Total Scalar Curvature and L 2 Harmonic 1-Forms on a Minimal Hypersurface in Euclidean Space

Let M ⁿ , n 3, be a complete oriented immersed minimal hypersurface in Euclidean space R ⁿ⁺¹. We show that if the total scalar curvature on M is less than the n/2 power of 1/C _s , where C _s is the Sobolev constant for M, then there are no L ² harmonic 1-forms on M. As corollaries, such a minimal hypersurface contains no nontrivial harmonic functions with finite Dirichlet integral and so it has only one end. This implies finally that M is a hyperplane.     

L 2-Harmonic 1-Forms on Submanifolds with Finite Total Curvature

Let $x:M^{m}\to\bar{M}$ , m≥3, be an isometric immersion of a complete noncompact manifold M in a complete simply connected manifold $\bar{M}$ with sectional curvature satisfying $-k^{2}\leq K_{\bar{M}}\leq0$ , for some constant k. Assume that the immersion has finite total curvature in the sense that the traceless second fundamental form has finite L ^m -norm. If $K_{\bar{M}}\not\equiv0$ , assume further that the first eigenvalue of the Laplacian of M is bounded from below by a suitable constant. We prove that the space of the L ² harmonic 1-forms on M has finite dimension. Moreover, there exists a constant Λ>0, explicitly computed, such that if the total curvature is bounded from above by Λ then there are no nontrivial L ²-harmonic 1-forms on M.     

Harmonic 1-Forms on Compact f-Manifolds

We consider a Riemannian manifold with a compatible f-structure which admits a parallelizable kernel. With some additional integrability conditions it is called -manifold. This class of manifolds is a natural generalization of the Sasakian manifolds. We study properties of harmonic 1-forms on such a manifold and deduce some topological properties. Research supported by the Italian MIUR 60% and GNSAGA. Professor J.J. Konderak died just during the preparation of this paper. His contribution has been substantial. Note of the Editor-in-Chief. Professor Jerzy J. Konderak passed away on September 14, 2005, at the age of 49. Born on February 12, 1956, in Krakow, Poland, he received his Ph.D. in 1986 from the Jagiellonian University of Krakow. Since 2003 he was associated professor of Mathematics at the Faculty of Sciences of the University of Bari. The main area of his scientific interests was Differential Geometry. Professor Konderak was a very gifted researcher and was appreciated by students and colleagues for his passion and devotion to Mathematics and its teaching. All the colleagues of the Department of Mathematics of the University of Bari will remember him not only as a clever mathematician but also as a men of rare quality.     

单位球面中极小子流形的曲率拼挤

本文证明了单位球面中极小子流形的一些拼挤定理,特别注意到单位球面中的极小超曲面、给出了截曲率的拼挤常数,我们也改进了由Ｎ．Ｅｊｉｒｉ得到的Ｒｉｃｃｉ曲率拼挤常数。     

A Characterization of Minimal Legendrian Submanifolds in S2n+1

Let x: L ⁿ S²ⁿ⁺¹ R²ⁿ⁺² be a minimal submanifold in S²ⁿ⁺¹. In this note, we show that L is Legendrian if and only if for any A su(n + 1) the restriction to L of Ax, (–1)x satisfies f = 2(n + 1)f. In this case, 2(n + 1) is an eigenvalue of the Laplacian with multiplicity at least (n(n + 3)). Moreover if the multiplicity equals to ;(n(n + 3)), then L ⁿ is totally geodesic.     

球面上的极小子流形

本文证明了,在一定条件下,单位球面上具有任意余维数p的n维极小子流形是球面极小积上的一个开子流形,其中,从而推广了文[1]中命题1,那里余维数为1.     

局部对称黎曼流形中的极小子流形

In this paper, we discuss the compact minimal submanifolds in locally symmetric Riemannian manifolds. Two Pinching theorems are obtained and two corresponding results of Chern, S. S. and Yau S. T. are generalized.     

欧氏空间中完备极小子流形

在一个类似于稳定不等式的条件下,得到了欧氏空间中完备极小子流形的Bernstein型定理.我们的结果部分推广了Li H.Z.和Wei G.X.的定理.     

Minimal Homogeneous Submanifolds in Euclidean Spaces

We prove that minimal (extrinsically) homogeneous submanifolds of the Euclidean space are totally geodesic. As an application, we obtain that a complex homogeneous submanifold of C ^N must be totally geodesic.     

Minimal Submanifolds in a Sphere II

We use the method of H. Gauchman [2] to show that if the sectionalcurvature of a compact minimal n-dimensional submanifold ina unit sphere is everywhere bigger than n/2(n+1) then it mustbe totally geodesic. We also discuss some related results andconjectures.     

Foliations of the Sasakian Space R2n+1 by Minimal Slant Submanifolds

Examples of slant submanifolds in the Sasakian space R²ⁿ⁺¹ are obtained as the leaves of a harmonic, Riemannian 3-dimensional foliation. With the exception of the anti-invariant ones, these leaves are all locally homogeneous manifolds with negative scalar curvature, whose Ricci tensor satisfies (S)(X, X) = 0 for all tangent vector fields.     

Uniqueness of Minimal Submanifolds in Euclidean Space

Let M be a properly immersed n-dimensional complete minimal submanifold in Euclidean space R^n+p of dimension n+p. Let A be the second fundamental form of the immersion, and r the extrinsic distance from the origin. Suppose M has one end and inf_t sup_r(x)>t r²(x) |A|²(x) < C(n,p), then M is an affine n-plane, where C(n,p) are constants given by C(n,1) = n – 1 and C(n,p) = (2/3)(n – 1) when p > 1.     

A Pinching Theorem for Minimal Submanifolds in Sphere

§１．ＩｎｔｒｏｄｕｃｔｉｏｎＬｅｔＭｂｅａｎｎ－ｄｉｍｅｎｓｉｏｎａｌｃｌｏｓｅｄｍｉｎｉｍａｌｙｉｍｍｅｒｓｅｄｓｕｂｍａｎｉｆｏｌｄｉｎｔｈｅｕｎｉｔｓｐｈｅｒｅＳｎ＋ｐ，Ｓｔｈｅｓｅｑｕｒｅｏｆｔｈｅｌｅｎｇｔｈｏｆｔｈｅｓｅｃｏｎｄｆｕｎｄ...     

单位球面的三维紧致极小子流形

本文得到高维球面上三维紧致极小子流形的若干刚性定理．在一个整体拚挤条件下，数量曲率的拚挤常数被改进了，也对李奇曲率的拚挤问题进行了讨论．     

局部对称黎曼流形中的紧致极小子流形的Ricci曲率

设N ^n+p是截面曲率K_N 满足1/2 <δ≤ K_N≤ 1 的n+p维局部对称完备的δ-Pinching黎曼流形. Mⁿ是N^n+p 的紧致极小子流形. 该文讨论了这类子流形关于Ricci曲率有关的Pinching定理.     

关于CPn中共圆平坦全实极小子流形的一点注记

Let M be a concircularly flat totally real minimal submanifold in CP~4. The infimum V_m of the volume V(M) of M is obtained, also the necessary and sufficient conditions of "V(M)=V_m" is given.     

Totally Real Minimal Submanifolds in a Quaternion Projective Space

51.IntroductionAquaternionKaehlermanifoldisdefinedasa4m-dimensionalRiemannianmanifoldwhoseholonomygroupiscontainedinSp(m).Sp(1)withtheadditionalconditionform=lthatitisaself-dualEinsteinspace.AquaternionprojectivespaceQP:)isaquaternionKeahlermanifoldwithconstantquaternionsectionalcurvaturec>O.AcomplexprojectivespaceCP2)withconstantholomorphicsectionalcurvatureccanbeisometricallyimbeddedinQP:)asatthtallygeodesicsubmanifold.LetMbeannsidimensionalRiemannianmanifoldander:M-Qer).-.'an1sometr1…     

A Remark on Harmonic Maps Between Pseudo-Riemannian Spheres

ＡＲｅｍａｒｋｏｎＨａｒｍｏｎｉｃＭａｐｓＢｅｔｗｅｅｎＰｓｅｕｄｏ－ＲｉｅｍａｎｎｉａｎＳｐｈｅｒｅｓＤｉｎｇＱｉｎｇ（丁青）（ＩｎｓｔｉｔｕｔｅｏｆＭａｔｈｅｍａｔｉｃｓ，ＦｕｄａｎＵｎｉｖｅｒｓｉｔｙ，Ｓｈａｇｈａｉ，２００４３３）ＷａｎｇＪｉ...     

A Remark on Concircularly Flat Totally Real Minimal Submanifolds in CP^n

Let M be a concircularly fiat totally real minimal submanifold in CP4. The infimum Vm of the volume V （M） of M is obtained, also the necessary and sufficient conditions of ＂V（M）=Vm＂ is given.