We study centroaffine minimal surfaces with non-semisimple centroaffine Tchebychev operator and classify such surfaces with constant centroaffine curvature. We also study the center map of such surfaces and show that it becomes a centroaffine surface if and only if the centroaffine curvature is not equal to 1.

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Centroaffine Minimal Surfaces with Non-Semisimple Centroaffine Tchebychev Operator

We study locally strongly convex, hyperbolic centroaffine Tchebychev hypersurfaces with complete centroaffine metric. We give a classification in case that the centroaffine Ricci tensor is semi-positive and the scalar curvature is constant. The classification extends known global results for hyperbolic affine hyperspheres and for complete centroaffine extremal hypersurfaces.

Complete Hyperbolic Tchebychev Hypersurfaces

In this paper, we study n-dimensional locally strongly convex hypersurfaces in 
$${\mathbb{R}}^{n+1}$$

 with the property that its center map is congruent to the original immersion. Our aim is to study such hypersurfaces in the generic case, i.e., when the affine shape operator has n different non zero eigenvalues. We will show that for such hypersurfaces the centroaffine metric is flat and the centroaffine difference tensor is parallel with respect to the Levi Civita connection of the centroaffine metric. In particular this means that they are canonical hypersurfaces in the sense of [3]. Extending slightly the classification of such hypersurfaces in [3], we obtain a complete classification.

Generic Affine Hypersurfaces with Self Congruent Center Map

In this paper, we calculate the Laplacian of the norm of the cubic Simon form for a hypersurface with a relative normalization. The method used here is developed in the rigidity theory of minimal submanifolds. As consequences, we obtain some local and global rigidity theorems about relative Tchebychev hypersurfaces.

Rigidity Theorems for Relative Tchebychev Hypersurfaces

We study centro-affine Tchebychev hyperbolic hypersurfaces M in $$\mathbb {R}^{4}$$
 , which satisfy the following conditions: for all $$X,Y,Z \in T_{p}M$$
 :

Flat Hyperbolic Centro-affine Tchebychev Hypersurfaces of $$\mathbb {R}^{4}$$ R 4

In this paper, we study n-dimensional improper affine spheres of which the center map is congruent to the original immersion. We show how to construct such hypersurfaces starting from (n − 1)-dimensional proper elliptic affine hyperspheres. We also show that they correspond to solutions of the Monge-Ampere equations which are homogeneous of degree 2.

Houda Trabelsi

Papers

Complete Hyperbolic Tchebychev Hypersurfaces

Generic Affine Hypersurfaces with Self Congruent Center Map

Improper affine hyperspheres with self-congruent center map