A NEW LAPLACIAN COMPARISON THEOREM AND THE ESTIMATE OF EIGENVALUES

This paper establishes a new Laplacian comparison theorem which is specially useful to the manifolds of nonpositive curvature.It leads naturally to the corresponding heat kernel comparison and eigenvalue comparison theorems. Furthermore, a lower estimate of L^2-spectrum of an n-dimensional non-compact complete Cartan-Hadamard manifold is given by (n-1)k/4,provided its Ricci curvature ≤-(n-1)k(k=const.≥0).     

SUBMANIFOLDS OF A HIGHER DIMENSIONAL SPHERE

Let M be an m-dimensional manifold immersed in S~(m+k)(r).Then △X=μH-(m/r~2)X,where X is the position vector of M and H is a unit normal vector field which is orthogonalto X everywhere.If M is a compact connected manifold with parallel mean curvature vector field ξimmersed inS~(m+k)(r),and the sectional curvature of M is not less than (1/2)((1/r~2)+|ξ|~2),thenM is a small sphere.For a compact connected hypersurface M in S~(m+1)(r),if the sectional curvature is non-nesative and the scalar curvature is proportional to the mean curvature everywhere,then M isa totally umbilical hypersurface or the multiplication of two totally umbilical submanifolds.     

一类射影平坦的多项式(α，β)度量

讨论了一类具有如下形式的Finsler度量F=α+εβ+kβ~2/α+k~2β~4/3α~3-k~3β~6/5α~5,其中α=(a_(ij)y~iy~j)~(1/2)是一个Riemann度量,β=b_iy~i是一个1-形式,ε和k≠0是常数,研究了这类度量的旗曲率性质,得到了F为局部射影平坦的充要条件.     

Liouville Type Theorem About p-Harmonic Function and p-Harmonic Map with Finite L-Energy

This paper deals with the p-harmonic function on a complete non-compact submanifold M isometrically immersed in an (n + k)-dimensional complete Riemannian manifold (M) of non-negative (n-1)-th Ricci curvature.The Liouville type theorem about the p-harmonic map with finite Lq-energy from complete submanifold in a partially nonnegatively curved manifold to non-positively curved manifold is also obtained.     

Comparing the relative volume with a revolution manifold as a model

Given a pair (P, M), whereM is ann-dimensional connected compact Riemannian manifold andP is a connected compact hypersurface ofM, the relative volume of (P, M) is the quotient volume(P)/volume(M). In this paper we give a comparison theorem for the relative volume of such a pair, with some bounds on the Ricci curvature ofM and the mean curvature ofP, with respect to that of a model pair where ℳ is a revolution manifold and a “parallel” of ℳ. Work partially supported by a DGICYT Grant No. PB91-0324.     

On the stability index of hypersurfaces with constant mean curvature in spheres

Barbosa, do Carmo and Eschenburg characterized the totally umbilical spheres as the only weakly stable compact constant mean curvature hypersurfaces in the Euclidean sphere . In this paper we prove that the weak index of any other compact constant mean curvature hypersurface in n+1 which is not totally umbilical and has constant scalar curvature is greater than or equal to , with equality if and only if is a constant mean curvature Clifford torus with radius .

     

The computations of Einstein-K(a)hler metric of Cartan-Hartogs domain

In this paper, we compute the complete Einstein-Kahler metric with explicit formula for the Cartan-Hartogs domain of the fourth type in some cases. Under this metric the holomorphic sectional curvature is given, which intervenes between -2k and -1. This is the sharp estimate.     

The computations of Einstein-Khler metric of Cartan-Hartogs domain

In this paper, we compute the complete Einstein-Kahler metric with explicit formula for the Cartan-Hartogs domain of the fourth type in some cases. Under this metric the holomorphic sectional curvature is given, which intervenes between -2k and -1. This is the sharp estimate.     

Soap films bounded by non-closed curves

This paper proves (i) every “geometrically knotted” non-closed curve bounds a soap-film, (ii) any non-closed curve bounding a soap-film must have total curvature greater than 2π, and (iii) for every k > 2π, there is a geometrically knotted non-closed curve with total curvature k.     

ON HOLOMORPHIC CURVES OF CONSTANT CURVATURE IN THE COMPLEX GRASSMANN MANIFOLD G（2,5）

In this article, it is proved that there doesn’t exist any nonsingular holomorphic sphere in complex Grassmann manifold G(2, 5) with constant curvature k = 4/7, 1/2, 4/9. Thus, from [7] it follows that if φ : S2 → G(2, 5) is a nonsingular holomorphic curve with constant curvature K, then, K = 4, 2, 4/3, 1 or 4/5.     

On the Geometry of Metric Measure Spaces with Variable Curvature Bounds

Motivated by a classical comparison result of J. C. F. Sturm, we introduce a curvature-dimension condition CD(k, N) for general metric measure spaces, variable lower curvature bound \(k\) and upper dimension bound \(N\ge 1\). In the case of non-zero constant lower curvature, our approach coincides with the celebrated condition that was proposed by Sturm (Acta Math 196(1):133–177, 2006). We prove several geometric properties as sharp Bishop–Gromov volume growth comparison or a sharp generalized Bonnet–Myers theorem (Schneider’s Theorem). In addition, the curvature-dimension condition is stable with respect to measured Gromov–Hausdorff convergence, and it is stable with respect to tensorization of finitely many metric measure spaces provided a non-branching condition is assumed. We also briefly describe possible extensions for variable dimension bounds.     

Mean Curvature and Volume of Extrinsic Spheres in Hypersurfaces of Real Space Forms

Given a hypersurface P^n-1 in a real space form of constantcurvature b, , we have obtained a lower bound for the norm of the mean curvature normal vector field of extrinsicspheres in P^n-1 in terms of the mean curvature of the geodesic spheres in , with the same radius, and the meancurvature of P^n-1, characterizing too the equality.     

On Saddle Submanifolds of Riemannian Manifolds

Saddle submanifolds are considered. A characterization of such submanifolds of Euclidean space is given in terms of sectional curvature. Extending results of T. Frankel, K. Kenmotsu and C. Xia, we determine under what conditions two complete saddle submanifolds of a complete connected Riemannian manifold M, with nonnegative k-Ricci curvature, must intersect. Moreover, if M has positive k -Ricci curvature and the dimension of a compact saddle submanifold satisfies a certain inequality then we show that the homomorphism of the fundamental groups _1(M) and ₁(M) is surjective.     

The Topoligical Sphere Theorem for Complete Submanifolds

A topological sphere theorem is obtained from the point of view of submanifold geometry. An important scalar is defined by the mean curvature and the squared norm of the second fundamental form of an oriented complete submanifold Mn in a space form of nonnegative sectional curvature. If the infimum of this scalar is negative, we then prove that the Ricci curvature of Mn has a positive lower bound. Making use of the Lawson–Simons formula for the nonexistence of stable k-currents, we eliminate Hk (Mn, Z) for all 1 ` k ` n – 1.We then observe that the fundamental group of Mn is trivial. It should be emphasized that our result is optimal.     

關閉撓曲線的全曲率

<正> §1.引言.大家知道,微分幾何學所討論的一般是關於圖形的局部性質.但是這些局部性質與圖形的整個性質間常存在有某些關係.討論圖形的整個性質的微分幾何學叫做整體性的.關於整體性微分幾何學有這樣的一個著名定理:設一關閉撓曲線C     

Some comparison theorems for K?hler manifolds

In this work, we will verify some comparison results on K?hler manifolds. They are: complex Hessian comparison for the distance function from a closed complex submanifold of a K?hler manifold with holomorphic bisectional curvature bounded below by a constant, eigenvalue comparison and volume comparison in terms of scalar curvature. This work is motivated by comparison results of Li and Wang (J Differ Geom 69(1):43–47, 2005).     

Riemannian Metrics with the Prescribed Curvature Tensor and all Its Covariant Derivatives at One Point

On an n-dimensional vector space, equipped with a scalar product, we prescribe (0, 4) -, (0, 5)-, … type tensors R⁽⁰⁾, R⁽¹⁾, …, satisfying the well-known conditions for a curvature tensor and its derivatives and furthermore certain inequalities for the absolute values of the components of R^(k). Then there is an analytic Riemannian metric g on an open ball of the Cartesian space Rⁿ[u¹, …, uⁿ] for which u¹, …, uⁿ are normal coordinates and (▽^(k)R)₀ = R^(k) (k = 0, 1, 2, …) hold under an identification of the tangent space T₀Rⁿ at the origin with the vector space; ▽^(k)R denote the curvature tensor and its covariant derivatives with respect to the Levi-Civita connection ▽ of g, respectively.     

Curvatures and Currents for Unions of Sets with Positive Reach, II

A class of subsets of ^d which can berepresented as locally finite unions of sets with positive reach isconsidered. It plays a role in PDE's on manifolds with singularities.For such a set, the unit normal cycle (determining the d – 1curvature measures) is introduced as a (d – 1)-currentsupported by the unit normal bundle and its properties are established.It is shown that, under mild additional assumptions, the unit normalcycle (and, hence, also the curvature measures) of such a set can beapproximated by that of a close parallel body or, alternatively, by themirror image of that of the closure of the complement of the parallelbody (which has positive reach). Finally, the mixed curvature measuresof two sets of this class are introduced and a translative integralgeometric formula for curvature measures is proved.     

Liouville Type Theorem About p-Harmonic Function and p-Harmonic Map with Finite L~q-Energy

This paper deals with the p-harmonic function on a complete non-compact submanifold M isometrically immersed in an(n + k)-dimensional complete Riemannian manifold M of non-negative(n-1)-th Ricci curvature. The Liouville type theorem about the p-harmonic map with finite L~q-energy from complete submanifold in a partially nonnegatively curved manifold to non-positively curved manifold is also obtained.     

Totally umbilical manifolds inG(2,n). I

It is well known that every totally umbilical submanifold of a space of constant curvature is either a small sphere or is totally geodesic. B.-Y. Chen has classified totally umbilical submanifolds of compact, rank one, symmetric spaces ([4], [5]): in particular, they are all extrinsic spheres, that is, they have a parallel mean curvature vector H (or are totally geodesic). In this paper totally umbilical submanifolds F^l of dimension l 3 are classified in the irreducible symmetric space that is "next in complexity": Grassmann manifold G(2, n). Such submanifolds are either 1) totally geodesic [3] or 2) extrinsic spheres [small spheres in totally geodesic spheres; their position in G(2, n) is described here] or 3) essentially totally umbilical (H 0, H 0). If the submanifold is of type 3), then it is either a) an umbilical hypersurface of nonconstant mean curvature in totally geodesic S¹ × S¹ G(2, n) or b) an "oblique diagonal," a diagonal of the product of two small spheres of different radii in totally geodesic S^l+1 × S^l+1 G(2, n) (it has constant mean and sectional curvatures). Submanifolds 3a) and 3b) are described completely. The latter of the two negates two of Chen's conjectures. It is shown that submanifold F^l E^l+2 (l 3) with a totally umbilical Grassmannian image has a totally geodesic Grassmannian image and is classifiable [11].Translated from Ukrainskii Geometricheskii Sbornik, No. 34, pp. 83–98, 1991.