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1.
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
2.
In this paper, we study cyclic surfaces in E5
generated by equiform motions of a circle. The properties
of this cyclic surfaces up to the first order are discussed. We prove
the following new result: A cyclic 2-surfaces in E5
in general are contained in canal hypersurfaces. Finally we give an example. 相似文献
3.
We solve Blaschke’s problem for hypersurfaces of dimension . Namely, we determine all pairs of Euclidean hypersurfaces that induce conformal metrics on M
n
and envelop a common sphere congruence in . 相似文献
4.
It is known that the totally umbilical hypersurfaces in the (n + 1)-dimensional spheres are characterized as the only hypersurfaces with weak stability index 0. That is, a compact hypersurface with constant mean curvature, cmc, in S n+1, different from an Euclidean sphere, must have stability index greater than or equal to 1. In this paper we prove that the weak stability index of any non-totally umbilical compact hypersurface ${M \subset S^{{n+1}}}$ with cmc cannot take the values 1, 2, 3 . . . , n. 相似文献
5.
We construct new examples of embedded, complete, minimal hypersurfaces in complex hyperbolic space, including deformations
of bisectors and some minimal foliations.
Received: 20 March 2000 / Revised version: 21 July 2000 相似文献
6.
Jocelino Sato Vicente Francisco De Souza Neto 《Annals of Global Analysis and Geometry》2006,29(3):221-240
We classify the zero scalar curvature O(p+1)×O(q+1)-invariant hypersurfaces in the euclidean space ℝ
p+q+2, p,q > 1, analyzing whether they are embedded and stable. The Morse index of the complete hypersurfaces show the existence of embedded, complete and globally stable zero scalar curvature O(p+1)×O(q+1)-invariant hypersurfaces in ℝ
p+q+2, p+q≥ 7, which are not homeomorphic to ℝ
p+q+1. Such stable examples provide counter-examples to a Bernstein-type conjecture in the stable class, for immersions with zero scalar curvature.
Mathematics Subject Classifications (2000): 53A10, 53C42,49005. 相似文献
7.
Let M be a compact oriented minimal
hypersurface of the unit n-dimensional sphere
Sn.
It is known that if the norm squared of the second fundamental form,
, satisfies that
for all
, then M is isometric to a Clifford
minimal hypersurface ([2], [5]). In this paper we will generalize this result
for minimal hypersurfaces with two principal curvatures and dimension greater
than 2. For these hypersurfaces we will show that if the average of the function
is n - 1, then M
must be a Clifford hypersurface.
Received: 24 December 2002 相似文献
8.
The n-dimensional complex hyperquadric is a compact complex algebraic hypersurface defined by the quadratic equation in the (n+1)-dimensional complex projective space, which is isometric to the real Grassmann manifold of oriented 2-planes and is a
compact Hermitian symmetric space of rank 2. In this paper, we study geometry of compact Lagrangian submanifolds in complex
hyperquadrics from the viewpoint of the theory of isoparametric hypersurfaces in spheres. From this viewpoint we provide a
classification theorem of compact homogeneous Lagrangian submanifolds in complex hyperquadrics by using the moment map technique.
Moreover we determine the Hamiltonian stability of compact minimal Lagrangian submanifolds embedded in complex hyperquadrics
which are obtained as Gauss images of isoparametric hypersurfaces in spheres with g(= 1, 2, 3) distinct principal curvatures.
Dedicated to Professor Hajime Urakawa on his sixtieth birthday.
H. Ma was partially supported by NSFC grant No. 10501028, SRF for ROCS, SEM and NKBRPC No. 2006CB805905. Y. Ohnita was partially
supported by JSPS Grant-in-Aid for Scientific Research (A) No. 17204006. 相似文献
9.
The theory of Ribaucour transformations for hypersurfaces in space forms is established. For any such hypersurface M, that admits orthonormal principal vector fields, it was shown the existence of a totally umbilic hypersurface locally associated to M by a Ribaucour transformation. A method of obtaining linear Weingarten surfaces in a three-dimensional space form is provided. By applying the theory, a new one-parameter family of complete constant mean curvature (cmc) surfaces in the unit sphere, locally associated to the flat torus, is obtained. The family contains a class of complete cmc cylinders in the sphere. In particular, one gets a family of complete minimal surfaces and minimal cylinders, locally associated to the Clifford torus.Mathematics Subject Classifications (2000): 53C20. 相似文献
10.
Peter Krauter 《Geometriae Dedicata》1994,51(3):287-303
We give a complete list of affine minimal surfaces inA
3 with Euclidean rotational symmetry, completing the treatise given in [1] and prove that these surfaces have maximal affine surface area within the class of all affine surfaces of rotation satisfying suitable boundary conditions. Besides we show that for rotationally symmetric locally strongly convex affine minimal hypersurfaces inA
n
,n4, the second variation of the affine surface area is negative definite under certain conditions on the meridian. 相似文献
11.
Zejun Hu Haizhong Li Udo Simon Luc Vrancken 《Differential Geometry and its Applications》2009,27(2):188-205
In this paper, we study locally strongly convex affine hypersurfaces of Rn+1 that have parallel cubic form with respect to the Levi-Civita connection of the affine Berwald-Blaschke metric; it is known that they are affine spheres. In dimension n?7 we give a complete classification of such hypersurfaces; in particular, we present new examples of affine spheres. 相似文献
12.
In this paper, we classify complete spacelike hypersurfaces in the anti-de Sitter space (n?3) with constant scalar curvature and with two principal curvatures. Moreover, we prove that if Mn is a complete spacelike hypersurface with constant scalar curvature n(n−1)R and with two distinct principal curvatures such that the multiplicity of one of the principal curvatures is n−1, then R<(n−2)c/n. Additionally, we also obtain several rigidity theorems for such hypersurfaces. 相似文献
13.
Let M
n
be an immersed umbilic-free hypersurface in the (n + 1)-dimensional unit sphere
, then M
n
is associated with a so-called M?bius metric g, a M?bius second fundamental form B and a M?bius form Φ which are invariants of M
n
under the M?bius transformation group of
. A classical theorem of M?bius geometry states that M
n
(n ≥ 3) is in fact characterized by g and B up to M?bius equivalence. A M?bius isoparametric hypersurface is defined by satisfying two conditions: (1) Φ ≡ 0; (2) All
the eigenvalues of B with respect to g are constants. Note that Euclidean isoparametric hypersurfaces are automatically M?bius isoparametrics, whereas the latter
are Dupin hypersurfaces.
In this paper, we determine all M?bius isoparametric hypersurfaces in
by proving the following classification theorem: If
is a M?bius isoparametric hypersurface, then x is M?bius equivalent to either (i) a hypersurface having parallel M?bius second fundamental form in
; or (ii) the pre-image of the stereographic projection of the cone in
over the Cartan isoparametric hypersurface in
with three distinct principal curvatures; or (iii) the Euclidean isoparametric hypersurface with four principal curvatures
in
. The classification of hypersurfaces in
with parallel M?bius second fundamental form has been accomplished in our previous paper [7]. The present result is a counterpart
of the classification for Dupin hypersurfaces in
up to Lie equivalence obtained by R. Niebergall, T. Cecil and G. R. Jensen.
Partially supported by DAAD; TU Berlin; Jiechu grant of Henan, China and SRF for ROCS, SEM.
Partially supported by the Zhongdian grant No. 10531090 of NSFC.
Partially supported by RFDP, 973 Project and Jiechu grant of NSFC. 相似文献
14.
We prove that there exist (n − 1)-dimensional compact embedded rotational hypersurfaces with constant scalar curvature (n − 1)(n − 2)S (S > 1) of S
n
other than product of spheres for 4 ≤ n ≤ 6. As a result, we prove that Leite’s Assertion was incorrect.The project is supported by the grant No. 10531090 of NSFC. 相似文献
15.
We study the center map of an equiaffine immersion which is introduced using the equiaffine support function. The center map
is a constant map if and only if the hypersurface is an equiaffine sphere. We investigate those immersions for which the center
map is affine congruent with the original hypersurface. In terms of centroaffine geometry, we show that such hypersurfaces
provide examples of hypersurfaces with vanishing centroaffine Tchebychev operator. We also characterize them in equiaffine
differential geometry using a curvature condition involving the covariant derivative of the shape operator. From both approaches,
assuming the dimension is 2 and the surface is definite, a complete classification follows.
Received: May 24, 2006. Revised: July 26, 2006. Accepted: July 28, 2006. 相似文献
16.
The complex two-plane Grassmannian carries a K?hler structure J and also a quaternionic K?hler structure ?. For we consider the classes of connected real hypersurfaces (M, g) with normal bundle such that and are invariant under the action of the shape operator. We prove that the corresponding unit Hopf vector fields on these hypersurfaces
always define minimal immersions of (M, g), and harmonic maps from (M, g), into the unit tangent sphere bundle with Sasaki metric . The radial unit vector fields corresponding to the tubular hypersurfaces are also minimal and harmonic. Similar results
hold for the dual space .
(Received 27 August 1999; in revised form 18 November 1999) 相似文献
17.
In this paper we characterize the spacelike hyperplanes in the Lorentz–Minkowski space L
n
+1 as the only complete spacelike hypersurfaces with constant mean curvature which are bounded between two parallel spacelike
hyperplanes. In the same way, we prove that the only complete spacelike hypersurfaces with constant mean curvature in L
n
+1 which are bounded between two concentric hyperbolic spaces are the hyperbolic spaces. Finally, we obtain some a priori estimates
for the higher order mean curvatures, the scalar curvature and the Ricci curvature of a complete spacelike hypersurface in
L
n
+1 which is bounded by a hyperbolic space. Our results will be an application of a maximum principle due to Omori and Yau, and
of a generalization of it.
Received: 5 July 1999 相似文献
18.
A submanifold M
n
r
of Minkowski space
is said to be of restricted type if its shape operator with respect to the mean curvature vector is the restriction of a fixed linear transformation of
to the tangent space of M
n
r
at every point of M
n
r
. In this paper we completely classify hypersurfaces of restricted type in
. More precisely, we prove that a hypersurface of
is of restricted type if and only if it is either a minimal hypersurface, or an open part of one of the following hypersurfaces: S
k
×
, S
k
1
×
, H
k
×
, S
n
1
, H
n
, with 1kn–1, or an open part of a cylinder on a plane curve of restricted type.This work was done when the first and fourth authors were visiting Michigan State University.Aangesteld Navorser N.F.W.O., Belgium. 相似文献
19.
Julien Roth 《Differential Geometry and its Applications》2007,25(5):485-499
We prove some pinching results for the extrinsic radius of compact hypersurfaces in space forms. In the hyperbolic space, we show that if the volume of M is 1, then there exists a constant C depending on the dimension of M and the L∞-norm of the second fundamental form B such that the pinching condition (where H is the mean curvature) implies that M is diffeomorphic to an n-dimensional sphere. We prove the corresponding result for hypersurfaces of the Euclidean space and the sphere with the Lp-norm of H, p?2, instead of the L∞-norm. 相似文献
20.
A hypersurface without umbilics in the (n+1)-dimensional Euclidean space f:Mn→Rn+1 is known to be determined by the Möbius metric g and the Möbius second fundamental form B up to a Möbius transformation when n?3. In this paper we consider Möbius rigidity for hypersurfaces and deformations of a hypersurface preserving the Möbius metric in the high dimensional case n?4. When the highest multiplicity of principal curvatures is less than n−2, the hypersurface is Möbius rigid. When the multiplicities of all principal curvatures are constant, deformable hypersurfaces and the possible deformations are also classified completely. In addition, we establish a reduction theorem characterizing the classical construction of cylinders, cones, and rotational hypersurfaces, which helps to find all the non-trivial deformable examples in our classification with wider application in the future. 相似文献