Embedded Constant Mean Curvature Hypersurfaces on Spheres

where Ai(p) and A2(p) are the principal curvatures of M at p. When H is constant, M is called a surface of constant mean curvature. A surface is said to have finite type if it is homeomorphic to a closed surface with a finite number of points removed. An important problem in classical differential geometry is the classification of properly embedded finite type surfaces M of constant mean curvature in R . If M is a closed embedded surface of constant mean curvature, then it follows from Alexandrov [1] that M must be a round sphere. The classical examples of properly embedded surfaces with zero mean curvature are the plane, the helicoid and the catenoid. Surfaces of zero mean curvature are usually called minimal surfaces. The remaining classical examples of properly embedded surfaces of constant mean curvature were found by Delaunay [4]. The Delaunay surfaces are surfaces of revolution. Recently Hoffman and Meeks [6, 7] have found examples of properly embedded minimal surfaces which are homeomorphic to closed surfaces of positive genus with 3 points removed. Callahan, Hoffman and Meeks [3] have found other examples with more ends. An annular end E of & properly embedded surface in R 3 is a properly embedded annulus E in M where E is homeomorphic to S x [0,1). When M has finite type, then every end of M is annular. Hoffman and Meeks have developed a theory to deal with global problems concerning the geometry of properly embedded minimal surfaces M and, in particular, they show that most annular ends of M converge at infinity in R to a flat plane or to the end of a catenoid. Recently N. Kapouleas [8] in his thesis has shown that for every positive integer k > 2, there exists a properly embedded surface M& of finite type with nonzero mean curvature and with k ends. He also has constructed highergenus examples. As in the case of minimal surfaces, the annular ends of a properly embedded surface of nonzero constant mean curvature have a special geometry and play an important role in global theorems.

Biconservative submanifolds, with important role in mathematical physics and differential geometry, arise as the conservative stress-energy tensor associated to the variational problem of biharmonic submanifolds. Many examples of biconservative hypersurfaces have constant mean curvature. A famous conjecture of Bang-Yen Chen on Euclidean spaces says that everybiharmonic submanifold has null mean curvature. Inspired by Chen conjecture, we study biconservative Lorentz submanifolds of the Minkowski spaces. Although the conjecture has not been generally confirmed, it has been proven in many cases, and this has led to its spread to various types of submenifolds. As an extension, we consider a advanced version of the conjecture (namely, $L_1$-conjecture) on Lorentz hypersurfaces of the pseudo-Euclidean space $\mathbb{M}^5 :=\mathbb{E}^5_1$ (i.e. the Minkowski 5-space). We show every $L_1$-biconservative Lorentz hypersurface of $\mathbb{M}^5$ with constant mean curvature and at least three principal curvatures has constant second mean curvature.

We prove that every biharmonic hypersurface having constant higher order mean curvature Hr for r > 2 in a space form M 5(c) is of constant mean curvature. In particular, every such biharmonic hypersurface in 𝕊5(1) has constant mean curvature. There exist no such compact proper biharmonic isoparametric hypersurfaces M in 𝕊5(1) with four distinct principal curvatures. Moreover, there exist no proper biharmonic hypersurfaces in hyperbolic space ℍ5 or in E 5 having constant higher order mean curvature Hr for r > 2.

Constant mean curvature surfaces constructed by fusing Wente tori

Let M_s^2 be an orientable surface immersed in the De Sitter space mathbb {S}_1^3subset mathbb {R}^4_1 or anti de Sitter space mathbb {H}_1^3subset mathbb {R}^4_2. In the case that M_s^2 is of L_1-2-type we prove that the following conditions are equivalent to each other: M_s^2 has a constant principal curvature; M_s^2 has constant mean curvature; M_s^2 has constant second mean curvature. As a consequence, we also show that an L_1-2-type surface is either an open portion of a standard pseudo-Riemannian product, or a B-scroll over a null curve, or else its mean curvature, its Gaussian curvature and its principal curvatures are all non-constant.

As first noted in Korevaar, Kusner and Solomon ("KKS"), constant mean\ncurvature implies a homological conservation law for hypersurfaces in ambient\nspaces with Killing fields.In Theorem 3.5 here, we generalize that law by\nrelaxing the topological restrictions assumed in [KKS] and by allowing a\nweighted mean curvature functional. We also prove a partial converse (Theorem\n4.1) which roughly says that when flux is conserved along a Killing field, a\nhypersurface splits into two regions: one with constant (weighted) mean\ncurvature, and one preserved by the Killing field. We demonstrate our theory by\nusing it to derive a first integral for helicoidal surfaces of constant mean\ncurvature in Euclidean 3-space, i.e., "twizzlers."\n

One of the most important results in differential geometry is that the only closed hypersurfaces of constant mean curvature and in general constant higher order mean curvature) embedded in Euclidean space are round spheres [1]. This result is not true for the case of immersed (and non-embedded hypersurfaces [11, 14]. Many generalizations of this result have been obtained later, for example constant scalar curvatures or constant higher order mean curvatures hypersurfaces [2,3,7,9]. As a natural generalization of hypersurfaces with constant mean curvature or with constant higher order mean curvature, linear Weingarten and more general generalized, Weingarten hypersurfaces hypersurface has been studied in many places. [5],[10]. The aim of our work is to establish a characterization theorem concerning complete generalized Weingarten hypersurfaces embedded in Euclidean space. That is an hypersurfaces where some of the higher order mean curvature are linearly related. We prove that the only closed, oriented generalized Weingarten hypersurfaces embedded in Euclidean space with non-vanishing higher order mean curvature are the round spheres. This result generalizes the cases of constant higher order mean curvature hypersurfaces and linear Weingarten hypersurfaces embedded in Euclidean space.

The aim of this work is to show that a compact smooth star-shaped hypersurface Σ n in the Euclidean sphere S n+1 whose second function of curvature S 2 is a positive constant must be a geodesic sphere S n (ρ). This generalizes a result obtained by Jellett in 1853 for surfaces Σ 2 with constant mean curvature in the Euclidean space ℝ 3 as well as a recent result of the authors for this type of hypersurface in the Euclidean sphere S n+1 with constant mean curvature. In order to prove our theorem we shall present a formula for the operator L r (g) = div(P r ∇g) associated with a new support function g defined over a hypersurface M n in a Riemannian space form M n+1 c .

The isometric deformation of surfaces preserving the principal curvatures was first studied by O. Bonnet in 1867. Bonnet restricted himself to the complex case, so that his surfaces are analytic, and the results are different from the real case. After the works of a number of mathematicians, W. C. Graustein took up the real case in 1924-, without completely settling the problem. An authoritative study of this problem was carried out by Elie Cartan in [2], using moving frames. Based on this work, we wish to prove the following: Theorem: The non-trivial families of isometric surfaces having the same principal curvatures are the following: 1) a family of surfaces of constant mean curvature; 2) a family of surfaces of non-constant mean curvature. Such surfaces depend on six arbitrary constants, and have the properties: a) they are W-surfaces; b) the metric $$d{s^2} = {\left( {gradH} \right)^2}d{s^2}/\left( {{H^2} - K} \right)$$ , where d s 2 is the metric of the surface and H and K are its mean curvature and Gaussian curvature respectively, has Gaussian curvature equal to — 1.

We prove the rigidity of rectifiable boundaries with constant distributional mean curvature in the Brendle class of warped product manifolds (which includes important models in general relativity, like the de Sitter–Schwarzschild and Reissner–Nordstrom manifolds). As a corollary, we characterize limits of rectifiable boundaries whose mean curvatures converge, as distributions, to a constant. The latter result is new, and requires the full strength of distributional CMC-rigidity, even when one considers smooth boundaries whose mean curvature oscillations vanish in arbitrarily strong C k C^{k} -norms. Our method also establishes that rectifiable boundaries of sets of finite perimeter in the hyperbolic space with constant distributional mean curvature are finite unions of possibly mutually tangent geodesic spheres.

In [5], Chen gives a classification of null 2-type surfaces in the Euclidean 3-space and he shows in [6] that a similar characterization cannot be given for a surface in the Euclidean 4-space. In fact, helical cylinders in Euclidean 4-space are characterized as those surfaces of null 2-type and constant mean curvature. In this paper we give a characterization of null 2-type hypersurfaces in a space of constant sectional curvature Mn+1(k) and an approach to hypersurfaces of null 3-type. Indeed, we get a generalization of Chen’s paper [5] not only by considering hypersurfaces, but also taking them in space forms. In spherical and hyperbolic cases we show that there is no null 2-type hypersurface, so that the Euclidean case becomes the most attractive situation where our classification works on. Actually, we show that Euclidean hypersurfaces of null 2-type and having at most two distinct principal curvatures are locally isometric to a generalized cylinder. Why the hypothesis on principal curvatures? First, we think this is the most natural one, because, after Chen’s paper, we know that cylinders are the only surfaces of null 2-type in Euclidean 3-space. Secondly, it is well-known that a Euclidean isoparametric hypersurface has at most two distinct principal curvatures, so that if it has exactly two, then one of them has to be zero. Our classification depends strongly on that isoparametricity condition. Finally, bounding the number of principal curvatures is not as restrictive as one could hope. As a matter of fact, the families of conformally flat and rotational hypersurfaces satisfy that hypothesis and both are sufficiently large so that it is worth trying to give a characterization of some subfamily of them in order to get along in their classifications. To this effect, we characterize rotational and conformally flat hypersurfaces of null 2-type. As for hypersurfaces of null 3-type one immediately sees that they are not difficult to handle when they have constant mean curvature, because a nice formula for ∆2H can be given. In that case, we show that there is no spherical or hyperbolic hypersurface of null 3-type. It turns out again that our only hope to get some more information concerns with Euclidean hypersurfaces. Now, following a similar reasoning as in the null 2-type case, we are able to say that there is no Euclidean hypersurface of null 3-type having constant mean curvature and at most two distinct principal curvatures. We wish to thank to Prof. M. Barros for many valuable comments and suggestions.

It is shown that timelike surfaces of constant mean curvature ± in anti-de Sitter 3-space ℍ3 1(−1) can be constructed from a pair of Lorentz holomorphic and Lorentz antiholomorphic null curves in ℙSL2ℝ via Bryant type representation formulae. These Bryant type representation formulae are used to investigate an explicit one-to-one correspondence, the so-called Lawson–Guichard correspondence, between timelike surfaces of constant mean curvature ± 1 and timelike minimal surfaces in Minkowski 3-space E 3 1. The hyperbolic Gaus map of timelike surfaces in ℍ3 1(−1), which is a close analogue of the classical Gaus map is considered. It is discussed that the hyperbolic Gaus map plays an important role in the study of timelike surfaces of constant mean curvature ± 1 in ℍ3 1(−1). In particular, the relationship between the Lorentz holomorphicity of the hyperbolic Gaus map and timelike surface of constant mean curvature ± 1 in ℍ3 1(−1) is studied.

We are concerned with hypersurfaces of ℝ N {\mathbb{R}^{N}} with constant nonlocal (or fractional) mean curvature. This is the equation associated to critical points of the fractional perimeter under a volume constraint. Our results are twofold. First we prove the nonlocal analogue of the Alexandrov result characterizing spheres as the only closed embedded hypersurfaces in ℝ N {\mathbb{R}^{N}} with constant mean curvature. Here we use the moving planes method. Our second result establishes the existence of periodic bands or “cylinders” in ℝ 2 {\mathbb{R}^{2}} with constant nonlocal mean curvature and bifurcating from a straight band. These are Delaunay-type bands in the nonlocal setting. Here we use a Lyapunov–Schmidt procedure for a quasilinear type fractional elliptic equation.

The surfaces of revolution with constant mean curvature in 3 are classified by Delaunay [2] in 1841. They are locally plane, catenoid, sphere, circular cylinder, unduloid, and nodoid up to isometries of 3. On these 10 years, new and interesting examples of non-zero constant mean curvature surfaces are discovered. In the global study of complete surfaces with constant mean curvature, unduloids and nodoids play important role as the models of ends of such surfaces (see [5], [6]). The work by Delaunay is now revived after 150 years of his discovery. The purpose of this paper is to study surfaces of revolution with periodic mean curvature in order to extend the theory of constant mean curvature surfaces. In general such a surface is not periodic, because the catenoid gives the counter-example. First we show the criterion for a periodic function to be the mean curvature of a periodic surface of revolution and second describe a method how to construct these periodic surfaces of revolution whose mean curvatures are periodic functions satisfying the criterion. I thank Professor Yusuke Sakane for his interest and plotting of the beautiful pictures by the computer of Osaka University which I used in the seminar talk of Granada University on March, 2001, and in the AMS meeting held in Hoboken on April, 2001. The figures of this paper are all programmed by the author using the symbolic manipulation program Mathematica. I also thank my graduate student Shinya Hirakawa for his help to simplify the programing.

We study semi-discrete surfaces in three dimensional euclidean space which are defined on a parameter domain consisting of one smooth and one discrete parameter. More precisely, we consider only those surfaces which are glued together from individual developable surface strips. In particular we investigate minimal surfaces and constant mean curvature (CMC) surfaces with non vanishing mean curvature in the setting of Koenigs nets and Christoffel duality. We obtain incidence-geometric characterizations of the dualizability of Koenigs nets as well as for the Gauss image of CMC surfaces. We also consider isothermic semi-discrete CMC surfaces and a specific type of Cauchy problem in this regard.