define the discrete quasi-Einstein metric, which is a combinatorial analogue of the constant scalar curvature metric in smooth case. We find that discrete quasi-Einstein metric is critical point of both the combinatorial Yamabe functional and the quadratic energy functional we defined on triangulated 3-manifolds. We introduce combinatorial curvature flows, including a new type of combinatorial Yamabe flow, to study the discrete quasi-Einstein metrics and prove that the flows produce solutions converging to discrete quasi-Einstein metrics if the initial normalized quadratic energy is small enough. As a corollary, we prove that nonsingular solution of the combinatorial Yamabe flow with nonpositive initial curvatures converges to discrete quasi-Einstein metric. The proof relies on a careful analysis of the discrete dual-Laplacian, which we interpret as the Jacobian matrix of curvature map.     

On Almost Hermitian 4-manifolds with J-invariant Ricci Tensor

We prove that every almost Hermitian 4-manifold with J-invariant Ricci tensor which is conformally flat or has harmonic curvature is either a space of constant curvature or a Kähler manifold. We also obtain analogous results on almost Kähler 4-manifolds.     

A curvature theory for discrete surfaces based on mesh parallelity

We consider a general theory of curvatures of discrete surfaces equipped with edgewise parallel Gauss images, and where mean and Gaussian curvatures of faces are derived from the faces’ areas and mixed areas. Remarkably these notions are capable of unifying notable previously defined classes of surfaces, such as discrete isothermic minimal surfaces and surfaces of constant mean curvature. We discuss various types of natural Gauss images, the existence of principal curvatures, constant curvature surfaces, Christoffel duality, Koenigs nets, contact element nets, s-isothermic nets, and interesting special cases such as discrete Delaunay surfaces derived from elliptic billiards.     

Conformal metrics on the ball with zero scalar curvature and prescribed mean curvature on the boundary

This paper considers the prescribed zero scalar curvature and mean curvature problem on the n-dimensional Euclidean ball for n?3. We consider the limits of solutions of the regularization obtained by decreasing the critical exponent. We characterize those subcritical solutions which blow-up at the least possible energy level, determining the points at which they can concentrate, and their Morse indices. We show that when n=3 this is the only blow-up which can occur for solutions. We use this in combination with the Morse inequalities for the subcritical problem to obtain a general existence theorem for the prescribed zero scalar curvature and mean curvature on the three-dimensional Euclidean ball. In the higher-dimensional case n?4, we give conditions on the function h to guarantee there is only one simple blow-up point.     

A volume convergence theorem for Alexandrov spaces with curvature bounded above

We obtain a volume convergence theorem for Alexandrov spaces with curvature bounded above with respect to the Gromov-Hausdorff distance. As one of the main tools proving this, we construct an almost isometry between Alexandrov spaces with curvature bounded above, with weak singularities, which are close to each other. Furthermore, as an application of our researches of convergence phenomena, for given positive integer , we prove that, if a compact, geodesically complete, n-dimensional CAT(1)-space has the volume sufficiently close to that of the unit n-sphere, then it is bi-Lipschitz homeomorphic to the unit n-sphere. Received: 30 January 2001; in final form: 30 October 2001 / Published online: 4 April 2002