# Surfaces with constant mean curvature

by K. Kenmotsu

Publisher: American Mathematical Society in Providence, R.I

Written in English

## Subjects:

• Minimal surfaces.,
• Curvature.,
• Geometry, Differential.

## Edition Notes

Includes bibliographical references (p. 137-138) and index.

The study of surfaces with constant mean curvature (CMC) is one of the main topics in classical differential geometry. Moreover, CMC surfaces are important mathematical models for the physics of interfaces in the absence of gravity, where they separate two different media or for capillary : Capa dura. Loosely speaking, the curvature •of a curve at the point P is partially due to the fact that the curve itself is curved, and partially because the surface is curved. In order to somehow disentangle these two eﬁects, it it useful to deﬂne the two concepts normal curvature and geodesic curvature. We follow Kreyszig  in our Size: 1MB. Let ψ: M → N be a stable constant mean curvature immersion of a complete orientable surface into a three dimensional oriented Riemannian manifold. Then the following interesting, although partial results, are known Theorem 1. If N is compact and has positive Ricci curvature, then M is compact and connected, and genus(M)≤ 3. Theorem 2. minimal surfaces and surfaces of constant mean curvature. We discuss various types of natural Gauss images, the existence of principal curvatures, constant curvature sur-faces, Christoﬀel duality, Koenigs nets, contact element nets, s-isothermic nets, and interesting special cases such as discrete Delaunay surfaces derived from elliptic bil.

orienting such a surface with constant mean curvature H, we will assume H ≥0 andwillrefertothesurfaceasanH-surface. We next brieﬂy explain the contents of the three lectures in the course. The ﬁrst lecture introduced the notation, deﬁnitions and examples, as well as the ba-sic tools. In differential geometry, constant-mean-curvature (CMC) surfaces are surfaces with constant mean curvature. This includes minimal surfaces as a subset, but typically they are treated as special case.. Note that these surfaces are generally different from constant Gaussian curvature surfaces, with the important exception of the sphere. Biharmonic surfaces of constant mean curvature. We present a new method for modeling discrete constant mean curvature (CMC) surfaces, which arise frequently in nature and are highly demanded in architecture and other engineering applications. Our method is based on a novel use of the CVT (centroidal Voronoi tessellation) optimization : PanHao, ChoiYi-King, LiuYang, HuWenchao, DuQiang, PolthierKonrad, ZhangCaiming, WangWenping.

Bang-Yen Chen, in Handbook of Differential Geometry, Stability of surfaces with constant mean curvature. Since a compact constant mean curvature surface in E 3 is a critical point of the area functional with respect to volume-preserving normal variations, one can define the stability of such surfaces: A compact constant mean curvature surface in E 3 is called stable if A″(0) > 0. Constant mean curvature (CMC) surfaces have played a prominent role in di erential geometry. In , Charles Eug ene Delaunay introduced a way of constructing rotationally symmetric CMC surfaces in R3, by proving that a surface of revolution in R3 is a CMC surface if . complete, properly immersed, or embedded, surfaces of nonzero constant mean curvature in EV. Very little is known in this regard. For a long time, the only known examples of such surfaces were, besides the round sphere and the cylinder, a family of rotationally invariant surfaces discovered in by.

## Surfaces with constant mean curvature by K. Kenmotsu Download PDF EPUB FB2

: Surfaces With Constant Mean Curvature (Translations of Mathematical Monographs) (): Katsuei Kenmotsu: BooksCited by: The study of surfaces with constant mean curvature (CMC) is one of the main topics in classical differential geometry.

Moreover, CMC surfaces are important mathematical models for the physics of interfaces in the absence of gravity, where they separate two different media, or for capillary by:   The mean curvature of a surface is an extrinsic parameter measuring how the surface is curved in the three-dimensional space.

A surface whose mean curvature is zero at each point is a minimal surface, and it is known that such surfaces are models for soap film. An easy example of a surface of constant mean curvature is the sphere. A nontrivial example is provided by the constant curvature torus, whose discovery in gave a powerful incentive for studying such surfaces.

Later, many examples of constant mean curvature surfaces were discovered using various methods of analysis, differential geometry, and differential equations. It is now becoming clear that. The mean curvature of a surface is an extrinsic parameter measuring how the surface is curved in the three-dimensional space.

A surface whose mean curvature is zero at each point is a minimal surface, and it is known that such surfaces are models for soap film. There is a rich and well-known theory of minimal surfaces. Surfaces with constant mean curvature.

[K Kenmotsu] -- "The mean curvature of a surface is an extrinsic parameter measuring how the surface is curved in the three-dimensional space. A surface whose mean curvature is zero at each point is a minimal.

Surfaces with constant mean curvature will arise as solutions of a variational problem associated to the area functional and related with the classical isoperimetric problem. We state the first and second variation formula for the area and we give the notion of stability of a cmc by: 2.

of constant mean curvature (CMC) in R 3. Such surfaces are often called soap bubbles since a soap ﬁlm in equilibrium between two regions is characterized by having constant mean curvature. The surface area of these surfaces is critical under volume-preserving deformations.

CMC surfaces may also be characterized by the fact that their Gauss map N: S. S2 is harmonic i.e. it satisﬁes t (N) = 0. Deﬁnition A constant mean curvature surface is a surface whose mean curvatures equal some constant at any point. We denote the constant h. We call the surface a CMC h-surface.

When h ≡ 0, we call it a minimal surface. History Generally constant mean curvature surfaces are not as well understood as minimal surfaces. ALEKSANDROV’S THEOREM: CLOSED SURFACES WITH CONSTANT MEAN CURVATURE 2 1. Introduction Recall that a closed surface is one that is compact and without boundary.

Aleksan-drov proved that if a closed, connected C2 surface has constant mean curvature, then the surface is a sphere.

In this paper, we present his Size: KB. The study of surfaces with constant mean curvature (CMC) is one of the main topics in classical differential geometry. Moreover, CMC surfaces are important mathematical models for the physics of interfaces in the absence of gravity, where they separate two different media, or for capillary phenomena.

A surface whose mean curvature is constant but nonzero is obtained when we try to minimize the area of a closed surface without changing the volume it encloses.

A trivial example of a surface of constant mean curvature is the sphere. A nontrivial example is provided by the constant curvature torus, whose discovery in gave a powerfu.

the slices, provided its mean curvature satis es some lower bound. More generally, we prove that stable, compact without boundary, oriented, constant mean curvature surfaces in a large class of three dimensional warped product manifolds are embedded topological spheres, provided the mean curvature satis es a lower bound depending only on the.

Let me add some uniqueness theorems for CMC and minimal surfaces: 1) A classical theorem of Hopf says that any immersed CMC sphere in is the round sphere. 2) A classical theorem of Aleksandrov says that any embedded closed hypersurface in Euclidean space with constant mean curvature is the round sphere.

The study of surfaces with constant mean curvature (CMC) is one of the main topics in classical differential geometry. Moreover, CMC surfaces are important mathematical models for the physics of interfaces in the absence of gravity, where they separate two different media or for capillary phenomena.

In this article we survey recent developments in the theory of constant mean curvature surfaces in homogeneous 3-manifolds, as well as some related aspects on existence and descriptive results for H-laminations and CMC foliations of Riemannian n-manifolds.

Mathematics Subject Classification: Primary 53A10, Secondary 49Q05, 53C nected surfaces of the same constant mean curvature is a congru-ence ;2 (ii) Gauss curvature on 5 is set up as a solution to a nonlinear el-liptic boundary value problem; and (iii) construction of local surfaces of any given constant mean curvature.

Notation. 5 denotes a surface with a fixed immersion v: S-+ by: Unduloid, a surface with constant mean curvature. In differential geometry, constant-mean-curvature (CMC) surfaces are surfaces with constant mean curvature.

This includes minimal surfaces as a subset, but typically they are treated as special case. Constant Mean Curvature Surfaces at the Intersection of Integrable Geometries Quintino, Aurea, ; Constant mean curvature surfaces in sub-Riemannian geometry Hladky, R.

and Pauls, S. D., Journal of Differential Geometry, ; On surfaces with constant mean curvature in hyperbolic space de Lima, Ronaldo F., Illinois Journal of Cited by:   The authors also cover Alexandrov's theorem on embedded compact surfaces in $$\mathbb{R}^3$$ with constant mean curvature.

The last chapter addresses the global geometry of curves, including periodic space curves and the four-vertices theorem for plane curves that are not necessarily convex. Then we consider the existence of constant mean curvature surface with prescribed Neumann boundary value on a strictly convex bounded domain ω in Rn,(){div(Du1+|Du|2)=λinΩ,uν=φ(x)on∂ω, where φ(x) and ν are the same as stated by: The purpose of this paper is twofold.

First, by means of computer graphics, we construct pictures of the representative cases of the helicoidal surfaces of constant mean curvature. At the same time, we depict their periodic isometric deformation into Delaunay surfaces under preservation of the mean curvature.

Second, we prove new properties of these surfaces--we find necessary and sufficient. Constant mean curvature surfaces and integrable equations 3 integrability of equation () only in the recent works of Hitchin  and Pinkall and Sterling .

In , which is devoted to a classification of CMC tori, there is implicitly the important theorem that all doubly periodic. Are there surfaces that have (except for cusps or borders) a constant positive gaussian curvature but that do not have not a constant mean curvature.

2 Neglected constant curvature difference surfaces. Surfaces with Constant Mean Curvature Katsuei Kenmotsu Publication Year: ISBN ISBN Translations of Mathematical Monographs, vol.

Constant Mean Curvature Surfaces, Harmonic Maps and Integrable Systems by Frederic Helein,available at Book Depository with free delivery worldwide.

The study of surfaces with constant mean curvature (CMC) is one of the main topics in classical differential geometry. Moreover, CMC surfaces are important mathematical models for the physics of interfaces in the absence of gravity, where they separate two different media, or for capillary : Kindle.

It is a well known fact that the meridian curve of a rotational constant mean curvature (cmc) surface (which determines the surface completely) can be obtained as the trace of a focal point of an Author: Konrad Polthier.

Example 3. If the Gaussian curvature K of a surface S is constant, then the total Gaussian curvature is KA(S), where A(S) is the area of the surface.

Thus a sphere of radius r has total Gaussian curvature 1 r2 4πr 2 = 4π, which is independent of the radius Size: KB. Constant Mean Curvature Surfaces at the Intersection of Integrable Geometries Quintino, Aurea, ; Constant mean curvature surfaces in sub-Riemannian geometry Hladky, R.

and Pauls, S. D., Journal of Differential Geometry, ; On surfaces with constant mean curvature in hyperbolic space de Lima, Ronaldo F., Illinois Journal of Cited by: 5. surfaces of possibly varying constant mean curvature (the case of minimal leaves is included as well).

Many of these results extend to the case of codimension one laminations and folia-tions in n-dimensional Riemannian manifolds by hypersurfaces of possibly varying constant mean by: In mathematics, constant curvature is a concept from differential geometry.

Here, curvature refers to the sectional curvature of a space (more precisely a manifold) and is a single number determining its local geometry. The sectional curvature is said to be constant if it has the same value at every point.The study of surfaces with constant mean curvature (CMC) is one of the main topics in classical differential geometry.

Moreover, CMC surfaces are important mathematical models for the physics of interfaces in the absence of gravity, where they separate two different media or for capillary : Pasta blanda.