共查询到20条相似文献,搜索用时 468 毫秒
1.
《Nonlinear Analysis: Theory, Methods & Applications》2003,52(4):1069-1077
We study the prescribed mean curvature equation for nonparametric surfaces, obtaining existence and uniqueness results in the Sobolev space W2,p. We also prove that under appropriate conditions the set of surfaces of mean curvature H is a connected subset of W2,p. Moreover, we obtain existence results for a boundary value problem which generalizes the one-dimensional periodic problem for the mean curvature equation. 相似文献
2.
It is known that a compact Riemannian surface can admit at most two (2) geometrically distinct, i. e., non-congruent isometric immersions into R 3 with given non-constant mean curvature. If the genus is zero, then there is at most one such immersion. Here, we show that there is at most one such immersion in each of the following cases for surfaces of genus one: 1) there exists a point p such that (H 2 ? K)(p) = 0, where K is the curvature of the Riemannian metric and H is the given non-constant mean curvature (umbilic point); 2) the surface is a surface of revolution; 3) the surface is a tube formed by moving a circle in such a way that its center describes a smooth plane curve and its plane is constantly perpendicular to this curve. We also indicate the difficulties as to why the so-far existing methodologies cannot give a clear-cut answer to the question if it is possible to reduce the at most two immersions to at most one for any compact Riemannian surface of genus greater than zero. 相似文献
3.
It has been recently shown by Abresch and Rosenberg that a certain Hopf differential is holomorphic on every constant mean curvature surface in a Riemannian homogeneous 3-manifold with isometry group of dimension 4. In this paper we describe all the surfaces with holomorphic Hopf differential in the homogeneous 3-manifolds isometric to H2×R or having isometry group isomorphic either to the one of the universal cover of PSL(2,R), or to the one of a certain class of Berger spheres. It turns out that, except for the case of these Berger spheres, there exist some exceptional surfaces with holomorphic Hopf differential and non-constant mean curvature. 相似文献
4.
Makoto Sakaki 《Results in Mathematics》2014,66(3-4):343-362
Deforming rotation surfaces with constant mean curvature in S 3 and H 3 to S 3 × R and H 3 × R respectvely, we give four classes of surfaces with mean curvature vector of constant length in S 3 × R and H 3 × R. We have complete minimal surfaces in S 3 × R and H 3 × R. Also we obtain minimal 2-tori in S 3 × S 1, some of which are embedded. 相似文献
5.
John A. Velling 《Journal of Geometric Analysis》1999,9(3):457-489
A set of conditions are given, each equivalent to the constancy of mean curvature of a surface in
H
3.It is shown that analogs of these equivalences exist for surfaces in
S
∞
2
,the bounding ideal sphere of
H
3,leading to a notion of constant mean curvature at infinity of
H
3.A parametrization of all complete constant mean curvature surfaces at infinity of
H
3
is given by holomorphic quadratic differentials on Ĉ,C, and
D. 相似文献
6.
In this paper, we construct helicoidal surfaces under the cubic screw motion with prescribed mean or Gauss curvature in Minkowski 3-space . We also find explicitly the relation between the mean curvature and Gauss curvature of them. Furthermore, we discuss helicoidal surfaces under the cubic screw motion with H2=K and prove that these surfaces have equal constant principal curvatures. 相似文献
7.
Bang-Yen Chen 《Results in Mathematics》1998,33(1-2):65-78
A slant immersion is an isometric immersion of a Riemannian manifold into an almost Hermitian manifold with constant Wirtinger angle. A slant submanifold is called proper if it is neither holomorphic nor totally real. In [2], the author proved that, for any proper slant surface M with slant angle θ in a complex-space-form $?detilde M^2(4?silon)$ with constant holomorphic sectional curvature 4?, the squared mean curvature and the Gauss curvature of M satisfy the following basic inequality: H2(p) ≥ 2K(p) ? 2(1 + 3 cos2θ)?. Every proper slant surface satisfying the equality case of this inequality is special slant. One purpose of this article is to completely classify proper slant surfaces which satisfy the equality case of this inequality. Another purpose of this article is to completely classify special slant surfaces with constant mean curvature. Further results on special slant surfaces are also presented. 相似文献
8.
Let M be a Riemannian manifold equipped with two complementary orthogonal distributions D and D ⊥. We introduce the conformal flow of the metric restricted to D with the speed proportional to the divergence of the mean curvature vector H, and study the question: When the metrics converge to one for which D enjoys a given geometric property, e.g., is harmonic, or totally geodesic? Our main observation is that this flow is equivalent to the heat flow of the 1-form dual to H, provided the initial 1-form is D ⊥-closed. Assuming that D ⊥ is integrable with compact and orientable leaves, we use known long-time existence results for the heat flow to show that our flow has a solution converging to a metric for which H = 0; actually, under some topological assumptions we can prescribe the mean curvature H. 相似文献
9.
We give an algorithm for finding finite element approximations to surfaces of prescribed variable mean curvature, which span
a given boundary curve. We work in the parametric setting and prove optimal estimates in the H1 norm. The estimates are verified computationally. 相似文献
10.
Rafael López 《Calculus of Variations and Partial Differential Equations》2010,39(3-4):333-359
A stationary rotating surface is a compact surface in Euclidean space whose mean curvature H at each point x satisfies 2H(x) = a r(x)2 + b, where r(x) denotes the distance from x to a fixed straight-line L, and a and b are constants. These surfaces are solutions of a variational problem that describes the shape of a drop of incompressible fluid in equilibrium by the action of surface tension when it rotates about L with constant angular velocity. The effect of gravity is neglected. In this paper we study the geometric configurations of such surfaces, focusing the relationship between the geometry of the surface and the one of its boundary. As special cases, we will consider two families of such surfaces: axisymmetric surfaces and embedded surfaces with planar boundary. 相似文献
11.
Karlheinz Goldhorn 《manuscripta mathematica》1973,8(2):189-207
In recent papers HILDEBRANDT [11] and HARTH [5] proved the existence of solutions of the problem of Plateau for surfaces of bounded mean curvature with fixed and free boundaries in E3 and for minimal surfaces with free boundaries in a Riemannian manifold, respectively. Here their methods will be combined to solve the problem of Plateau for surfaces of bounded mean curvature in a Riemannian manifold. This will be done for fixed and free boundaries. Moreover, isoperimetric inequalities for the solutions will be given.
Diese Arbeit beruht auf meiner Dissertation (Mainz 1971) 相似文献
Diese Arbeit beruht auf meiner Dissertation (Mainz 1971) 相似文献
12.
Yi Fang 《Archiv der Mathematik》1999,72(6):473-480
13.
Karlheinz Schüffler 《manuscripta mathematica》1982,40(1):1-15
In the Sobolev space Hm(B,?3), B the open unit disc in ?2, we consider the set Mn of all conformally parametrized surfaces of constant mean curvature H with exactly n simple interior branch points (and no others). We denote by M*n the set of all xεMn with the following properties:
- in every branch point the geometrical condition KG¦xZ¦≡O holds (KG is the Gauss curvature and xz is the complex gradient of the surface x).
- the corresponding boundary value problem Δh+×z{2(2H2-KG)h=O,hδB=O, is uniquely solvable.
14.
Peter Kohlmann 《Results in Mathematics》1999,36(1-2):75-101
A hyperbolic analogon to Hartman’s characterization of orthogonal sphere cylinders is proved: Let Mn ? Hn+1 be a noncompact closed hypersurface with sectional curvature K ≥ 0 which bounds a convex set. Assume further Hr ≡ c for one normalized mean curvature. Then Mn is a horosphere or a geodesic cylinder if $r{\leq}\ {2\over 3}\ (n+1)$ . For $r >\ {2\over 3}\ (n+1)$ the same follows but only if c lies in a specified interval which however covers the case of a horosphere. The argumentation is based on results of S.B. Alexander and R.B. Currier on the infinity set of certain convex hypersurfaces, the comparison with interior spindle surfaces, first eigenvalue estimates for Voss operators and variational properties of relevant curvature expressions. 相似文献
15.
Sungwook Lee 《Annals of Global Analysis and Geometry》2006,29(4):355-401
It is shown that timelike surfaces of constant mean curvature ± in anti-de Sitter 3-space ?3 1(?1) can be constructed from a pair of Lorentz holomorphic and Lorentz antiholomorphic null curves in ?SL2? 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 Gauß map of timelike surfaces in ?3 1(?1), which is a close analogue of the classical Gauß map is considered. It is discussed that the hyperbolic Gauß map plays an important role in the study of timelike surfaces of constant mean curvature ± 1 in ?3 1(?1). In particular, the relationship between the Lorentz holomorphicity of the hyperbolic Gauß map and timelike surface of constant mean curvature ± 1 in ?3 1(?1) is studied. 相似文献
16.
Rafael López 《Mathematische Nachrichten》2008,281(8):1169-1181
In three‐dimensional Lorentz–Minkowski space ??3, we consider a spacelike plane Π and a round disc Ω over Π. In this article we seek the shapes of unbounded surfaces whose boundary is ? Ω and its mean curvature is a linear function of the distance to Π. These surfaces, called stationary surfaces, are solutions of a variational problem and governed by the Young–Laplace equation. In this sense, they generalize the surfaces with constant mean curvature in ??3. We shall describe all axially symmetric unbounded stationary surfaces with special attention in the case that the surface is asymptotic to Π at the infinity. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim) 相似文献
17.
By using the method of integrable system, we study the deformation of constant mean curvature surfaces in three-dimensional hyperbolic space form H3. We also obtain a Weierstrass representation formula of the constant mean curvature surfaces with mean curvature greater than 1 相似文献
18.
By using the method of integrable system, we study the deformation of constant mean curvature surfaces in three-dimensional
hyperbolic space form H3. We also obtain a Weierstrass representation formula of the constant mean curvature surfaces with mean curvature greater
than 1 相似文献
19.
Armando V. Corro Francisco Milán 《Journal of Mathematical Analysis and Applications》2010,366(2):582-592
We construct examples of flat surfaces in H3 which are graphs over a two-punctured horosphere and classify complete embedded flat surfaces in H3 with only one end and at most two isolated singularities. 相似文献
20.
Tooru Sasahara 《Results in Mathematics》2003,43(1-2):168-180
We study surfaces in a Sasakian manifold R2n++1(?3) whose mean curvature vector fields admit a finite spectral decomposition with respect to certain elliptic linear differential operators. 相似文献