Nonzero Constant Mean Curvature Research Articles

Geometry of constant mean curvature surfaces in $\mathbb{R}^{3}$

The crowning achievement of this paper is the proof that round spheres are the only complete, simply-connected surfaces embedded in \mathbb{R}^{3} with nonzero constant mean curvature. Fundamental to this proof are new results including the existence of intrinsic curvature and radius estimates for compact disks embedded in \mathbb{R}^{3} with nonzero constant mean curvature. We also prove curvature estimates for compact annuli embedded in \mathbb{R}^{3} with nonzero constant mean curvature and apply them to obtain deep results on the global geometry of complete surfaces of finite topology embedded in \mathbb{R}^{3} with constant mean curvature.

Basic classes of timelike general rotational surfaces in the four-dimensional Minkowski space

In the present paper, we consider timelike general rotational surfaces in the Minkowski 4- space which are analogous to the general rotational surfaces in the Euclidean 4-space introduced by C. Moore. We study two types of such surfaces (with timelike and spacelike meridian curve, respectively) and describe analytically some of their basic geometric classes: flat timelike general rotational surfaces, timelike general rotational surfaces with flat normal connection, and timelike general rotational surfaces with non-zero constant mean curvature. We give explicitly all minimal timelike general rotational surfaces and all timelike general rotational surfaces with parallel normalized mean curvature vector field.

Discrete constant mean curvature surfaces on general graphs

The contribution of this paper is twofold. First, we generalize the definition of discrete isothermic surfaces. Compared with the previous ones, it covers more discrete surfaces, e.g., the associated families of discrete isothermic minimal and non-zero constant mean curvature (CMC in short) surfaces, whose counterpart in smooth case are isothermic surfaces. Second, we show that the discrete isothermic CMC surfaces can be obtained by the discrete holomorphic data (a solution of the additive rational Toda system) via the discrete generalized Weierstrass type representation.

On regular algebraic hypersurfaces with non-zero constant mean curvature in Euclidean spaces

We prove that there are no regular algebraic hypersurfaces with non-zero constant mean curvature in the Euclidean space $\mathbb {R}^{n+1},\,\;n\geq 2,$ defined by polynomials of odd degree. Also we prove that the hyperspheres and the round cylinders are the only regular algebraic hypersurfaces with non-zero constant mean curvature in $\mathbb {R}^{n+1}, n\geq 2,$ defined by polynomials of degree less than or equal to three. These results give partial answers to a question raised by Barbosa and do Carmo.

Limit lamination theorem for H-disks

We describe the lamination limits of sequences of compact disks M_n embedded in {mathbb {R}}^3 with constant mean curvature H_n, when the boundaries of these disks tend to infinity. This theorem generalizes to the non-zero constant mean curvature case Theorem 0.1 by Colding and Minicozzi (Ann Math 160:573–615, 2004) for minimal disks. We apply this theorem to prove the existence of a chord arc result for compact disks embedded in {mathbb {R}}^3 with constant mean curvature; this chord arc result generalizes Theorem 0.5 by Colding and Minicozzi (Ann Math 167:211–243, 2008) for minimal disks.

A characteristic property of Delaunay surfaces

We prove that Delaunay surfaces, except the plane and the catenoid, are the only surfaces in Euclidean space with nonzero constant mean curvature that can be expressed as an implicit equation of type $f(x)+g(y)+h(z)=0$, where $f$, $g$ and $h$ are smooth real functions of one variable.

Curvature estimates for constant mean curvature surfaces

We derive extrinsic curvature estimates for compact disks embedded in $\mathbb{R}^3$ with nonzero constant mean curvature.

Existence of conformal metrics with constant scalar curvature and constant boundary mean curvature on compact manifolds

We study the problem of deforming a Riemannian metric to a conformal one with nonzero constant scalar curvature and nonzero constant boundary mean curvature on a compact manifold of dimension [Formula: see text]. We prove the existence of such conformal metrics in the cases of [Formula: see text] or the manifold is spin and some other remaining ones left by Escobar. Furthermore, in the positive Yamabe constant case, by normalizing the scalar curvature to be [Formula: see text], there exists a sequence of conformal metrics such that their constant boundary mean curvatures go to [Formula: see text].

Translation surfaces with non-zero constant mean curvature in Euclidean space

We prove that the only surface in 3-dimensional Euclidean space $${\mathbb {R}}^3$$ with constant and non-zero mean curvature H, constructed by the sum of a planar curve and a space curve, is the circular cylinder of radius $$\frac{1}{2|H|}$$ .

Stability of constant mean curvature surfaces in three-dimensional warped product manifolds

In this paper we prove that stable, compact without boundary, oriented, nonzero constant mean curvature surfaces in the de Sitter-Schwarzschild and Reissner-Nordstrom manifolds are the slices, provided its mean curvature satisfies some positive lower bound. More generally, we prove that stable, compact without boundary, oriented nonzero constant mean curvature surfaces in a large class of three dimensional warped product manifolds are embedded topological spheres, provided the mean curvature satisfies a positive lower bound depending only on the ambient curvatures. We conclude the paper proving that a stable, compact without boundary, nonzero constant mean curvature surface in a general Riemannian is a topological sphere provided its mean curvature has a lower bound depending only on the scalar curvature of the ambient space and the squared norm of the mean curvature vector field of the immersion of the ambient space in some Euclidean space.

Algebraic CMC hypersurfaces of order 3 in Euclidean spaces

Understanding and finding of general algebraic constant mean curvature surfaces in the Euclidean spaces is a hard open problem. The basic examples are the standard spheres and the round cylinders, all defined by a polynomial of degree 2. In this paper, we prove that there are no algebraic hypersurfaces of degree 3 in mathbb {R}^n, nge 3, with nonzero constant mean curvature.

Fold singularities on spacelike CMC surfaces in Lorentz-Minkowski space

Fold singular points play important roles in the theory of maximal surfaces. For example, if a maximal surface admits fold singular points, it can be extended to a timelike minimal surface analytically. Moreover, there is a duality between conelike singular points and folds. In this paper, we investigate fold singular points on spacelike surfaces with non-zero constant mean curvature (spacelike CMC surfaces). We prove that spacelike CMC surfaces do not admit fold singular points. Moreover, we show that the singular point set of any conjugate CMC surface of a spacelike Delaunay surface with conelike singular points consists of $(2,5)$-cuspidal edges.

Non-lightlike ruled surfaces with constant curvatures in Minkowski 3-space

We study the non-lightlike ruled surfaces in Minkowski 3-space with non-lightlike base curve [Formula: see text], where [Formula: see text], [Formula: see text], [Formula: see text] are the tangent, principal normal and binormal vectors of an arbitrary timelike curve [Formula: see text]. Some important results of flat, minimal, II-minimal and II-flat non-lightlike ruled surfaces are studied. Finally, the following interesting theorem is proved: the only non-zero constant mean curvature (CMC) non-lightlike ruled surface is developable timelike ruled surface generated by binormal vector.

Meridian Surfaces with Constant Mean Curvature in Pseudo-Euclidean 4-Space with Neutral Metric

In the present paper we consider a special class of Lorentz surfaces in the four-dimensional pseudo-Euclidean space with neutral metric which are one-parameter systems of meridians of rotational hypersurfaces with timelike or spacelike axis and call them meridian surfaces. We give the complete classification of minimal and quasi-minimal meridian surfaces. We also classify the meridian surfaces with non-zero constant mean curvature.

On complete hypersurfaces with constant mean and scalar curvatures in Euclidean spaces

Generalizing a theorem of Huang, Cheng and Wan classified the complete hypersurfaces of $\mathbb R^4$ with non-zero constant mean curvature and constant scalar curvature. In our work, we obtain results of this nature in higher dimensions. In particular, we prove that if a complete hypersurface of $\mathbb R^5$ has constant mean curvature $H\neq 0$ and constant scalar curvature $R\geq\frac{2}{3}H^2$, then $R=H^2$, $R=\frac{8}{9}H^2$ or $R=\frac{2}{3}H^2$. Moreover, we characterize the hypersurface in the cases $R=H^2$ and $R=\frac{8}{9}H^2$, and provide an example in the case $R=\frac{2}{3}H^2$. The proofs are based on the principal curvature theorem of Smyth-Xavier and a well known formula for the Laplacian of the squared norm of the second fundamental form of a hypersurface in a space form.

On surfaces that are intrinsically surfaces of revolution

We consider surfaces in Euclidean space parametrized on an annular domain such that the first fundamental form and the principal curvatures are rotationally invariant, and the principal curvature directions only depend on the angle of rotation (but not the radius). Such surfaces generalize the Enneper surface. We show that they are necessarily of constant mean curvature, and that the rotational speed of the principal curvature directions is constant. We classify the minimal case. The (non-zero) constant mean curvature case has been classified by Smyth.

On Submanifolds with 2-Type Pseudo-Hyperbolic Gauss Map in Pseudo-Hyperbolic Space

In this paper, we study pseudo-Riemannian submanifolds of a pseudo-hyperbolic space \(\mathbb H^{m-1}_s (-1) \subset \mathbb E^m_{s+1}\) with 2-type pseudo-hyperbolic Gauss map. We give a characterization of proper pseudo-Riemannian hypersurfaces in \(\mathbb H^{n+1}_s (-1) \subset \mathbb E^{n+2}_{s+1}\) with non-zero constant mean curvature and 2-type pseudo-hyperbolic Gauss map. For \(n=2\), we prove classification theorems. In addition, we show that the hyperbolic Veronese surface is the only maximal surface fully lying in \(\mathbb H^4_2 (-1) \subset \mathbb H^{m-1}_2 (-1)\) with 2-type pseudo-hyperbolic Gauss map. Moreover, we prove that a flat totally umbilical pseudo-Riemannian hypersurface \(M^n_t\) of the pseudo-hyperbolic space \(\mathbb {H}^{n+1}_t(-1) \subset \mathbb E^{n+2}_{t+1}\) has biharmonic pseudo-hyperbolic Gauss map.

Height estimates and half-space type theorems in weighted product spaces with nonnegative Bakry–Émery–Ricci curvature

We prove height estimates concerning compact hypersurfaces with nonzero constant weighted mean curvature and whose boundary is contained into a slice of a weighted product space of nonnegative Bakry–Emery–Ricci curvature. As applications of our estimates, we obtain half-space type results related to complete noncompact hypersurfaces properly immersed in such an ambient space.

Affine translation surfaces with constant mean curvature in Euclidean 3-space

Using classical methods and solving certain differential equations we classify a kind of new surface, affine translation surfaces, which has the non-zero constant mean curvature in three dimensional Euclidean space $${{\bf E}^3}$$ . Therefore a kind of solutions for the constant mean curvature equation is given.

Limit lamination theorems for H-surfaces

Abstract In this paper we prove some general results for constant mean curvature lamination limits of certain sequences of compact surfaces M n M_{n} embedded in ℝ 3 \mathbb{R}^{3} with constant mean curvature H n H_{n} and fixed finite genus, when the boundaries of these surfaces tend to infinity. Two of these theorems generalize to the non-zero constant mean curvature case, similar structure theorems by Colding and Minicozzi in [6, 8] for limits of sequences of minimal surfaces of fixed finite genus.