Totally umbilical manifolds inG(2,n). I

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.

Local space time constant mean curvature and constant expansion foliations

Discretizing constant curvature surfaces via loop group factorizations: The discrete sine- and sinh-Gordon equations

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.

In this work we consider a complete submanifold M with parallel mean curvature vector h immersed in a space form of constant sectional curvature \(c\leq 0\). If M has finite total curvature and \(|H|^2>-c\), we prove that M must be compact.

The aim of this work is to show that a compact smooth star-shaped hypersurface <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Sigma Superscript n"> <mml:semantics> <mml:msup> <mml:mi mathvariant="normal"> Σ </mml:mi> <mml:mi>n</mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">\Sigma ^n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in the Euclidean sphere <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper S Superscript n plus 1"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">S</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>n</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">\mathbb {S}^{n+1}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> whose second function of curvature <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S 2"> <mml:semantics> <mml:msub> <mml:mi>S</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">S_2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a positive constant must be a geodesic sphere <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper S Superscript n Baseline left-parenthesis rho right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">S</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>n</mml:mi> </mml:mrow> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:mi> ρ </mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {S}^{n}(\rho )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . This generalizes a result obtained by Jellett in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="1853"> <mml:semantics> <mml:mn>1853</mml:mn> <mml:annotation encoding="application/x-tex">1853</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for surfaces <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Sigma squared"> <mml:semantics> <mml:msup> <mml:mi mathvariant="normal"> Σ </mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">\Sigma ^2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with constant mean curvature in the Euclidean space <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper R cubed"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mn>3</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">\mathbb {R}^3</mml:annotation> </mml:semantics> </mml:math> </inline-formula> as well as a recent result of the authors for this type of hypersurface in the Euclidean sphere <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper S Superscript n plus 1"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">S</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>n</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">\mathbb {S}^{n+1}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with constant mean curvature. In order to prove our theorem we shall present a formula for the operator <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L Subscript r Baseline left-parenthesis g right-parenthesis equals d i v left-parenthesis upper P Subscript r Baseline nabla g right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>L</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>r</mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>g</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mi>d</mml:mi> <mml:mi>i</mml:mi> <mml:mi>v</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>P</mml:mi> <mml:mi>r</mml:mi> </mml:msub> <mml:mi mathvariant="normal"> ∇ </mml:mi> <mml:mi>g</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">L_{r}(g)=div\left (P_r\nabla g\right )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> associated with a new support function <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="g"> <mml:semantics> <mml:mi>g</mml:mi> <mml:annotation encoding="application/x-tex">g</mml:annotation> </mml:semantics> </mml:math> </inline-formula> defined over a hypersurface <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M Superscript n"> <mml:semantics> <mml:msup> <mml:mi>M</mml:mi> <mml:mi>n</mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">M^n</mml:annotation>

In this paper, we study the intrinsic structures of high-dimensional data sets for analyzing their geometrical properties, where the core message of the high-dimensional data is hiding on some nonlinear manifolds. Using the manifold learning technique with a particular focus on the mean curvature, we develop new methods to investigate the uniqueness of constant mean curvature spacelike hypersurfaces in the Lorentzian warped product manifolds. Furthermore, we extend the uniqueness of stochastically complete hypersurfaces using the weak maximum principle. For the more general cases, we propose some non-existence results and a priori estimates for the constant higher-order mean curvature spacelike hypersurface.

We find a Simons type formula for submanifolds with parallel mean curvature vector (pmc submanifolds) in product spaces M n (c)×ℝ, where M n (c) is a space form with constant sectional curvature c, and then we use it to prove a gap theorem for the mean curvature of certain complete proper-biharmonic pmc submanifolds, and classify proper-biharmonic pmc surfaces in $\mathbb{S}^{n}(c)\times\mathbb{R}$ .

Low-type submanifolds of real space forms via the immersions by projectors

We consider hypersurfaces of products $$M\times {\mathbb{R}}$$ with constant rth mean curvature $${H_{r}} \ge 0$$ (to be called $${{H}_{r}}$$ -hypersurfaces), where M is an arbitrary Riemannian n-manifold. We develop a general method for constructing them and employ it to produce many examples for a variety of manifolds M, including all simply connected space forms and the hyperbolic spaces $${\mathbb{H}}_{{\mathbb{F}}}^{m}$$ (rank one symmetric spaces of noncompact type). We construct and classify complete rotational $${{H}_{r}}(\ge 0)$$ -hypersurfaces in $${\mathbb{H}}_{{\mathbb{F}}}^{m}\times {\mathbb{R}}$$ and in $${\mathbb{S}}^{n}\times {\mathbb{R}}$$ as well. They include spheres, Delaunay-type annuli and, in the case of $${\mathbb{H}}_{{\mathbb{F}}}^{m}\times {\mathbb{R}},$$ entire graphs. We also construct and classify complete $${{H}_{r}}(\ge 0)$$ -hypersurfaces of $${\mathbb{H}}_{{\mathbb{F}}}^{m}\times {\mathbb{R}}$$ which are invariant by either parabolic isometries or hyperbolic translations. We establish a Jellett–Liebmann-type theorem by showing that a compact, connected and strictly convex $${{H}_{r}}$$ -hypersurface of $${\mathbb{H}}^{n}\times {\mathbb{R}}$$ or $${\mathbb{S}}^{n}\times {\mathbb{R}}$$ $$(n\ge 3)$$ is a rotational embedded sphere. Other uniqueness results for complete $${{H}_{r}}$$ -hypersurfaces of these ambient spaces are obtained.

Introduction. In the last two decades there has been a substantial interest in studying spacelike hypersurfaces with constant mean curvature in Lorentz-Minkowski space R In +1 and in de Sitter space Sin +1, n -2. This interest is motivated by certain needs in general relativity [CFM], [G], [St], and also by the fact that complete maximal and constant mean curvature hypersurfaces in these spaces exhibit the property, that is, they are of a very special type; for example linear subspaces, totally umbilic, or totally geodesic; [Ca], [CY], [G], [R], [A], [M]. It is a natural and interesting problem to investigate stability of a Bernstein-type property relative to perturbation of the data. One of the main purposes of this paper is to give estimates of the spread of the principal curvatures on a compact spacelike hypersurface in SIn+l in terms of the mean curvature and its derivatives (up to second order). Using these estimates we show that if the mean curvature H on a complete spacelike hypersurface F in SIn+ I is such that H2 < 4 (n 1)/n2 and H is close to a constant (in C2-norm) then F is close to being umbilic. A simple consequence of our estimates is the result of Akutagawa [A] and Montiel [M] showing that complete spacelike hypersurfaces in SI' with constant mean curvature H such that H2 < 4 (n 1)/n2 are umbilic. Another aim of this paper is to study duality relations between hypersurfaces in hyperbolic space Hn+l and spacelike hypersurfaces in SIn+l. Apart from the well known fact that the Gauss image of a hypersurface in Hn+l lies in SIn+1 (when the two spaces are considered as

We show that the discrete principal nets in quadrics of constant curvature that have constant mixed area mean curvature can be characterized by the existence of a K\"onigs dual in a concentric quadric.

In this paper we study a surface of revolution in the Euclidean three space Λ 3 .The generating curve of the surface satisfies a nonlinear differential equation which describes the mean curvature.The purpose of this note is to solve the differential equation by an elementary method.Solutions are represented explicitly by generalized FresnePs integrals which involve the mean curvature.Therefore, for a given continuous function H(s), we can construct a 3-parameter family of surfaces of revolution admitting H(s) as the mean curvature.We shall remark that these computations are different from the one in Delaunay [1].About 140 years ago, he solved the differential equation under the constancy of the mean curvature and gave a method of geometric constructions for such surfaces.For the proof, he first obtains a solution of an evolute of the generating curve.By making use of this solution, he found a representation formula of the generating curve.Therefore these solutions hold only on some intervals on which the evolute can be defined.It seems not to be simple to obtain a global solution from his method.Our calculations will easily find global solutions and the corollary of the main theorem of this note describes all complete surfaces of revolution with constant mean curvature.During this research I stayed in Kδln and received many nice advices from Professors Peter Dombrowski and Helmut Reckziegel.By their suggestions, my original computations could be simplified and generalized.And also I shall mention that Reckziegel got interested in drawing graphs of these generating curves by making use of the computer and obtained many beautiful pictures.He permitted me with favor to include some of the graphs by his programming in this paper.

It is still an open question whether a compact embedded hypersurface in the Euclidean space with constant mean curvature and spherical boundary is necessarily a hyperplanar ball or a spherical cap, even in the simplest case of a compact constant mean curvature surface in \mathbb{R}^3 bounded by a circle. In this paper we prove that this is true for the case of the scalar curvature. Specifically we prove that the only compact embedded hypersurfaces in the Euclidean space with constant scalar curvature and spherical boundary are the hyperplanar round balls (with zero scalar curvature) and the spherical caps (with positive constant scalar curvature). The same applies in general to the case of embedded hypersurfaces with constant r -mean curvature, with r \geq 2 .

We consider codimension 2 sphere congruences in pseudo-conformal geometry that are harmonic with respect to the conformal structure of an orthogonal surface. We characterise the orthogonal surfaces of such congruences as either $S$-Willmore surfaces, quasi-umbilical surfaces, constant mean curvature surfaces in 3-dimensional space forms or surfaces of constant lightlike mean curvature in 3-dimensional lightcones. We then investigate Bryant's quartic differential in this context and show that generically this is divergence free if and only if the surface under consideration is either superconformal or orthogonal to a harmonic congruence of codimension 2 spheres. We may then apply the previous result to characterise surfaces with such a property.