Konjugierte Netze mit p-ebenen u-Kurven,welche von den v-Kurven projektiv aufeinander bezogen werden

In this paper we consider conjugate nets in projective space P ⁿwith the following property: The u-curves of the net belong to p-dimensional subspaces of P ⁿand are projectively related to each other by the v-curves. We show that these nets form two classes. The first class consists of conjugate nets for which the u-curves from conic shadow boundaries. The u-curves of the nets of the second class, which we call C _p-nets, are rational normal curves of order p. Each u-curve possesses the characteristic (p–1)-space of its ambient p-space as a (p–1)-dimensional osculating space. This generalizes a result found by Degen [3] in the case p=2, n=3. By means of the Laplace transformation we get a construction of C _p-nets without integration.     

REGULARITY OF THE FREE BOUNDARY FOR SOME ELLIPTIC AND PARABOLIC PROBLEMS. I

We study the free boundary of solutions to some obstacle problems in the elliptic and parabolic cases. In the one-phase Stefan problem, the parabolic case, we prove that the points where the zero set has no density lie in a Lipschitz surface in space and time.

For some fully nonlinear elliptic equations of second order, we get similar results.

Furthermore, we prove the C ¹ regularity for singular points with some (n ? 1)-dimensional density.     

Cohomogeneity one Alexandrov spaces

We obtain a structure theorem for closed, cohomogeneity one Alexandrov spaces and we classify closed, cohomogeneity one Alexandrov spaces in dimensions 3 and 4. As a corollary we obtain the classification of closed, n-dimensional, cohomogeneity one Alexandrov spaces admitting an isometric T ⁿ⁻¹ action. In contrast to the one- and two-dimensional cases, where it is known that an Alexandrov space is a topological manifold, in dimension 3 the classification contains, in addition to the known cohomogeneity one manifolds, the spherical suspension of \mathbbRP² \mathbb{R}{P^2} , which is not a manifold.     

Certain condition on the second fundamental form of CR submanifolds of maximal CR dimension of complex Euclidean space

We treat m-dimensional real submanifolds M of complex space forms ̿M when the maximal holomorphic tangent subspace is (m−1)-dimensional. On these manifolds there exists an almost contact structure F which is naturally induced from the ambient space and in this paper we study the condition h(FX,Y)−h(X,FY) = g(FX,Y)η, η∊ T⊥(M), on the structure F and on the second fundamental form h of these submanifolds. Especially when the ambient space ̿M is a complex Euclidean space, we obtain a complete classification of submanifolds M which satisfy these conditions.Mathematics Subject Classifications (2000): 53C15, 53C40, 53B20.     

Translating Solutions of the Nonparametric Mean Curvature Flow with Nonzero Neumann Boundary Data in Product Manifold Mn × R*

In this paper, the authors can prove the existence of translating solutions to the nonparametric mean curvature flow with nonzero Neumann boundary data in a prescribed product manifold Mⁿ × R, where Mⁿ is an n-dimensional (n ≥ 2) complete Riemannian manifold with nonnegative Ricci curvature, and R is the Euclidean 1-space.     

Topological and metric rigidity teorems for hypersurfaces in a hyperbolic space

In this paper we study the topological and metric rigidity of hypersurfaces in ℍ ⁿ⁺¹, the (n + 1)-dimensional hyperbolic space of sectional curvature −1. We find conditions to ensure a complete connected oriented hypersurface in ℍ ⁿ⁺¹ to be diffeomorphic to a Euclidean sphere. We also give sufficient conditions for a complete connected oriented closed hypersurface with constant norm of the second fundamental form to be totally umbilic.     

Conformally Invariant Operators,Differential Forms,Cohomology and a Generalisation of Q-Curvature

On conformal manifolds of even dimension n ≥ 4 we construct a family of new conformally invariant differential complexes, each containing one coboundary operator of order greater than 1. Each bundle in each of these complexes appears either in the de Rham complex or in its dual (which is a different complex in the non-orientable case). Each of the new complexes is elliptic in case the conformal structure has Riemannian signature. We also construct gauge companion operators which (for differential forms of order k ≤ n/2) complete the exterior derivative to a conformally invariant and (in the case of Riemannian signature) elliptically coercive system. These (operator, gauge) pairs are used to define finite dimensional conformally stable form subspaces which are are candidates for spaces of conformal harmonics. This generalizes the n/2-form and 0-form cases, in which the harmonics are given by conformally invariant systems. These constructions are based on a family of operators on closed forms which generalize in a natural way Branson's Q-curvature. We give a universal construction of these new operators and show that they yield new conformally invariant global pairings between differential form bundles. Finally we give a geometric construction of a family of conformally invariant differential operators between density-valued differential form bundles and develop their properties (including their ellipticity type in the case of definite conformal signature). The construction is based on the ambient metric of Fefferman and Graham, and its relationship to the tractor bundles for the Cartan normal conformal connection. For each form order, our derivation yields an operator of every even order in odd dimensions, and even order operators up to order n in even dimension n. In the case of unweighted (or true) forms as domain, these operators are the natural form analogues of the critical order conformal Laplacian of Graham et al., and are key ingredients in the new differential complexes mentioned above.     

n维双曲空间和n维球面空间中的正弦定理及应用

本文研究了n维双曲空间和n维球面空间中单形的正弦定理和相关几何不等式. 应用距离几何的理论和方法, 给出了n维双曲空间和n维球面空间中一种新形式的正弦定理, 利用建立的正弦定理获得了Hadamard 型和Veljan-Korchmaros型不等式. 另外, 建立了涉及两个n维双曲单形和n维球面单形的"度量加"的一些几何不等式.     

Complete noncompact manifolds with harmonic curvature

Let (M ⁿ , g) be an n-dimensional complete noncompact Riemannian manifold with harmonic curvature and positive Sobolev constant. In this paper, by employing an elliptic estimation method, we show that (M ⁿ , g) is a space form if it has sufficiently small L ^n/2-norms of trace-free curvature tensor and nonnegative scalar curvature. Moreover, we get a gap theorem for (M ⁿ , g) with positive scalar curvature.     

A basic inequality and new characterization of Whitney spheres in a complex space form

LetN ⁿ (4c) be ann-dimensional complex space form of constant holomorphic sectional curvature 4c and letx:M ⁿ →N ⁿ (4c) be ann-dimensional Lagrangian submanifold inN ⁿ (4c). We prove that the following inequality always hold onM ⁿ: whereh is the second fundamental form andH is the mean curvature of the submanifold. We classify all submanifolds which at every point realize the equality in the above inequality. As a direct consequence of our Theorem, we give, a new characterization of theWhitney spheres in a complex space form. Partially supported by a research fellowship of the Alexander von Humboldt Stiftung.     

The classification of N-dimensional non-associative Jordan algebras with (N − 3)-dimensional annihilator

The paper is devoted to give a complete classification of all n-dimensional non-associative Jordan algebras with (n?3)-dimensional annihilator over an algebraically closed field of characteristic ≠2. We also give a complete classification of all n-dimensional Jordan algebras with (n?1)- and (n?2)-dimensional annihilator.     

Linear controllability by piecewise constant controls with assigned switching times

For a linear time-dependent control process inR ⁿ, we prove that the complete controllability by means of the space of all the admissible controls is equivalent to the complete controllability by means of a suitablen-dimensional space of piecewise constant controls with at mostn preassigned switching times. An analogous result is also established for more general controllability problems.     

Engel groups II

We study m-dimensional real submanifolds M with (m − 1)-dimensional maximal holomorphic tangent subspace in complex projective space. On these manifolds there exists an almost contact structure F which is naturally induced from the ambient space and in this paper we study the condition h(FX, Y) − h(X, FY) = g(FX, Y)η, η ∈ T ^⊥(M), on the almost contact structure F and on the second fundamental form h of these submanifolds and we characterize certain model spaces in complex projective space.     

Rigidity of minimal submanifolds with flat normal bundle

Let M ⁿ (n ≥ 3) be an n-dimensional complete immersed $ \frac{{n - 2}} {n} $ \frac{{n - 2}} {n} -super-stable minimal submanifold in an (n + p)-dimensional Euclidean space ℝ ^n+p with flat normal bundle. We prove that if the second fundamental form of M satisfies some decay conditions, then M is an affine plane or a catenoid in some Euclidean subspace.     

Certain condition on the second fundamental form of CR submanifolds of maximal CR dimension of complex hyperbolic space

Studying the condition \({h(FX,Y)-h(X,FY)=g(FX,Y)\eta, 0\ne\eta\in T^\perp(M)}\) on the almost contact structure F and on the second fundamental form h of n-dimensional real submanifolds M of complex hyperbolic space \({\mathbb {CH}^{\frac{n+p}{2}}}\) when their maximal holomorphic tangent subspace is (n ? 1)-dimensional, we obtain the complete classification of such submanifolds M and we characterize certain model spaces in complex hyperbolic space.     

Minimal two-spheres with constant curvature in the quaternionic projective space

In this paper we completely classify the homogeneous two-spheres, especially, the minimal homogeneous ones in the quaternionic projective space HPn. According to our classification, more minimal constant curved two-spheres in HPnare obtained than what Ohnita conjectured in the paper "Homogeneous harmonic maps into complex projective spaces. Tokyo J Math, 1990, 13: 87–116".     

双曲空间形式中具有常平均曲率的超曲面(英文)

本文研究了双曲空间形式H~(n+1)(—1)中具有常平均曲率及两个离散主曲率(其中一个主曲率是1-重)的完备连通可定向的n-维超曲面M~n.利用活动标架,得到如果M~n的基本形式的模长满足刚性条件(1.3),那么M~n同构双曲柱面.     

The classification of n-dimensional anticommutative algebras with (n − 3)-dimensional annihilator

We give the classification of all n-dimensional anticommutative complex algebras with (n???3)-dimensional annihilator. Namely, we describe all central extensions of all 3-dimensional anticommutative complex algebras.     

On the -equation¶over pseudoconvex Kähler manifolds

The Picard variety Pic⁰(? ⁿ ) of a complex n-dimensional torus? ⁿ is the group of all holomorphic equivalence classes of topologically trivial holomorphic (principal) line bundles on ? ⁿ . The total space of a topologically trivial holomorphic (principal) line bundle on a compact K?hler manifold is weakly pseudoconvex. Thus we can regard Pic⁰(? ⁿ ) as a family of weakly pseudoconvex K?hler manifolds. We consider a problem whether the Kodaira's -Lemma holds on a total space of holomorphic line bundle belonging to Pic⁰(? ⁿ ). We get a criterion for this problem using a dynamical system of translations on Pic⁰(? ⁿ ). We also discuss the problem whether the -Lemma holds on strongly pseudoconvex K?hler manifolds or not. Using the result of ColColţoiu, we find a 1-convex complete K?hler manifold on which the -Lemma does not hold. Received: 11 June 1999 / Revised version: 22 November 1999     

Complete submanifolds in Euclidean spaces¶with parallel mean curvature vector

In this paper, we prove that n-dimensional complete and connected submanifolds with parallel mean curvature vector H in the (n+p)-dimensional Euclidean space E ^{n + p} are the totally geodesic Euclidean space E ⁿ , the totally umbilical sphere S ⁿ (c) or the generalized cylinder S ^{n − 1} (c) ×E ¹ if the second fundamental form h satisfies <h>²≤n ²|H|²/ (n− 1). Received: 28 November 2000 / Revised version: 7 May 2001