相关论文: Parallel surfaces in affine 4-space

We provide an explicit classification of the following four families of surfaces in any homogeneous 3-manifold with 4-dimensional isometry group: isoparametric surfaces, surfaces with constant principal curvatures, homogeneous surfaces, and…

The surface of lines in a cubic fourfold intersecting a fixed line splits motivically into two parts, one of which resembles a K3 surface. We define the analogue of the Beauville-Voisin class and study the push-forward map to the Fano…

We classify surjective self-maps (of degree at least two) of affine surfaces according to the log Kodaira dimension.

We classify meromorphic affine connections on compact complex surfaces with algebraic dimension one, extending the work of Inoue,Kobayashi and Ochiai (1981) in the holomorphic case. The motivation is to investigate possible extension of the…

We first define a complex angle between two oriented spacelike planes in 4-dimensional Minkowski space, and then study the constant angle surfaces in that space, i.e. the oriented spacelike surfaces whose tangent planes form a constant…

This article is a survey article that gives detailed constructions and illustrations of some of the standard examples of non-orientable surfaces that are embedded and immersed in 4-dimensional space. The illustrations depend upon their…

We study the integral Brauer--Manin obstruction for affine diagonal cubic surfaces, which we employ to construct the first counterexamples to the integral Hasse principle in this setting. We then count in three natural ways how such…

We classify complete biharmonic surfaces with parallel mean curvature vector field and non-negative Gaussian curvature in complex space forms.

Global isothermic immersions are defined and studied with the aid of a connection between quadratic differentials and immersions. The applications are two problems stemming from the fundamental question: how much data is needed to identify…

We give geometric interpretations of certain affine invariants of convex bodies. The affine invariants are the p-affine surface areas introduced by Lutwak. The geometric interpretations involve generalizations of the Santal\'o-bodies…

In this paper we study curvature types of immersed surfaces in three-dimensional (normed or) Minkowski spaces. By endowing the surface with a normal vector field, which is a transversal vector field given by the ambient Birkhoff…

The isometric immersion of two-dimensional Riemannian manifolds or surfaces in the three-dimensional Euclidean space is a fundamental problem in differential geometry. When the Gauss curvature is negative, the isometric immersion problem is…

Motivated by the construction of Bach flat neutral signature Riemannian extensions, we study the space of parallel trace free tensors of type $(1,1)$ on an affine surface. It is shown that the existence of such a parallel tensor field is…

We define submersions f between manifolds M and N modelled on locally convex spaces. If the range N is finite-dimensional or a Banach manifold, then these coincide with the naive notion of a submersion. We study pre-images of submanifolds…

We construct new examples of embedded, complete minimal hypersurfaces in quaternionc hyperbolic space and also some minimal foliations. We introduce fans an construct analytic deformations of bisectors.

In this work we study the affine principal lines of surfaces in 3-space. We consider the binary differential equation of the affine curvature lines and obtain the topological models of these curves near the affine umbilic points (elliptic…

This article explains a program to study complete and properly embedded minimal surfaces in $\mathbb{R}^3$ developed jointly with W.H. Meeks and A. Ros in the last three decades. It follows closely the structure of my invited ICM talk with…

In (equi-)affine differential geometry, the most important algebraic invariants are the affine (Blaschke) metric h, the affine shape operator S and the difference tensor K. A hypersurface is said to admit a pointwise symmetry if at every…

In this paper, we study the problem of finding the affine factorable surfaces in a 3-dimensional isotropic space with prescribed Gaussian (K) and mean (H) curvature. Because the absolute figure two different types of these surfaces appear…

We study complex spatial quartic surfaces with simple singularities up to equisingular deformations; as a first step, give a complete equisingular deformation classification of the so-called non-special simple quartic surfaces.