共查询到20条相似文献,搜索用时 46 毫秒
1.
We study the topology of (properly) immersed complete minimal surfaces P 2 in Hyperbolic and Euclidean spaces which have finite total extrinsic curvature, using some isoperimetric inequalities satisfied by the extrinsic balls in these surfaces (see [10]). We present an alternative and unified proof of the Chern-Osserman inequality satisfied by these minimal surfaces (in ? n and in ? n (b)), based in the isoperimetric analysis mentioned above. Finally, we show a Chern-Osserman-type equality attained by complete minimal surfaces in the Hyperbolic space with finite total extrinsic curvature. 相似文献
2.
Yi Fang 《Archiv der Mathematik》1999,72(6):473-480
3.
We give a complete classification of complete noncompact oriented surfaces with nonnegative Gaussian curvature and finite total mean curvature in R3. 相似文献
4.
Qiaoling Wang 《Annals of Global Analysis and Geometry》2010,37(2):113-124
We prove that a complete non-compact submanifold in a complete manifold of partially non-negative sectional curvature has
only one end if the Sobolev inequality holds on it and if its total curvature is not very big by showing a Liouville theorem
for harmonic maps and by using a existence theorem of constant harmonic functions with finite energy. We also generalize a
result by Cao–Shen–Zhu saying that a complete orientable stable minimal hypersurface in a Euclidean space has only one end
to submanifolds in manifolds of partially non-negative sectional curvature. Some related results about the structure of the
same kind of submanifolds are also obtained. 相似文献
5.
本文把Berard P.,do Carmo M.,Santos W.在1998年所得的结果,分别推广到局部对称的Cartan-Hadamard流形中具有常平均曲率和有限全曲率的完备超曲面,以及球面上具有平行平均曲率和有限全曲率的完备子流形. 相似文献
6.
Paralleling what has been done for minimal surfaces in ℝ3, we develop a gluing procedure to produce, for any k≥ 2 and any n≥ 3 complete immersed minimal hypersurfaces of ℝ
n
+1 which have k planar ends. These surfaces are of the topological type of a sphere with k punctures and they all have finite total curvature.
Received: 1 July 1999 / Revised version: 31 May 2000 相似文献
7.
We prove existence theorems for two-dimensional, noncompact, complete minimal surfaces in ℝn of annular type, which span a given contour and have a finite total curvature end and prescribed asymptotical behavior. For
arbitrary rectifiable Jordan curves, we show the existence of such surfaces with a flat end, i.e., within a bounded distance
from a 2-plane. For more restricted classes of curves, we prove the existence of minimal surfaces with higher multiplicity
flat ends as well as of surfaces with polynomial-type nonflat ends.
__________
Translated from Sovremennaya Matematika. Fundamental’nye Napravleniya (Contemporary Mathematics. Fundamental Directions),
Vol. 17, Differential and Functional Differential Equations. Part 3, 2006. 相似文献
8.
In this paper, we introduce the class of hypersurfaces of finitegeometric type. They are defined as the ones that share the basicdifferential topological properties of minimal surfaces of finite totalcurvature. We extend to surfaces in this class the classical theorem ofOsserman on the number of omitted points of the Gauss mapping ofcomplete minimal surfaces of finite total curvature. We give aclassification of the even-dimensional catenoids as the only even-dimensional minimal hypersurfaces of R
n
of finite geometric type. 相似文献
9.
F. J. Ló pez Francisco Martin Santiago Morales 《Transactions of the American Mathematical Society》2006,358(9):3807-3820
The existence of complete minimal surfaces in a ball was proved by N. Nadirashvili in 1996. However, the construction of such surfaces with nontrivial topology remained open. In 2002, the authors showed examples of complete orientable minimal surfaces with arbitrary genus and one end. In this paper we construct complete bounded nonorientable minimal surfaces in with arbitrary finite topology. The method we present here can also be used to construct orientable complete minimal surfaces with arbitrary genus and number of ends.
10.
Francisco Torralbo 《Annals of Global Analysis and Geometry》2012,41(4):391-405
In this article, we construct compact, arbitrary Euler characteristic, orientable and non-orientable minimal surfaces in the
Berger spheres. Also, we show an interesting family of surfaces that are minimal in every Berger sphere, characterizing them
by this property. Finally we construct, via the Daniel correspondence, new examples of constant mean curvature surfaces in
\mathbb S2 ×\mathbb R, \mathbb H2 ×\mathbb R{\mathbb S^2 \times \mathbb R,\; \mathbb H^2 \times \mathbb R} and the Heisenberg group with many symmetries. 相似文献
11.
Chikako Mese 《Proceedings of the American Mathematical Society》2001,129(2):573-580
In this paper, we investigate the global geometric behavior of lagrangian stationary surfaces which are lagrangian surfaces whose area is critical with respect to lagrangian variations. We find that if a complete oriented immersed lagrangian surface has quadratic area growth, one end and finite topological type, then it is minimal and hence holomorphic. The key to the proof is the mean curvature estimate of Schoen and Wolfson combined with the observation that a complete immersed surface of quadratic area growth, finite topology and mean curvature has finite total absolute curvature.
12.
Juncheol Pyo 《Annals of Global Analysis and Geometry》2011,40(2):167-176
We construct three kinds of complete embedded minimal surfaces in \({\mathbb {H}^2\times \mathbb {R}}\) . The first is a simply connected, singly periodic, infinite total curvature surface. The second is an annular finite total curvature surface. These two are conjugate surfaces just as the helicoid and the catenoid are in \({\mathbb {R}^3}\) . The third one is a finite total curvature surface which is conformal to \({\mathbb {S}^2\setminus\{p_1,\ldots,p_k\}, k\geq3.}\) 相似文献
13.
Ralf Zimmermann 《Mathematische Annalen》2010,346(1):85-105
Using a method by Traizet (J Differ Geom 60:103–153, 2002), which reduces the construction of minimal surfaces via the Weierstraß Theorem and the implicit function theorem to solving algebraic equations in several complex variables, we will show the existence of complete embedded minimal surfaces of finite total curvature with planar ends of least possible order. 相似文献
14.
Abstract—In this paper, we consider connected minimal surfaces in R3 with isothermal coordinates and with a family of geodesic coordinates curves, these surfaces will be called GICM-surfaces. We give a classification of the GICM-surfaces. This class of minimal surfaces includes the catenoid, the helicoid and Enneper’s surface. Also, we show that one family of this class of minimal surfaces has at least one closed geodesic and one 1-periodic family of this class has finite total curvature. As application we show other characterization of catenoid and helicoid. Finally, we show that the class of GICM-surfaces coincides with the class of minimal surfaces whose the geodesic curvature k g 1 and k g 2 of the coordinates curves satisfy αk g 1 + βk g 2 = 0, α, β ∈ R. 相似文献
15.
Lu Jin 《Differential Geometry and its Applications》2007,25(6):701-712
In this paper, we first establish a second main theorem for algebraic curves into the n-dimensional projective space. We then use it to study the ramified values for the Gauss map of the complete (regular) minimal surfaces in Rm with finite total curvature, as well as the uniqueness problem. 相似文献
16.
We consider Ribaucour transformations between minimal surfaces and we relate such transformations to generating planar embedded ends. Applying Ribaucour transformations to Enneper's surface and to the catenoid, we obtain new families of complete, minimal surfaces, of genus zero, immersed in R
3, with infinitely many embedded planar ends or with any finite number of such ends. Moreover, each surface has one or two nonplanar ends. A particular family is obtained from the catenoid, for each pair (n,m), nm, such that n
m0 is an irreducible rational number. For any such pair, we get a 1-parameter family of finite total curvature, complete minimal surfaces with n+2 ends, n embedded planar ends and two nonplanar ends of geometric index m, whose total curvature is –4(n+m). The analytic interpretation of a Ribaucour transformation as a Bäcklund type transformation and a superposition formula for the nonlinear differential equation = e-2 is included. 相似文献
17.
Chen Qing 《manuscripta mathematica》1997,92(1):135-142
LetM be an immersed complete minimal surface inR
n. We show that the total curvature ofM is finite if and only ifM is of quadratic area growth and finite topological type. 相似文献
18.
In this paper we study complete orientable surfaces with a constant principal curvature R in the 3‐dimensional hyperbolic space H 3. We prove that if R2 > 1, such a surface is totally umbilical or umbilically free and it can be described in terms of a complete regular curve in H 3. When R2 ≤ 1, we show that this result is not true any more by means of several examples. This contradicts a previous statement by Zhisheng [6]. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim) 相似文献
19.
In this paper we prove that two lines bounding an immersed minimal surface in a slab in R
3 homeomorphic to a compact Riemann surface minus two disks and a finite number of points must be parallel. This theorem is
extended to a higher dimensional minimal hypersurface. Also it is proved that if the Gauss map of a complete embedded minimal
surface of finite total curvature at a planar end has order two, then the intersection of the surface with the plane asymptotic
to the planar end cannot admit a one-to-one orthogonal projection onto any line in the plane.
Received: November 26, 1998 相似文献
20.
We give an estimate of the smallest spectral value of the Laplace operator on a complete noncompact stable minimal hypersurface
M in a complete simply connected Riemannian manifold with pinched negative sectional curvature. In the same ambient space,
we prove that if a complete minimal hypersurface M has sufficiently small total scalar curvature then M has only one end. We also obtain a vanishing theorem for L
2 harmonic 1-forms on minimal hypersurfaces in a Riemannian manifold with sectional curvature bounded below by a negative constant.
Moreover, we provide sufficient conditions for a minimal hypersurface in a Riemannian manifold with nonpositive sectional
curvature to be stable. 相似文献