Semi-Discrete Isothermic Surfaces

We study mappings of the form ${x : \mathbb{Z}\times\mathbb{R}\to\mathbb{R}^3}$ which can be seen as a limit case of purely discrete surfaces, or as a semi-discretization of smooth surfaces. In particular we discuss circular surfaces, isothermic surfaces, conformal mappings, and dualizability in the sense of Christoffel. We arrive at a semi-discrete version of Koenigs nets and show that in the setting of circular surfaces, isothermicity is the same as dualizability. We show that minimal surfaces constructed as a dual of a sphere have vanishing mean curvature in a certain well-defined sense, and we also give an incidence-geometric characterization of isothermic surfaces.     

Bonnet Surfaces and Isothermic Surfaces

By the study of the relations between Bonnet surfaces and isothermic surfaces, we obtain classification results of Bonnet surfaces in 3-dimensional space form R³(c) and of the spacelike Bonnet surfaces in indefinite space form R₁ ³(c), which generalize the results in Bobenko’s [1] and Peng-Lu’s [11]. It is remarkable that there exist always Bonnet surfaces which are not Weingarten surfaces, if the ambient space is not R³(c) for c ≥ 0.     

Isothermic constrained Willmore tori in 3-space

Annals of Global Analysis and Geometry - We show that the homogeneous and the 2-lobe Delaunay tori in the 3-sphere provide the only isothermic constrained Willmore tori in 3-space with Willmore...     

Minimal Submanifolds

This paper provides a survey of both old and new results aboutminimal surfaces and submanifolds. 2000 Mathematics SubjectClassification 54C40, 14E20 (primary), 46E25, 20C20 (secondary).     

Generic Submanifolds

Summary In this paper we give some examples of generic submanifolds of complex space forms and prove some theorems which give the characterizations of these examples. For this purpose we study the relations between a submanifold of a Kählerian manifold and a submanifold of a Sasakian manifold by using the method of Riemannian fibre bundles.     

Diametrical Submanifolds

A submanifold M of a Euclidean space is called diametrical with respect to a centre p if it admits a tangent preserving diffeomorphism such that the chords connecting the points on M with their images pass through p. Characterisations are given for the obvious situation, when M is reflectionally symmetric with respect to p and when M is spherical in addition to this. Moreover, non-obvious examples are obtained and the structure of diametrical submanifolds is investigated in more general cases.     

Submanifolds associated with graphs

In this paper we present an interesting relationship between graph theory and differential geometry by defining submanifolds of almost Hermitian manifolds associated with certain kinds of graphs. We show some results about the possibility of a graph being associated with a submanifold and we use them to characterize CR-submanifolds by means of trees. Finally, we characterize submanifolds associated with graphs in a four-dimensional almost Hermitian manifold.

     

子流形刚性定理

该文改进了Ｓｈｉｏｈａｍａ和Ｘｕ的常曲率空间中子流形的刚性定理,得到了一个新的整体拼级常数犆狀,并且犆狀只与子流形的维数有关,而与其余维数无关．     

Submanifolds of restricted type

A submanifoldM ⁿ of the Euclidean space ^m is said to be of restricted type if the shape operator of the mean curvature vector is the tangent part of a fixed linear transformation of ^m. We show that a hypersurface of restricted type is either minimal, a part of the product of a sphere and a linear subspace or a cylinder on a plane curve of restricted type. Finally we classify plane curves of restricted type.Supported by a research fellowship of the Research Council of the Katholieke Universiteit LeuvenResearch Assistant of the National Fund for Scientific Research (Belgium)     

Finite Type Non—Minimal Submanifolds

The notion of finite type xubmanifolds was introduced by B.Y.Chen.In this paper we consider the characteristics and the classifications of finite type non-minimal submanifolds.The characteristic theorems of 2-type Chen submanifolds、mass-symmetric hypersurfaces and Dupin hypersurfaces in Es^m are obtained.The classification theorems of 3-type hypersurfaces and null 2-type curves in Es^m are also proved.     

Brownian Bridges to Submanifolds

We introduce and study Brownian bridges to submanifolds. Our method involves proving a general formula for the integral over a submanifold of the minimal heat kernel on a complete Riemannian manifold. We use the formula to derive lower bounds, an asymptotic relation and derivative estimates. We also see a connection to hypersurface local time. This work is motivated by the desire to extend the analysis of path and loop spaces to measures on paths which terminate on a submanifold.     

Isothermic and S-Willmore Surfaces as Solutions to a Problem of Blaschke

We consider the generalization of a classical problem of Blaschke to the higher codimensional case, characterizing Darboux pairs of isothermic surfaces and dual S-Willmore surfaces as the only non-trivial surface pairs that envelop a 2-sphere congruence and conformally correspond to each other. When the sphere congruence consists of the mean curvature spheres of one enveloping surface f, f must be a CMC-1 surface in hyperbolic 3-space, or an S-Willmore surface.     

Universal Models For Real Submanifolds

In previous papers by the present author, a machinery for calculating automorphisms, constructing invariants, and classifying real submanifolds of a complex manifold was developed. The main step in this machinery is the construction of a “nice” model surface. The nice model surface can be treated as an analog of the osculating paraboloid in classical differential geometry. Model surfaces suggested earlier possess a complete list of the desired properties only if some upper estimate for the codimension of the submanifold is satisfied. If this estimate fails, then the surfaces lose the universality property (that is, the ability to touch any germ in an appropriate way), which restricts their applicability. In the present paper, we get rid of this restriction: for an arbitrary type (n,K) (where n is the dimension of the complex tangent plane, and K is the real codimension), we construct a nice model surface. In particular, we solve the problem of constructing a nondegenerate germ of a real analytic submanifold of a complex manifold of arbitrary given type (n,K) with the richest possible group of holomorphic automorphisms in the given class.     

Submanifolds in Cauchy Riemann Geometry

In this paper I would like to make a report on the results about hypersurfaces in the Heisenberg group and invariant curves and surfaces in CR geometry. The results are contained in the papers [8, 9, 16] and [14]. Besides, I would also report on the results about the strong maximum principle for a class of mean curvature type operators in [10].