Self-intersections of curves on surfaces

Let M be a compact orientable surface with nonempty boundary (x(M)<0) and fundamental group . Let be a geodesic on M (with a fixed hyperbolic structure), and let W be a (cyclically reduced) word in a fixed set of generators of which represents . In this paper, we give an algorithm to count the number of self-intersections of in terms of W, generalizing a result of Birman and Series, where an algorithm was given to decide if was simple. Some applications of the algorithm to surfaces with one boundary and the Markoff spectrum are also given.     

Invariant smoothness measures for surfaces

Two novel smoothness measures for surfaces are presented in this paper. The second and third order smoothness are defined as the squared normal curvature and the squared variation in normal curvature integrated over all directions in the tangent plane. Both quantities are truly geometric in the sense that they are invariant with respect to the actual parametrization of the surface. All the same, all formulae are derived in terms of an arbitrary parametrization. In addition to providing a basis for variational surface construction, the second and third order smoothness can also be used for evaluation and assessment of the quality of an existing surface. This revised version was published online in August 2006 with corrections to the Cover Date.     

A Problem of Blocking Light Rays

Let K be an n-dimensional convex body with interior int(K). This article mainly deals with the following problem: How many nonoverlapping translates of K (or int(K)) are enough to block all the light rays (straight lines) starting from K?     

Degeneracy of holomorphic curves in surfaces

Let X be a complex projective algebraic manifold of dimension 2 and let D1,…,Du be distinct irreducible divisors on X such that no three of them share a common point. Let f: C→X\(U1≤i≤uDi) be a holomorphic map. Assume that u≥4 and that there exist positive integers n1,…,nu, c such that ninj(Di.Dj) = c for all pairs i, j. Then f is algebraically degenerate, i.e. its image is contained in an algebraic curve on X.     

Flocking and collision avoidance of a Cucker-Smale type system with singular weights

The dynamical behavior of a flock model with a singular communication rate and extra interaction terms is investigated in this paper. A rigorous theoretical proof of collision avoidance between any two agents is obtained which guarantees the existence of global solutions. Moreover, a sufficient condition for the existence of time-asymptotic flocking is also acquired and numerical simulations verified these results which show that a compact equilibrium configuration may emerge.     

A note on general blow-ups of abelian surfaces

In the paper we give numerical conditions for a line bundle on a general blow-up of abelian surface to be k-very ample.     

Mixed modulus of smoothness with Muckenhoupt weights and approximation by angle

Mixed modulus of smoothness in weighted Lebesgue spaces with Muckenhoupt weights are investigated. Using mixed modulus of smoothness we obtain Potapov type direct and inverse estimates of angular trigonometric approximation of functions in these spaces. Also we obtain equivalences between mixed modulus of smoothness and K-functional and realization functional. Fractional order modulus of smoothness is considered as well.     

A martingale approach to minimal surfaces

We provide a probabilistic approach to studying minimal surfaces in R³. After a discussion of the basic relationship between Brownian motion on a surface and minimality of the surface, we introduce a way of coupling Brownian motions on two minimal surfaces. This coupling is then used to study two classes of results in minimal surface theory, maximum principle-type results, such as weak and strong halfspace theorems and the maximum principle at infinity, and Liouville theorems.     

Embeddings of general blowups of abelian surfaces

Blowups of algebraic surfaces polarized by tensor powers of ample line bundles were studied by Coppens [8]. In this note we complete the picture in the case of abelian surfaces studying blowups of surfaces polarized by primitive line bundles. Received: 16 November 1999 / Revised version: 13 June 2000     

Godeaux,Campedelli, and surfaces of general type with and

We construct simply connected surfaces of general type with invariants and . We use ‐Gorenstein deformations in conjunction with explicit constructions that express the canonical rings by generators and relations. The canonical rings of the surfaces are described as projections. The whole construction is simplified by the use of key varieties based on Steiner 3‐folds. As a consequence of the construction we find two families, each family in a different connected component of the moduli stack , and each linking a Campedelli surface with a Godeaux surface.     

Quasi-isometries of groups, graphs and surfaces

We give a characterization of virtual surface groups as groups quasi-isometric to complete simply-connected Riemannian surfaces. Results on the equivalence up to quasi-isometry of various bounded geometry conditions for Riemannian surfaces are also obtained. Received: January 18, 2000     

On a non-abelian invariant on complex surfaces of general type

In this paper, we give certain homotopy and diffeomorphism versions as a generalization to an earlier result due to W.S. Cheung, Bun Wong and Stephen S. T. Yau concerning a local rigidity problem of the tangent bundle over compact surfaces of general type.     

Embeddable anticonformal automorphisms of Riemann surfaces

Let S be a Riemann surface and f be an automorphism of finite order of S. We call f embeddable if there is a conformal embedding such that is the restriction to e(S) of a rigid motion. In this paper we show that an anticonformal automorphism of finite order is embeddable if and only if it belongs to one of the topological conjugation classes here described. For conformal automorphisms a similar result was known by R.A. Rüedy [R3]. Received: February 8, 1996     

Splitting and cone avoidance in the D.C.E. degrees

It is shown that the Cooper splitting theorem for the n-c.e. degrees is not compatible with cone avoidance: For any n > 1, there exist n-c.e. degree a, c.e. degree b such that 0 < b < a and such that for any n-c.e. degrees x,y, if x ∨ y = a, then either b ≤ x or b ≤ y. This provides a new type of elementary difference between the classes of c.e. and d.c.e. degrees, implementable at lower levels of the high/low hierarchy.     

Cubic surfaces and Borcherds products

We apply Borcherds' methods for constructing automorphic forms to embed the moduli space of marked complex cubic surfaces into . Specifically, we construct 270 automorphic forms on the complex 4-ball , automorphic with respect to a particular discrete group . We use the identification from [ACT2] of with the Baily-Borel compactification of . Our forms span a 10-dimensional space, and we exhibit the image of in as the intersection of 270 cubic hypersurfaces. Finally, we interpret the pairwise ratios of our forms as the original invariants of cubic surfaces, the cross-ratios introduced by Cayley. It turns out that this model of was found by Coble [C] in an entirely different way; see [vG]. Received: December 5, 2000     

On a non-abelian invariant on complex surfaces of general type Dedicated to Professor Sheng GONG on the occasion of his 75th birthday

In this paper, we give certain homotopy and diffeomorphism versions as a generalization to an earlier result due to W.S. Cheung, Bun Wong and Stephen S. T. Yau concerning a local rigidity problem of the tangent bundle over compact surfaces of general type.     

Nonexistence of certain complete minimal surfaces with planar ends

In this paper we prove that two lines bounding an immersed minimal surface in a slab in R ³ homeomorphic to a compact Riemann surface minus two disks and a finite number of points must be parallel. This theorem is extended to a higher dimensional minimal hypersurface. Also it is proved that if the Gauss map of a complete embedded minimal surface of finite total curvature at a planar end has order two, then the intersection of the surface with the plane asymptotic to the planar end cannot admit a one-to-one orthogonal projection onto any line in the plane. Received: November 26, 1998     

Complexity of Chow varieties and number of morphisms on surfaces of general type

We provide a general estimate for the number of irreducible components of a Chow variety, the variety that parametrizes algebraic cycles of given dimension and degree contained in a projective variety. The result is then applied to obtain an upper bound for the finite number of surfaces of general type that are images of a fixed surface. Received: 29 January 1998 / Revised version: 24 June 1998     

Closed incompressible surfaces in the complements of positive knots

We show that any closed incompressible surface in the complement of a positive knot is algebraically non-split from the knot, positive knots cannot bound non-free incompressible Seifert surfaces and that the splittability and the primeness of positive knots and links can be seen from their positive diagrams. Received: June 28, 2000