Minimal surfaces,Hopf differentials and the Ricci condition

We deal with minimal surfaces in a sphere and investigate certain invariants of geometric significance, the Hopf differentials, which are defined in terms of the complex structure and the higher fundamental forms. We discuss the holomorphicity of Hopf differentials and provide a geometric interpretation for it in terms of the higher curvature ellipses. This motivates the study of a class of minimal surfaces, which we call exceptional. We show that exceptional minimal surfaces are related to Lawson’s conjecture regarding the Ricci condition. Indeed, we prove that, under certain conditions, compact minimal surfaces in spheres which satisfy the Ricci condition are exceptional. Thus, under these conditions, the proof of Lawson’s conjecture is reduced to its confirmation for exceptional minimal surfaces. In fact, we provide an affirmative answer to Lawson’s conjecture for exceptional minimal surfaces in odd dimensional spheres or in S ^4m .     

Rotational linear Weingarten surfaces of hyperbolic type

A linear Weingarten surface in Euclidean space ℝ³ is a surface whose mean curvature H and Gaussian curvature K satisfy a relation of the form aH + bK = c, where a, b, c ∈ ℝ. Such a surface is said to be hyperbolic when a ² + 4bc < 0. In this paper we study rotational linear Weingarten surfaces of hyperbolic type giving a classification under suitable hypothesis. As a consequence, we obtain a family of complete hyperbolic linear Weingarten surfaces in ℝ³ that consists of surfaces with self-intersections whose generating curves are periodic. Partially supported by MEC-FEDER grant no. MTM2007-61775.     

Automorphisms of Maps with a Given Underlying Graph and Their Application to Enumeration

A graph is called a semi-regular graph if its automorphism group action on its ordered pair of adjacent vertices is semi-regular. In this paper, a necessary and sufficient condition for an automorphism of the graph F to be an automorphism of a map with the underlying graph F is obtained. Using this result, all orientation-preserving automorphisms of maps on surfaces (orientable and non-orientable) or just orientable surfaces with a given underlying semi-regular graph F are determined. Formulas for the numbers of non-equivalent embeddings of this kind of graphs on surfaces (orientable, non-orientable or both) are established, and especially, the non-equivalent embeddings of circulant graphs of a prime order on orientable, non-orientable and general surfaces are enumerated.     

Welschinger invariants of toric Del Pezzo surfaces with nonstandard real structures

The Welschinger invariants of real rational algebraic surfaces are natural analogs of the Gromov-Witten invariants, and they estimate from below the number of real rational curves passing through prescribed configurations of points. We establish a tropical formula for the Welschinger invariants of four toric Del Pezzo surfaces equipped with a nonstandard real structure. Such a formula for real toric Del Pezzo surfaces with a standard real structure (i.e., naturally compatible with the toric structure) was established by Mikhalkin and the author. As a consequence we prove that for any real ample divisor D on a surface Σ under consideration, through any generic configuration of c ₁(Σ)D − 1 generic real points, there passes a real rational curve belonging to the linear system |D|. To Vladimir Igorevich Arnold on the occasion of his 70th birthday     

Higher Order Parallel Surfaces in Bianchi–Cartan–Vranceanu Spaces

We give a full classification of higher order parallel surfaces in three-dimensional homogeneous spaces with four-dimensional isometry group, i.e., in the so-called Bianchi–Cartan–Vranceanu family. This gives a positive answer to a conjecture formulated in [2]. As a partial result, we prove that totally umbilical surfaces only exist if the ambient Bianchi–Cartan–Vranceanu space is a Riemannian product of a surface of constant Gaussian curvature and the real line, and we give a local parametrization of all totally umbilical surfaces. Received: December 20, 2006. Revised: March 15, 2007.     

Laguerre Geometry of Surfaces in R^3

Let f ： M → R3 be an oriented surface with non-degenerate second fundamental form. We denote by H and K its mean curvature and Gauss curvature. Then the Laguerre volume of f, defined by L（f） = f（H2 - K）/KdM, is an invariant under the Laguerre transformations. The critical surfaces of the functional L are called Laguerre minimal surfaces. In this paper we study the Laguerre minimal surfaces in R^3 by using the Laguerre Gauss map. It is known that a generic Laguerre minimal surface has a dual Laguerre minimal surface with the same Gauss map. In this paper we show that any surface which is not Laguerre minimal is uniquely determined by its Laguerre Gauss map. We show also that round spheres are the only compact Laguerre minimal surfaces in R^3. And we give a classification theorem of surfaces in R^3 with vanishing Laguerre form.     

A Bishop surface with a vanishing Bishop invariant

We give a solution to the equivalence problem for Bishop surfaces with the Bishop invariant λ=0. As a consequence, we answer, in the negative, a problem that Moser asked in 1985 after his work with Webster in 1983 and his own work in 1985. This will be done in two major steps: We first derive the formal normal form for such surfaces. We then show that two real analytic Bishop surfaces with λ=0 are holomorphically equivalent if and only if they have the same formal normal form (up to a trivial rotation). Our normal form is constructed by an induction procedure through a completely new weighting system from what is used in the literature. Our convergence proof is done through a new hyperbolic geometry associated with the surface. As an immediate consequence of the work in this paper, we will see that the modular space of Bishop surfaces with the Bishop invariant vanishing and with the Moser invariant s<∞ is of infinite dimension. This phenomenon is strikingly different from the celebrated theory of Moser–Webster for elliptic Bishop surfaces with non-vanishing Bishop invariants where the surfaces only have two and one half invariants. Notice also that there are many real analytic hyperbolic Bishop surfaces, which have the same Moser–Webster formal normal form but are not holomorphically equivalent to each other as shown by Moser–Webster and Gong. Hence, Bishop surfaces with the Bishop invariant λ=0 behave very differently from hyperbolic Bishop surfaces and elliptic Bishop surfaces with non-vanishing Bishop invariants.     

On injective homomorphisms between Teichmüller modular groups I

In this paper and its sequel, we prove that injective homomorphisms between Teichmüller modular groups of compact orientable surfaces are necessarily isomorphisms, if an appropriately measured “size” of the surfaces in question differs by at most one. In particular, we establish the co-Hopfian property for modular groups of surfaces of positive genus. Oblatum 24-III-1995 & 22-X-97 / Published online: 14 October 1998     

A generalization of Steiner symmetrization for immersed surfaces and its applications

We generalize the classical Steiner symmetrization to surfaces with self-intersections. Then we apply the generalized Steiner symmetrization to several isoperimetric problems. For example, let Г⊂ℝ³ be an analytic plane Jordan curve which is symmetric with respect to a plane ϖ (ϖ⊅Г). LetS be a compact immersed surface bounded by Λ which has the smallest area among all compact surfaces bounded by Λ with a fixed volume. In this situation, under some additional assumptions, the wholeS is proved to be symmetric with respect to ϖ. When Λ is a round circle,S is proved to be a spherical cap or the flat disk bounded by Λ without any additional assumptions. Dedicated to Professor Hideki Ozeki on his 60th birthday This article was processed by the author using the Springer-Verlag T_EX P Jourlg macro package 1991.     

Complete classification of parallel Lorentz surfaces in neutral pseudo hyperbolic 4-space

A Lorentz surface of an indefinite space form is called a parallel surface if its second fundamental form is parallel with respect to the Van der Waerden-Bortolotti connection. Such surfaces are locally invariant under the reflection with respect to the normal space at each point. Parallel surfaces are important in geometry as well as in general relativity since extrinsic invariants of such surfaces do not change from point to point. Recently, parallel Lorentz surfaces in 4D neutral pseudo Euclidean 4-space $ \mathbb{E}_2^4 $ \mathbb{E}_2^4 and in neutral pseudo 4-sphere S ₂⁴ (1) were classified in [14] and in [10], respectively. In this paper, we completely classify parallel Lorentz surfaces in neutral pseudo hyperbolic 4-space H ₂⁴ (−1). Our main result states that there are 53 families of parallel Lorentz surfaces in H ₂⁴ (−1). Conversely, every parallel Lorentz surface in H ₂⁴ (−1) is obtained from the 53 families. As an immediate by-product, we achieve the complete classification of all parallel Lorentz surfaces in 4D neutral indefinite space forms.     

The mixed problem for a transtropic cylinder under local heating of the surfaces

By the method of homogeneous solutions we study the stressed state of a transversally isotropic cylinder under local heating of the lateral surfaces. We describe the results of numerical studies as functions of the elastic properties of the material, the geometric characteristics, and the variation of the heating zone on one of the lateral surfaces of the cylinder. Five figures, 3 tables. Bibliography: 3 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, Vol. 27, 1997, pages 3–10.     

Asymmetric and pseudo-symmetric hyperelliptic surfaces

 In this paper we examine hyperelliptic Riemann surfaces which possess an anticonformal automorphism but are not symmetric. We determine that all such surfaces must have a full automorphism group which is either cyclic, or an abelian extension of a cyclic group by ℤ₂. We give defining equations for all of these hyperelliptic surfaces and show how they can be constructed by using NEC groups. As a special case, we determine all hyperelliptic surfaces which are pseudo-symmetric but not symmetric. Received: 15 November 2001     

Complete algebraic vector fields on affine surfaces,Part I

In this work we study the singularities at infinity of algebraic vector fields in dimension 2.These singularities will be classified under a mild assumption. The general problem is also reduced to the study of the combinatorics of certain resolutions which will be developed in Part II. Our main results are local and therefore can be carried over more general surfaces. Whereas we deal with ℂ -complete vector fields, the results also apply to ℝ-complete ones thanks to a theorem of Forstneric [7].     

Pants Decompositions of Random Surfaces

Our goal is to show, in two different contexts, that “random” surfaces have large pants decompositions. First we show that there are hyperbolic surfaces of genus g for which any pants decomposition requires curves of total length at least g ^7/6−ε . Moreover, we prove that this bound holds for most metrics in the moduli space of hyperbolic metrics equipped with the Weil–Petersson volume form. We then consider surfaces obtained by randomly gluing euclidean triangles (with unit side length) together and show that these surfaces have the same property.     

A regularity theorem for deformations of a compact complex surface

Let ƒ:M →D ⊑C ⁿ be a holomorphic family of compact, complex surfaces, which is locally trivial onD∖Z, for an analytic subsetZ. Conditions are found under which ƒ extends trivially toD, if the fibers of ƒ|_D∖Z are either Hirzebruch surfaces (projective bundles overP ₁), Hopf surfaces (elliptic bundles overP ₁), hyperelliptic bundles, or any compact complex surface having one of these as minimal model under blowing-down. The results of this paper are motivated by the existence of non-Hausdorff moduli spaces in the deformation of complex structure for certain complex manifolds.     

A Discrete Laplace–Beltrami Operator for Simplicial Surfaces

We define a discrete Laplace–Beltrami operator for simplicial surfaces (Definition 16). It depends only on the intrinsic geometry of the surface and its edge weights are positive. Our Laplace operator is similar to the well known finite-elements Laplacian (the so called “cotan formula”) except that it is based on the intrinsic Delaunay triangulation of the simplicial surface. This leads to new definitions of discrete harmonic functions, discrete mean curvature, and discrete minimal surfaces. The definition of the discrete Laplace–Beltrami operator depends on the existence and uniqueness of Delaunay tessellations in piecewise flat surfaces. While the existence is known, we prove the uniqueness. Using Rippa’s Theorem we show that, as claimed, Musin’s harmonic index provides an optimality criterion for Delaunay triangulations, and this can be used to prove that the edge flipping algorithm terminates also in the setting of piecewise flat surfaces. Research for this article was supported by the DFG Research Unit 565 “Polyhedral Surfaces” and the DFG Research Center Matheon “Mathematics for key technologies” in Berlin.     

The Hadamard Theorem and Borel–Carathéodory Theorem on Riemann Surfaces

In this paper, we first define a kind of pseudo–distance function and annulus domain on Riemann surfaces, then prove the Hadamard Theorem and the Borel–Carathéodory Theorem on any Riemann surfaces. Supported by NSFC 10501052     

Constant angle surfaces in the Heisenberg group

In this article we extend the notion of constant angle surfaces in $ \mathbb{S}^2 $ \mathbb{S}^2 × ℝ and ℍ² × ℝ to general Bianchi-Cartan-Vranceanu spaces. We show that these surfaces have constant Gaussian curvature and we give a complete local classification in the Heisenberg group.     

General blowups of ruled surfaces

In this note we study blowups of algebraic surfaces of Kodaira dimension κ = - ∞ at general points, their embeddings and secant varieties of the embedded surfaces.     

Crystallography and Riemann surfaces

The level set of an elliptic function is a doubly periodic point set in ℂ. To obtain a wider spectrum of point sets, we consider, more generally, a Riemann surface S immersed in ℂ² and its sections (“cuts”) by ℂ More specifically, we consider surfaces S defined in terms of a fundamental surface element obtained as a conformai map of triangular domains in ℂ. The discrete group of isometries of ℂ² generated by reflections in the triangle edges leaves S invariant and generalizes double-periodicity. Our main result concerns the special case of maps of right triangles, with the right angle being a regular point of the map. For this class of maps we show that only seven Riemann surfaces, when cut, form point sets that are discrete in ℂ. Their isometry groups all have a rank 4 lattice subgroup, but only three of the corresponding point sets are doubly periodic in ℂ. The remaining surfaces form quasiperiodic point sets closely related to the vertex sets of quasiperiodic tilings. In fact, vertex sets of familiar tilings are recovered in all cases by applying the construction to a piecewise flat approximation of the corresponding Riemann surface. The geometry of point sets formed by cuts of Riemann surfaces is no less “rigid” than the geometry determined by a tiling, and has the distinct advantage in having a regular behavior with respect to the complex parameter which specifies the cut.