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. Published since 1878, the Journal has earned and CMC surfaces may also be characterized by the fact that their Gauss map N: S! Soviet. The Press is home to the largest journal publication program of any U.S.-based university press. 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. This item is part of a JSTOR Collection. For minimal hypersurfaces (H = 0), this was proved maintained its reputation by presenting pioneering Purchase this issue for $44.00 USD. theorem to constant mean curvature. constant curved manifold, then either the surface is minimal, a minimal surface. We denote the constant h. We call the surface a CMC h-surface. constant mean curvature hypersurfaces with boundary in a leaf. Equations of constant mean curvature surfaces in S 3 and H 3 15. We mean by it a path of shortest length, that is, a "geodesic." These examples solved the long-standing problem of Hopf [6]: Is a compact constant mean curvature surface in R3 necessarily a round sphere? The division also manages membership services for more than 50 scholarly and professional associations and societies. More precisely, x has nonzero constant mean curvature if and only if x is a critical point of the n-area A(t) Constant mean curvature spheres in S 3 and H 3 16. Secondary 53A10. (Basel)33, 91–104 (1979), D'Arcy Thompson: On growth and form. https://doi.org/10.1007/BF01215045, Over 10 million scientific documents at your fingertips, Not logged in In this context we say that the constant mean curvature immersion ψ is stable if the second variation formula of the Z.133, 1–29 (1973), Bolza, O.: Vorlesungen über Variationsrechnung. Math. option. A surface whose meancurvature is zero at each point is a minimal surface, and it is known that suchsurfaces are models for soap film. 3. J.32, 147–153 (1980), Lawson, B., Jr.: Lectures on Minimal Submanifolds, vol.1. Ann. volume 185, pages339–353(1984)Cite this article. form is covariant constant. 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. Could you provide some examples (It would be better with calculations). PubMed Google Scholar, Barbosa, J.L., do Carmo, M. Stability of hypersurfaces with constant mean curvature. 3 and inH of an umbilical hypersurface, or flat. Constant mean curvature surfaces in S 3 and H 3 14. If the ambient manifold is … In fact, Theorem 1.5 below can be proved. Acad. New constant mean curvature cylinders M. Kilian, I. McIntosh & N. Schmitt August 16, 1999. mathematical papers. 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. of an umbilical hypersurface, or flat. When h ≡ 0, we call it a minimal surface. They are classified by triples of points on the sphere whose distances are the asymptotic necksizes of the three ends. Equations of constant mean curvature surfaces in S 3 and H 3 15. Constant mean curvature tori in R3 were first discovered, in 1984, by Wente [14]. CMC surfaces may also be characterized by the fact that their Gauss map N: S! In 1841, Delaunay [2] classified all surfaces of revolution of constant mean curvature, with a beautifully simple description in terms of conics. 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 Master's thesis, IMPA 1982, Frid, H., Thayer, F.J.: The Morse index theorem for elliptic problems with a constraint. surface is immersed as a constant mean curved surface of a four-dimensional. The equations are derived from Bryant holomorphic representation (analogous to the Weierstrass representation of minimal surfaces), in terms of gamma … 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. MUSE delivers outstanding results to the scholarly community by maximizing revenues for publishers, providing value to libraries, and enabling access for scholars worldwide. In the last case, the second fundament. Check out using a credit card or bank account with. Share. Alexandrov [1] gave a 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. United States and abroad. An. as a basic reference work in academic libraries, both in the In Riemannian manifolds very few examples of constant k-curvature hypersurfaces are … Access supplemental materials and multimedia. 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. Constant mean curvature tori in S 3 17. Abstract: The mean curvature of a surface is an extrinsic parameter measuringhow the surface is curved in the three-dimensional space. ),1, 903–906 (1979), Fischer-Colbrie, D., Schoen, R.: The structure of complete stable minimal surfaces in 3-manifolds of nonnegative scalar curvature. Mathematische Zeitschrift : Stable complete minimal surfaces inR Bull. HFS clients enjoy state-of-the-art warehousing, real-time access to critical business data, accounts receivable management and collection, and unparalleled customer service. 5 denotes a surface with a fixed immersion v: S-+R3. 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. 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. The oldest mathematics journal in the Western Hemisphere in 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. Math Z 185, 339–353 (1984). The mean curvature would then give the mean effective mass for the two principal axes. Preprint, Pogorelov, A.V. Definition 0.1 A constant mean curvature surface is a surface whose mean curvatures equal some constant at any point. Constant mean curvature tori in H 3 19. This includes minimal surfaces as a subset, but typically they are treated as special case. We denote the constant h. We call the surface a CMC h-surface. differential-geometry curvature. A representation formula for spaeelike surfaces with prescribed 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; form is covariant constant. Math.35, 199–211 (1980), Frid, H.: O Teorema do índice de Morse. of Contents. Now suppose that our surface 5 has constant mean curvature H. Let z = ul + ( — l)ll2u2, complex local coordinate, and define 4>iz) = (611-622) + 2(-l)1'2Z>12. Part of Springer Nature. Comm. Definition 0.1 A constant mean curvature surface is a surface whose mean curvatures equal some constant at any point. In 1841, Delaunay [2] classified all surfaces of revolution of constant mean curvature, with a beautifully simple description in terms of conics. ©2000-2021 ITHAKA. Brasil. 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. Select a purchase American Journal of Mathematics Z.173, 13–28 (1980), Böhme, R., Tomi, F.: Zur Struktur der Lösungsmenge des Plateauproblems. The study of surfaces with constant mean curvature (CMC) is one of the main topics in classical differential geometry. mathematics. 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. Books