On constant mean curvature surfaces in the Heisenberg group

This work is devoted to the theory of surfaces of constant mean curvature in the three-dimensional Heisenberg group. It is proved that each surface of such a kind locally corresponds to some solution of the system of a sine-Gordon type equation and a first order partial differential equation.     

On the nondegeneracy of constant mean curvature surfaces

We prove that many complete, noncompact, constant mean curvature (CMC) surfaces are nondegenerate; that is, the Jacobi operator Δ_f + | A_f |² has no L² kernel. In fact, if ∑ has genus zero with k ends, and if f (∑) is embedded (or Alexandrov immersed) in a half-space, then we find an explicit upper bound for the dimension of the L² kernel in terms of the number of non-cylindrical ends. Our main tool is a conjugation operation on Jacobi fields which linearizes the conjugate cousin construction. Consequences include partial regularity for CMC moduli space, a larger class of CMC surfaces to use in gluing constructions, and a surprising characterization of CMC surfaces via spinning spheres. R.K. partially supported by NSF grants DMS-0076085 at GANG/UMass and DMS-9810361 at MSRI, and by a FUNCAP grant in Fortaleza, Brazil. J.R. partially supported by an NSF VIGRE grant at Utah. Received: January 2005; Accepted: June 2005     

Hypersurfaces with constant scalar curvature and constant mean curvature

Non-spherical hypersurfaces inE ⁴ with non-zero constant mean curvature and constant scalar curvature are the only hypersurfaces possessing the following property: Its position vector can be written as a sum of two non-constant maps, which are eigenmaps of the Laplacian operator with corresponding eigenvalues the zero and a non-zero constant.     

On the heat flow equation of surfaces of constant mean curvature in higher dimensions

In this paper, we consider the heat flow for the Hsystem with constant mean curvature in higher dimensions. We give sufficient conditions on the initial data such that the heat flow develops finite time singularity. We also provide a new set of initial data to guarantee the existence of global regular solution to the heat flow, that converges to zero in W 1,n with the decay rate t 2/(2-n) as time goes to infinity.     

On the scalar curvature of constant mean curvature hypersurfaces in space forms

In this paper we study the behavior of the scalar curvature S of a complete hypersurface immersed with constant mean curvature into a Riemannian space form of constant curvature, deriving a sharp estimate for the infimum of S. Our results will be an application of a weak Omori-Yau maximum principle due to Pigola, Rigoli, Setti (2005) [17].     

Constant angle surfaces in the Heisenberg group

In this article we extend the notion of constant angle surfaces in $ \mathbb{S}^2 $ \mathbb{S}^2 × ℝ and ℍ² × ℝ to general Bianchi-Cartan-Vranceanu spaces. We show that these surfaces have constant Gaussian curvature and we give a complete local classification in the Heisenberg group.     

Weakly stable constant mean curvature hypersurfaces

Let M^n be an n-dimensional complete noncompact oriented weakly stable constant mean curvature hypersurface in an （n ＋ 1）-dimensional Riemannian manifold N^n＋1 whose （n - 1）th Ricci curvature satisfying Ric^N（n-1） （n - 1）c. Denote by H and φ the mean curvature and the trace-free second fundamental form of M respectively. If ｜φ｜^2 - （n- 2）√n（n- 1）｜H｜｜φ｜＋ n（2n - 1）（H^2＋ c） 〉 0, then M does not admit nonconstant bounded harmonic functions with finite Dirichlet integral. In particular, if N has bounded geometry and c ＋ H^2 〉 0, then M must have only one end.     

Coarse classification of constant mean curvature cylinders

We give a coarse classification of constant mean curvature (CMC) immersions of cylinders into via the loop group method. Particularly for this purpose, we consider double loop groups and a new type of ``potentials' which are meromorphic 1-forms on Riemann surfaces.

     

A global pinching theorem for compact surfaces inS 3 with constant mean curvature

LetM be a compact minimal surface inS ³. Y. J. Hsu^[5] proved that if S₂22, thenM is either the equatorial sphere or the Clifford torus, whereS is the square of the length of the second fundamental form ofM, ·₂ denotes theL ²-norm onM. In this paper, we generalize Hsu's result to any compact surfaces inS ³ with constant mean curvature.Supported by NSFH.     

Spectral transformation of constant mean curvature surfaces in H3 and Weierstrass representation

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.     

Bernstein type result for constant mean curvature hypersurface

We prove a Bernstein type theorem for constant mean curvature hypersurfaces in ℝ ⁿ⁺¹ under certain growth conditions for n ⩽ 3. Our result extends the case when M is a minimal hypersurface in the same condition.      

局部对称黎曼流形中的超曲面

研究了局部对称黎曼流形中具有常平均曲率超曲面的几何性质,得到了关于其第二基本形式模长平方的拼挤定理.     

Stability of constant mean curvature hypersurfaces of revolution in hyperbolic space

In this article, by solving a nonlinear differential equation, we prove the existence of a one parameter family of constant mean curvature hypersurfaces in the hyperbolic space with two ends. Then, we study the stability of these hypersurfaces.     

Spectral transformation of constant mean curvature surfaces in H3 and Weierstrass representation

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     

The Dirichlet problem for constant mean curvature surfaces in Heisenberg space

We study constant mean curvature graphs in the Riemannian three- dimensional Heisenberg spaces . Each such is the total space of a Riemannian submersion onto the Euclidean plane with geodesic fibers the orbits of a Killing field. We prove the existence and uniqueness of CMC graphs in with respect to the Riemannian submersion over certain domains taking on prescribed boundary values. L. J. Alías was partially supported by MEC/FEDER project MTM2004-04934-C04-02 and Fundación Séneca project 00625/PI/04, Spain.     

The spaces of index one minimal surfaces and stable constant mean curvature surfaces embedded in flat three manifolds

It is proved that the spaces of index one minimal surfaces and stable constant mean curvature surfaces with genus greater than one in (non fixed) flat three manifolds are compact in a strong sense: given a sequence of any of the above surfaces we can extract a convergent subsequence of both the surfaces and the ambient manifolds in the topology. These limits preserve the topological type of the surfaces and the affine diffeomorphism class of the ambient manifolds. Some applications to the isoperimetric problem are given.

     

Non-preserved curvature conditions under constrained mean curvature flows

We provide explicit examples which show that mean convexity (i.e. positivity of the mean curvature) and positivity of the scalar curvature are non-preserved curvature conditions for hypersurfaces of the Euclidean space evolving under either the volume- or the area preserving mean curvature flow. The relevance of our examples is that they disprove some statements of the previous literature, overshadow a widespread folklore conjecture about the behaviour of these flows and bring out the discouraging news that a traditional singularity analysis is not possible for constrained versions of the mean curvature flow.     

Stable complete noncompact hypersurfaces with constant mean curvature

This article concerns the structure of complete noncompact stable hypersurfaces M ⁿ with constant mean curvature H > 0 in a complete noncompact oriented Riemannian manifold N ⁿ⁺¹. In particular, we show that a complete noncompact stable constant mean curvature hypersurface M ⁿ , n = 5, 6, in the Euclidean space must have only one end. Any such hypersurface in the hyperbolic space with , respectively, has only one end.     

Accessible domains in the Heisenberg group

We study the problem of accessibility of boundary points for domains in the sub-Riemannian setting of the first Heisenberg group. A sufficient condition for accessibility is given. It is a Dini-type continuity condition for the horizontal gradient of the defining function. The sharpness of this condition is shown by examples.