Topologically minimal surfaces versus self-amalgamated Heegaard surfaces

Let V ∪SW be a Heegaard splitting of M,such that αM = α-W = F1 ∪ F2 and g(S) = 2g(F1)= 2g(F2). Let V * ∪S*W * be the self-amalgamation of V ∪SW. We show if d(S) 3 then S* is not a topologically minimal surface.     

Tangential developable surfaces as bonnet surfaces

We study the real Bonnet surfaces which accept one unique nontrivial isometry that preserves the mean curvature, in the three-dimensional Euclidean space. We give a general criterion for these surfaces and use it to determine the tangential developable surfaces of this kind. They are determined implicitly by elliptic integrals of the third kind. Only the tangential developable surfaces of circular helices are explicit examples for which we completely determine the above unique nontrivial isometry. Dedication Dedicated to Siuping Ho for all her invaluable support and encouragement.     

Extensions of the duality between minimal surfaces and maximal surfaces

As a generalization of the classical duality between minimal graphs in E ³ and maximal graphs in L ³, we construct the duality between graphs of constant mean curvature H in Bianchi-Cartan-Vranceanu space E ³(κ, τ) and spacelike graphs of constant mean curvature τ in Lorentzian Bianchi-Cartan-Vranceanu space L ³(κ, H).     

On kth-order embeddings of K3 surfaces and Enriques surfaces

We give necessary and sufficient conditions for a big and nef line bundle L of any degree on a K3 surface or on an Enriques surface S to be k-very ample and k-spanned. Furthermore, we give necessary and sufficient conditions for a spanned and big line bundle on a K3 surface S to be birationally k-very ample and birationally k-spanned (our definition), and relate these concepts to the Clifford index and gonality of smooth curves in |L| and the existence of a particular type of rank 2 bundles on S. Received: 28 March 2000 / Revised version: 20 October 2000     

Porous surfaces

In fractal modeling, porous surfaces in the plane are usually described as the residual setE of a packing by connected open domains \(C_n\) . In the case whereE is nonempty, we investigate the relationships between the dimensionality ofE and the geometry of the complementary sets \(C_n\) . If they satisfy suitable regularity conditions, then the Bouligand dimension ofE is equal to the exponent of convergence of the series ∑(diam \(C_n\) ) ^α . We give here general conditions to obtain this equality, together with numerous examples and possible ways of developing this theory.     

Platonic surfaces

If S _O is a Riemann surface with a complete metric of finite area and constant curvature -1, let S _C denote the conformal compactification of S _O. We show that, under the assumption that the cusps of S _O are large, there is a close relationship between the hyperbolic metrics on S _O and S _C. We use this relationship to show that , where the Platonic surface P _k is the conformal compactification of the modular surface S _k. Received: November, 1996; revised: February, 1998     

Riemann surfaces

Since the classical work of Riemann, Plein, Chobe, and Poincaré, in mathematics the interest in the theory of Riemann surfaces and groups has not abated. The present survey covers papers reviewed in RZhMat during the period 1967–1976 primarily in the sections Algebra. Topology. Geometry. The following topics are considered most completely and thoroughly: the topology of Riemann surfaces and their automorphisms, Fuchsian groups, Teichmüller spaces, and spaces of moduli.Translated from Itogi Nauki i Tekhniki, Algebra, Topologiya, Geometriya, Vol. 16, pp. 191–245, 1978,     

Rational surfaces and moduli spaces of vector bundles on rational surfaces

Let X be a smooth algebraic surface, L ? Pic(X) L \in \textrm{Pic}(X) and H an ample divisor on X. Set M_X,H(2; L, c₂) the moduli space of rank 2, H-stable vector bundles F on X with det(F) = L and c₂(F) = c₂. In this paper, we show that the geometry of X and of M_X,H(2; L, c₂) are closely related. More precisely, we prove that for any ample divisor H on X and any L ? Pic(X) L \in \textrm{Pic}(X) , there exists n₀ ? \mathbbZ n_0 \in \mathbb{Z} such that for all n₀ \leqq c₂ ? \mathbbZ n_0 \leqq c_2 \in \mathbb{Z} , M_X,H(2; L, c₂) is rational if and only if X is rational.     

Embeddings of Danielewski surfaces

A Danielewski surface is defined by a polynomial of the form P=x ^nz –p(y). Define also the polynomial P =x ^nz –r(x)p(y) where r(x) is a non-constant polynomial of degree n–1 and r(0)=1. We show that, when n2 and deg p(y)2, the general fibers of P and P are not isomorphic as algebraic surfaces, but that the zero fibers are isomorphic. Consequently, for every non-special Danielewski surface S, there exist non-equivalent algebraic embeddings of S in ³. Using different methods, we also give non-equivalent embeddings of the surfaces xz=(y ^d _n >–1) for an infinite sequence of integers d _n . We then consider a certain algebraic action of the orthogonal group on ⁴ which was first considered by Schwarz and then studied by Masuda and Petrie, who proved that this action could not be linearized. This was done by comparing the strata of this action to those of the induced tangent space action. Inequivalent embeddings of a certain singular Danielewski surface S in ³ are found. We generalize their result and show how this leads to an example of two smooth algebraic hypersurfaces in ³ which are algebraically non-isomorphic but holomorphically isomorphic. Partially supported by NSF Grant DMS 0101836.     

The conformal structure of Riemann surfaces with boundary parametrizing minimal surfaces

Let F denote a surface with boundary F, being contained in a Riemann surface R, such that R\F is somedisk. If we vary the boiundary curve _o parametrizing F, we will get a manifold of real dimension 6g–3, such that any bounds some F and any local deformation of F is conformally equivalent to just one F for .This result also implies that none of the conformal invariants of R will be an invariant of this F, since its neighbors {F|} cover all possible deformations of F at all.     

Bi-Lipschitz parameterization of surfaces

We consider surfaces Z homeomorphic to the plane with complete, possibly singular Riemannian metrics. If we have _ZK⁺<2– for the positive and _ZK^–<C for the negative part of the integral curvature, then Z is L-bi-Lipschitz equivalent to ² with L depending only on >0 and C>0. This result implies a conjecture by J. Fu.Supported by NSF grant DMS-0200566.     

Immersions of non-orientable surfaces

Let F be a closed non-orientable surface. We classify all finite order invariants of immersions of F into R³, with values in any Abelian group. We show they are all functions of a universal order 1 invariant which we construct as T⊕P⊕Q, where T is a Z valued invariant reflecting the number of triple points of the immersion, and P,Q are Z/2 valued invariants characterized by the property that for any regularly homotopic immersions , P(i)−P(j)∈Z/2 (respectively, Q(i)−Q(j)∈Z/2) is the number mod 2 of tangency points (respectively, quadruple points) occurring in any generic regular homotopy between i and j.For immersion and diffeomorphism such that i and i○h are regularly homotopic we show:

P(i○h)−P(i)=Q(i○h)−Q(i)=(rank(h_∗−Id)+ε(deth∗∗))mod2     

Closed surfaces

It is established that closed surfaces of genus zero do not permit non-trivial deformation with retention of the mean curvature. There are four items in the Literature Cited.Translated from Matematicheskie Zametki, Vol. 5, No. 6, pp. 719–721, June, 1969.     

Unirationality of certain surfaces

Nonsingular cubic surfaces in P³ and nonsingular intersections of two quadrics in P⁴ are investigated. It is proved that if a k-point exists on a surface, there is a k-point not on a line; k is the field over which the surfaces are defined.Translated from Matematicheskie Zametki, Vol. 5, No. 2, pp. 155–159, February, 1969.The author expresses his gratitude to Yu. I. Manin, under whose direction this work has been written.     

Differential invariants of surfaces

The algebra of differential invariants of a suitably generic surface S⊂R³, under either the usual Euclidean or equi-affine group actions, is shown to be generated, through invariant differentiation, by a single differential invariant. For Euclidean surfaces, the generating invariant is the mean curvature, and, as a consequence, the Gauss curvature can be expressed as an explicit rational function of the invariant derivatives, with respect to the Frenet frame, of the mean curvature. For equi-affine surfaces, the generating invariant is the third order Pick invariant. The proofs are based on the new, equivariant approach to the method of moving frames.     

Non-classical Godeaux surfaces

A non-classical Godeaux surface is a minimal surface of general type with χ = K ² = 1 but with h ⁰¹ ≠ 0. We prove that such surfaces fulfill h ⁰¹ = 1 and they can exist only over fields of positive characteristic at most 5. Like non-classical Enriques surfaces they fall into two classes: the singular and the supersingular ones. We give a complete classification in characteristic 5 and compute their Hodge-, Hodge–Witt- and crystalline cohomology (including torsion). Finally, we give an example of a supersingular Godeaux surface in characteristic 5.     

Cylindrical surfaces of delaunay

For the cylindrical norm onR ³, for which the isoperimetric shape is a cylinder rather than a round ball, there are analogs of the classical Delaunay surfaces of revolution of constant mean curvature.     

Essential closed surfaces in surface sum of product I-bundle of closed surfaces

In this paper, we will give some results on the classification of essential closed surfaces in the surface sum of product I-bundle of closed surfaces and some applications of these results.