CMC surfaces may also be characterized by the fact that their Gauss map N: S! If the ambient manifold is … In 1841, Delaunay [2] classified all surfaces of revolution of constant mean curvature, with a beautifully simple description in terms of conics. Constant mean curvature tori in S 3 17. form is covariant constant. mathematics. … Berlin-Leipzig: Teubner 1909, do Carmo, M., Peng, C.K. constant mean curvature H = H 0 is known to be equivalent to the fact that x is a critical point of a variational problem. Select a purchase In mathematics, the mean curvature $${\displaystyle H}$$ of a surface $${\displaystyle S}$$ is an extrinsic measure of curvature that comes from differential geometry and that locally describes the curvature of an embedded surface in some ambient space such as Euclidean space. Math. Constant mean curvature spheres in S 3 and H 3 16. 2. Abstract We use the DPW construction [5] to present three new classes of immersed CMC cylinders, each of which includes surfaces with umbilics.The first class consists of cylinders with one end asymptotic to a Delaunay surface. Minimal tori in S 3 and Willmore tori 18. (Basel)33, 91–104 (1979), D'Arcy Thompson: On growth and form. Soviet. Abstract We use the DPW construction [5] to present three new classes of immersed CMC cylinders, each of which includes surfaces with umbilics.The first class consists of cylinders with one end asymptotic to a Delaunay surface. Alexandrov [1] gave a Access supplemental materials and multimedia. Equations of constant mean curvature surfaces in S 3 and H 3 15. The surface area of these surfaces is critical under volume-preserving deformations. Constant mean curvature tori in H 3 19. Math. Math. constant mean curvature hypersurfaces with boundary in a leaf. 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. We mean by it a path of shortest length, that is, a "geodesic." The surfaces of constant mean curvature or Gaussian curvature in 3-dimensional Euclidean space E s or 3-dimensional Minkowski space E~ have been studied extensively. 3 and inH 3 are planes. With a personal account, you can read up to 100 articles each month for free. Subscription will auto renew annually. Pure Appl. For the surface of revolution that maximizes volume for given surface area ( or for given volume contained within minimum surface area ) the optimal situation Lagrangian in R 3 are. Project MUSE is a leading provider of digital humanities and social sciences content, providing access to journal and book content from nearly 300 publishers. of Math.117, 609–625 (1983), Kenmotsu, K.: Surfaces of revolution with prescribed mean curvature. The oldest mathematics journal in the Western Hemisphere in surface is immersed as a constant mean curved surface of a four-dimensional. 1040 BO GUAN AND JOEL SPRUCK mean convex domain Ωin R n f 0 g, then for any H 2 (0,1) there is a unique function u 2 C 1 (Ω) whose graph is a hypersurface of constant mean curvature H with asymptotic boundary Γ. The Press is home to the largest journal publication program of any U.S.-based university press. Definition 0.1 A constant mean curvature surface is a surface whose mean curvatures equal some constant at any point. These spaces are defined in Section 2 and include basically all exam- Comm. An H(r)-torus in .S''l+1(l) is obtained by consid-ering the standard immersions Sn~x(r) c R" , Sl(\/l-r2) cR2, 0 < r < 1, where the value within the parentheses denotes the radius of the corresponding Chapter III. There are many scenarios where the effective mass fails to be defined, such as at band crossings (like in graphene), so the very minimal condition for a constant mean curvature surface is having a single band Fermi surface. Z.173, 13–28 (1980), Böhme, R., Tomi, F.: Zur Struktur der Lösungsmenge des Plateauproblems. Part of Springer Nature. The division also manages membership services for more than 50 scholarly and professional associations and societies. : On the stability of minimal surfaces. differential-geometry curvature. 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. These examples solved the long-standing problem of Hopf [6]: Is a compact constant mean curvature surface in R3 necessarily a round sphere? They are classified by triples of points on the sphere whose distances are the asymptotic necksizes of the three ends. Math.35, 199–211 (1980), Frid, H.: O Teorema do índice de Morse. Master's thesis, IMPA 1982, Frid, H., Thayer, F.J.: The Morse index theorem for elliptic problems with a constraint. The geometry of the surface of a sphere is the geometry of a surface with constant curvature: the surface of a sphere has the same curvature everywhere. Published since 1878, the Journal has earned and ),1, 903–906 (1979), Fischer-Colbrie, D., Schoen, R.: The structure of complete stable minimal surfaces in 3-manifolds of nonnegative scalar curvature. We need some notation. Further, as most techniques used in the theory of CMC surfaces not only involve geometric methods … Go to Table ©2000-2021 ITHAKA. Let u be the solution to the following mean curvature type equation with Neumann boundary value (3.2) {div (D u 1 + | D u | 2) = ε u in Ω, u ν = φ (x) on ∂ Ω, then there exists a constant C = C (n, Ω, L) such that sup Ω ‾ | D u | ≤ C. Constant mean curvature surfaces in S 3 and H 3 14. J.32, 147–153 (1980), Lawson, B., Jr.: Lectures on Minimal Submanifolds, vol.1. Section 4 describes the method of continuity to solve the Dirichlet problem in Equation (1). HFS provides print and digital distribution for a distinguished list of university presses and nonprofit institutions. In the last case, the second fundament. When h ≡ 0, we call it a minimal surface. In the last case, the second fundament. Acad. If the ambient manifold is … This interpolation algorithm is an essential ingredient in practical applica- I can't find a source for this. JSTOR is part of ITHAKA, a not-for-profit organization helping the academic community use digital technologies to preserve the scholarly record and to advance research and teaching in sustainable ways. of constant mean curvature (CMC) in R 3. It is positive curvature since two geodesics at right angles curve in … The surface area of these surfaces is critical under volume-preserving deformations. constant curved manifold, then either the surface is minimal, a minimal surface. More precisely, x has nonzero constant mean curvature if and only if x is a critical point of the n-area A(t) H-surface if it is embedded, connected and it has positive constant mean curvature H. We will call an H-surface an H-disk if the H-surface is homeomorphic to a closed unit disk in the Euclidean plane. Constant mean curvature surfaces in S 3 and H 3 14. Mathematics Subject Classification (2000). The study of surfaces with constant mean curvature (CMC) is one of the main topics in classical differential geometry. of constant mean curvature (CMC) in R 3. Purchase this issue for $44.00 USD. (N.S. surface is immersed as a constant mean curved surface of a four-dimensional. After Section 2 devoted to fix some definitions and notations, we derive the constant mean curvature equation in Section 3 obtaining some properties of the solutions showing differences in both ambient spaces. United States and abroad. An. New York: Cambridge at the University Press and The MacMillan Co 1945, Departamento de Matemática, Universidade Federal do Ceará, Campus do Pici, 60000, Fortaleza Ceará, Brasil, Instituto de Matemática Pura e Aplicada, Estrada D. Castorina 110, J. Botanico, 22460, Rio de Janeiro, Brasil, You can also search for this author in I want to see some examples on positive mean curvature surfaces (not necessary constant mean curvature). In fact, Theorem 1.5 below can be proved. Abstract: The mean curvature of a surface is an extrinsic parameter measuringhow the surface is curved in the three-dimensional space. Triunduloids are classified by triples of distinct labeled points in the two-sphere (up to rotations); the spherical distances of points in the triple are the necksizes of the unduloids asymptotic to the three ends. Project MUSE® Journals gravitational radiation. Constant mean curvature spacelike hypersurfaces in Generalized Robertson-Walker spacetimes All Rights Reserved. There is a rich and well-known theory ofminimal surfaces. © 2021 Springer Nature Switzerland AG. Download it once and read it on your Kindle device, PC, phones or tablets. Among many other results, these authors showed the existence of isoperimetric sets, and that, when considering the isoperimetric problem in the Heisenberg groups, if one restricts to the set of surfaces which are the union of Equations of constant mean curvature surfaces in S 3 and H 3 15. oriented Riemannian manifold. The mean curvature would then give the mean effective mass for the two principal axes. In differential geometry, constant-mean-curvature (CMC) surfaces are surfaces with constant mean curvature. - 45.123.144.16. The surfaces of constant mean curvature or Gaussian curvature in 3-dimensional Euclidean space E s or 3-dimensional Minkowski space E~ have been studied extensively. Secondary 53A10. A representation formula for spaeelike surfaces with prescribed mean curvature as a basic reference work in academic libraries, both in the History Generally constant mean curvature surfaces are not as well understood as minimal surfaces. possibly varying constant mean curvature has a bound on the norm of the second fundamental form of its leaves, that depends only on the geometry of N. Consequently, there is a uniform bound on the absolute value of the mean curvature function of all CMC foliations1 of N; we ranks as one of the most respected and celebrated journals There are many scenarios where the effective mass fails to be defined, such as at band crossings (like in graphene), so the very minimal condition for a constant mean curvature surface is having a single band Fermi surface. Constant mean curvature tori in H 3 19. A surface whose meancurvature is zero at each point is a minimal surface, and it is known that suchsurfaces are models for soap film. Mathematische Zeitschrift American Journal of Mathematics option. For minimal hypersurfaces (H = 0), this was proved Read your article online and download the PDF from your email or your account. Hopf proved that if the surface is topologically a sphere then it must be round A representation formula for spaeelike surfaces with prescribed mean curvature Then ψ has constant mean curvature if and only if it is a critical point of the area functional for any compactly supported variation that preserves the volume enclosed by the surface. Books In this paper, we consider the Dirichlet problem for the constant mean curvature equation on an unbounded convex planar domain Ω.Let H>0.We prove that there exists a graph with constant mean curvature H and with boundary ∂Ω if and only if Ω is included in an infinite strip of width 1 H.We also establish an existence result for convex bounded domains contained in a strip. JSTOR®, the JSTOR logo, JPASS®, Artstor®, Reveal Digital™ and ITHAKA® are registered trademarks of ITHAKA. Z.133, 1–29 (1973), Bolza, O.: Vorlesungen über Variationsrechnung. Request Permissions. Math Z 185, 339–353 (1984). MUSE delivers outstanding results to the scholarly community by maximizing revenues for publishers, providing value to libraries, and enabling access for scholars worldwide. Math. the computation of constant mean curvature surfaces via minimal surfaces in S3, joint with Oberknapp [86], and in Chapter 8 on the smooth interpolation between adaptively refined meshes using hier-archical data structures, joint with Friedrich and Schmies [47]. There is a rich and well-known theory ofminimal surfaces. This includes minimal surfaces as a subset, but typically they are treated as special case. This paper is organized as follows. Such surfaces are often called soap bubbles since a soap film in equilibrium between two regions is characterized by having constant mean curvature. In this context we say that the constant mean curvature immersion ψ is stable if the second variation formula of the maintained its reputation by presenting pioneering New constant mean curvature cylinders M. Kilian, I. McIntosh & N. Schmitt August 16, 1999. and constant mean curvature surfaces in Carnot groups. Dokl.24, 274–276 (1981), Ruchert, H.: Ein Eindeutigkeitssatz für Flächen konstanter mittlerer Krümmung. This item is part of a JSTOR Collection. Chapter III. https://doi.org/10.1007/BF01215045, Over 10 million scientific documents at your fingertips, Not logged in PubMed Google Scholar, Barbosa, J.L., do Carmo, M. Stability of hypersurfaces with constant mean curvature. Minimal tori in S 3 and Willmore tori 18. Bull. of an umbilical hypersurface, or flat. CMC surfaces may also be characterized by the fact that their Gauss map N: S! Berkeley: Publish or Perish 1980, Mori, H.: Stable constant mean curvature surfaces inR Trinoids with constant mean curvature are a family of surfaces that depend on the parameters , related to the monodromy group.When , the trinoid is symmetric [1].The trinoid is embedded when and the parameter is related to the embeddedness. A surface whose meancurvature is zero at each point is a minimal surface, and it is known that suchsurfaces are models for soap film. Definition 0.1 A constant mean curvature surface is a surface whose mean curvatures equal some constant at any point. 5 denotes a surface with a fixed immersion v: S-+R3. Constant mean curvature tori in S 3 17. Use features like bookmarks, note taking and highlighting while reading Constant Mean Curvature Surfaces with Boundary (Springer Monographs in Mathematics). Abstract: The mean curvature of a surface is an extrinsic parameter measuringhow the surface is curved in the three-dimensional space. Arch. In 1841, Delaunay [2] classified all surfaces of revolution of constant mean curvature, with a beautifully simple description in terms of conics. continuous publication, the American Journal of Mathematics in its field. Surfaces that minimize area under a volume constraint have constant mean curvature (CMC); this condition can be expressed as a nonlinear partial … The American Journal of Mathematics is used Soc. History Generally constant mean curvature surfaces are not as well understood as minimal surfaces. Notation. A triunduloid is an embedded surface of constant mean curvature with three ends, each asymptotic to a Delaunay unduloid. Math. Unduloid, a surface with constant mean curvature. The equations are derived from Bryant holomorphic representation (analogous to the Weierstrass representation of minimal surfaces), in terms of gamma … We are led to a constant value of curvature: w ″ ( 1 + w 2) 3 2 = 1 λ. volume 185, pages339–353(1984)Cite this article. of Contents. Constant mean curvature tori in R3 were first discovered, in 1984, by Wente [14]. Ci.55, 9–10 (1983), Hsiang, W.Y., Teng, Z.H., Yu, W.: New examples of constant mean curvature immersions of (2k-1)-spheres into euclidean 2k-space. We announce the classification of complete almost embedded surfaces of constant mean curvature, with three ends and genus zero. In this paper, we restrict ourselves to a large class of sub-Riemannian manifolds which we call vertically rigid sub-Riemannian (VR) spaces. theorem to constant mean curvature. Amer. form is covariant constant. Immediate online access to all issues from 2019. Hopkins Fulfillment Services (HFS) Published By: The Johns Hopkins University Press, Read Online (Free) relies on page scans, which are not currently available to screen readers. When h ≡ 0, we call it a minimal surface. Thank you. To access this article, please, Access everything in the JPASS collection, Download up to 10 article PDFs to save and keep, Download up to 120 article PDFs to save and keep. For terms and use, please refer to our Terms and Conditions © 1974 The Johns Hopkins University Press surfaces are characterized as zero mean curvature surfaces while isoperi-metric surfaces have constant mean curvature. ∫ π w 2 d x − λ ∫ 2 π w 1 + w 2 d x; F = w 2 − 2 λ w 1 + w 2; It does not specialize, but instead publishes Learn more about Institutional subscriptions, Barbosa, J.L., do Carmo, M.: Stability of minimal surfaces and eigenvalues of the Laplacian. Share. constant me an curvature H; our conven tion of mean curvature gives that a sphere S 2 in R 3 of radius 1 has H = 1 when oriented b y the inward pointing unit normal to the ball that it bounds. With critically acclaimed titles in history, science, higher education, consumer health, humanities, classics, and public health, the Books Division publishes 150 new books each year and maintains a backlist in excess of 3,000 titles. Preprint, Pogorelov, A.V. 1 Introduction It is a classical result that a compact hypersurface embedded in Euclidean space with constant mean curvature must be a round sphere. Hypersurfaces with constant mean curvature, constant scalar curvature or constant Gauss-Kronecker curvature in Euclidean space or space forms constitute an important class of submanifolds. Such surfaces are often called soap bubbles since a soap film in equilibrium between two regions is characterized by having constant mean curvature. In Riemannian manifolds very few examples of constant k-curvature hypersurfaces are … Check out using a credit card or bank account with. Tax calculation will be finalised during checkout. : Stable complete minimal surfaces inR One of the largest publishers in the United States, the Johns Hopkins University Press combines traditional books and journals publishing units with cutting-edge service divisions that sustain diversity and independence among nonprofit, scholarly publishers, societies, and associations.
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