A unimodularly invariant theory for immersions into the affine space

We develop a unimodularly invariant theory for immersions with higher codimension into the affine space. Received: 6 September 2001; in final form: 22 November 2001 / Published online: 29 April 2002 RID="*" ID="*" Supported by the Deutsche Forschungsgemeinschaft     

An affine analogue of the Hartman-Nirenberg cylinder theorem

Let X be a smooth, complete, connected submanifold of dimension in a complex affine space , and r is the rank of its Gauss map . The authors prove that if and in the pencil of the second fundamental forms of X, there are two forms defining a regular pencil all eigenvalues of which are distinct, then the submanifold X is a cylinder with -dimensional plane generators erected over a smooth, complete, connected submanifold Y of rank r and dimension r. This result is an affine analogue of the Hartman-Nirenberg cylinder theorem proved for and r = 1. For and , there exist complete connected submanifolds that are not cylinders. Received: 20 October 2000 / Revised version: 18 April 2001 / Published online: 18 January 2002     

Level surfaces of non-degenerate functions inR n+1

We study level surfaces of non-degenerate functions inR ⁿ⁺¹. Such level surfaces are non-degenerate in the sense of affine differential geometry. In affine differential geometry, the affine normal plays an important role for the study of a non-degenerate hypersurface. In this note, being motivated by Koszul's work we take a canonical vector field for level surfaces of a non-degenerate function and give certain characterizations of when is transversal, by the shape operatorS, the transversal connection , and consider the difference between and the affine normal.     

An affine characterization of the Veronese surface

IfM ² is a nondegenerate surface in a 4-dimensional Riemannian manifold , then there is a natural affine metricg defined onM ². It is shown that this affine metricg is conformal to the induced Riemannian metric onM ² if and only ifM ² is a minimal submanifold of in the usual Riemannian sense. If the conformal factor is a constant, then the two metrics are said to be homothetic. It is shown that there does not exist a nondegenerate surface in Euclidean space ⁴ or hyperbolic spaceH ⁴ whose affine metric is homothetic to the induced Riemannian metric. Furthermore, ifM ² is a nondegenerate surface in the standard 4-sphereS ⁴ whose affine metric is homothetic to the induced Riemannian metric, thenM ² is a Veronese surface.T. Cecil was supported by NSF Grant No. DMS-9101961.     

Equivalence theorems in affine differential geometry

In this paper we establish an affine equivalence theorem for affine submanifolds of the real affine space with arbitrary codimension. Next, this theorem is used to prove the classical congruence theorem for submanifolds of the Euclidean space, and to prove some results on affine hypersurfaces of the real affine space.Research Assistant of the National Fund for Scientific Research (Belgium).     

The Cauchy problem for indefinite improper affine spheres and their Hessian equation

We give a conformal representation for indefinite improper affine spheres which solve the Cauchy problem for their Hessian equation. As consequences, we can characterize their geodesics and obtain a generalized symmetry principle. Then, we classify the helicoidal indefinite improper affine spheres and find a new family with geodesically complete non-flat affine metric. Moreover, we present interesting examples with singular curves and isolated singularities.     

Enclosed convex hypersurfaces with maximal affine area

In this paper we study the regularity of closed, convex surfaces which achieve maximal affine area among all the closed, convex surfaces enclosed in a given domain in the Euclidean 3-space. We prove the C^1,α regularity for general domains and C^1,1 regularity if the domain is uniformly convex. This work is supported by the Australian Research Council. Research of Sheng was also supported by ZNSFC No. 102033. On leave from Zhejiang University.     

The mixed Lp affine surface areas for functions

In this paper, we propose a definition of a general mixed L_p Affine surface area, ?n ≠ p ∈ ?, for multiple functions. Our definition is di?erent from and is “dual” to the one in [11] by Caglar and Ye. In particular, our definition makes it possible to establish an integral formula for the general mixed L_p Affine surface area of multiple functions (see Theorem 3.1 for more precise statements). Properties of the newly introduced functional are proved such as affine invariance, and related affine isoperimetric inequalities are proved.     

A min-max theorem for complex symmetric matrices

We optimize the form Re x^tTx to obtain the singular values of a complex symmetric matrix T. We prove that for ,

     

A sextic holomorphic form of affine surfaces with constant affine mean curvature

We extend an original idea of Calabi for affine maximal surfaces and define a sextic holomorphic differential form for affine surfaces with constant affine mean curvature. We get some rigidity results for affine complete surfaces by using this sextic holomorphic form. Received: 17 May 2003     

Locally strongly convex hypersurfaces with constant affine mean curvature

Let be a locally strongly convex hypersurface, given by a strictly convex function xn+1=f(x₁,…,xn) defined in a convex domain Ω⊂An. We consider the Riemannian metric G# on M, defined by . In this paper we prove that if M is a locally strongly convex surface with constant affine mean curvature and if M is complete with respect to the metric G#, then M must be an elliptic paraboloid.     

Affine rotational surfaces with vanishing affine curvature

We give a classification of affine rotational surfaces in affine 3-space with vanishing affine Gauss-Kronecker curvature. Non-degenerated surfaces in three dimensional affine space with affine rotational symmetry have been studied by a number of authors (I.C. Lee. [3], P. Lehebel [4], P.A. Schirokow [10], B. Su [12], W. Süss [13]). In the present paper we study these surfaces with the additional property of vanishing affine Gauss-Kronecker curvature, that means the determinant of the affine shape operator is zero. We give a complete classification of these surfaces, which are the affine analogues to the cylinders and cones of rotation in euclidean geometry. These surfaces are examples of surfaces with diagonalizable rank one (affine) shape operator (cf. B. Opozda [8] and B. Opozda, T. Sasaki [7]). The affine normal images are curves.     

Harmonicity of gradient mappings of level surfaces in a real affine space

In our previous paper [4] we have investigated level surfaces of a non-degenerate function in a real affine space A ⁿ⁺¹ by using the gradient vector field . We gave characterizations of by means of the shape operatorS, the transversal connection , and studied the difference between and the affine normal. In particular we showed that a graph defined by a non-degenerate function satisfiesS=0 and =0. In this paper we consider harmonic gradient mappings of level surfaces and apply these results to a certain problem which is similar to the affine Bernstein problem conjectured by S. S. Chern [3].     

Focal points and support functions in affine differential geometry

The notions of focal point and support function are considered for a nondegenerate hypersurfaceM ⁿ in affine spaceR ⁿ⁺¹ equipped with an equiaffine transversal field. IfM ⁿ is locally strictly convex, these two concepts are related via an Index theorem concerning the critical points of the support functions onM ⁿ . This is used to obtain characterizations of spheres and ellipsoids in terms of the critical point behavior of certain classes of affine support functions.Research supported by NSF Grant No. DMS-9101961.     

On locally strongly convex affine hypersurfaces with parallel cubic form. Part I

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.     

Projectively flat affine surfaces

We study non-degenerate affine surfaces in A³ with a projectively flat induced connection. The curvature of the affine metric , the affine mean curvature H, and the Pick invariant J are related by . Depending on the rank of the span of the gradients of these functions, a local classification of three groups is given. The main result is the characterization of the projectively flat but not locally symmetric surfaces as a solution of a system of ODEs. In the final part, we classify projectively flat and locally symmetric affine translation surfaces.     

Affine immersion ofn-dimensional manifold intoR n+n(n+1)/2 and affine minimality

We formulate an affine theory of immersions of ann-dimensional manifold into the Euclidean space of dimensionn+n(n+1)/2 and give a characterization of critical immersions relative to the induced volume functional in terms of the affine shape operator.