Symplectic Packing in Dimension 4

We discuss closed symplectic 4-manifolds which admit full symplectic packings by N equal balls for large N's. We give a homological criterion for recognizing such manifolds. As a corollary we prove that can be fully packed by N equal balls for every . Submitted: June 1996, final version: October 1996     

Finiteness of the number of solutions of Plateau’s problem for polygonal boundary curves II

In this article, the author proves that a simple closed polygon can bound only finitely many immersed minimal surfaces of disc-type if it meets the following two requirements: firstly it has to bound only minimal surfaces without boundary branch points, and secondly its total curvature, i.e. the sum of the exterior angles at its N + 3 vertices, has to be smaller than 6π.      

Minimal shells containing a convex surface in Minkowski space

Summary LetK be the unit ball of a Minkowski space (finite dimensional Banach space). AK-shell is the closed set of all points between two concentric balls of the space. We consider different assignments of size to aK-shell and investigate theK-shells with minimum size which contain a given convex surface. Our results extend to Minkowski geometry classical results on minimal shells in Euclidean space. This article was processed by the author using the LATEX style file from Springer-Verlag.     

A General Existence Theorem for Embedded Minimal Surfaces with Free Boundary

In this paper, we prove a general existence theorem for properly embedded minimal surfaces with free boundary in any compact Riemannian 3‐manifold M with boundary ?M. These minimal surfaces are either disjoint from ?M or meet ?M orthogonally. The main feature of our result is that there is no assumptions on the curvature of M or convexity of ?M. We prove the boundary regularity of the minimal surfaces at their free boundaries. Furthermore, we define a topological invariant, the filling genus, for compact 3‐manifolds with boundary and show that we can bound the genus of the minimal surface constructed above in terms of the filling genus of the ambient manifold M. Our proof employs a variant of the min‐max construction used by Colding and De Lellis on closed embedded minimal surfaces, which were first developed by Almgren and Pitts.© 2014 Wiley Periodicals, Inc.     

A stability property of symplectic packing

We prove that for any closed symplectic 4-manifold (M,Ω) with [Ω]∈H ²(M, Q) there exists a number N ₀ such that for every N≥N ₀, (M,Ω) admits full symplectic packing by N equal balls. We also indicate how to compute this N ₀. Our approach is based on Donaldson's symplectic submanifold theorem and on tools from the framework of Taubes theory of Gromov invariants. Oblatum 9-I-1998 & 1-VII-1998 / Published online: 14 January 1999     

Occupancy with two types of balls

The classical occupancy problem is extended to the case where two types of balls are thrown. In particular, the probability that no urn contains both types of balls is studied. This is a birthday problem in two groups of boys and girls to consider the coincidence of a boy's and a girl's birthday. Let N ₁ and N ₂ denote the numbers of balls of each type thrown one by one when the first collision between the two types occurs in one of m urns. Then N ₁ N ₂/m is asymptotically exponentially distributed as m tends to infinity.This problem is related to the security evaluation of authentication procedures in electronic message communication.     

Minimal helix surfaces in <Emphasis Type="Italic">N</Emphasis><Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>×ℝ

An immersed surface M in N ⁿ ×ℝ is a helix if its tangent planes make constant angle with ∂ _t . We prove that a minimal helix surface M, of arbitrary codimension is flat. If the codimension is one, it is totally geodesic. If the sectional curvature of N is positive, a minimal helix surfaces in N ⁿ ×ℝ is not necessarily totally geodesic. When the sectional curvature of N is nonpositive, then M is totally geodesic. In particular, minimal helix surfaces in Euclidean n-space are planes. We also investigate the case when M has parallel mean curvature vector: A complete helix surface with parallel mean curvature vector in Euclidean n-space is a plane or a cylinder of revolution. Finally, we use Eikonal f functions to construct locally any helix surface. In particular every minimal one can be constructed taking f with zero Hessian.     

Cusps of Picard modular surfaces

We determine the number of cusps of minimal Picard modular surfaces. The proof also counts cusps of other Picard modular surfaces of arithmetic interest. Consequently, for each N > 0 there are finitely many commensurability classes of nonuniform arithmetic lattices in SU(2, 1) that contain an N-cusped surface. We also discuss a higher-rank analogue.     

Coverings of 3-manifolds by open balls and two open solid tori

Hempel and McMillan showed that a closed 3-manifold that can be covered by three open balls is a connected sum of S³- and S²-bundles over S¹. In this paper we obtain a classification of all closed 3-manifolds that can be covered by two open balls and one open solid torus or by one open ball and two open solid tori.     

Isoperimetric Inequalities for Extrinsic Balls in Minimal Submanifolds and their Applications

S.-Y. Cheng, P. Li and S.-T. Yau proved comparison theoremsfor the volume of extrinsic balls in minimal submanifolds ofspace forms. These results were extended by S. Markvorsen forminimal submanifolds of a riemannian manifold with just an upperbound on the sectional curvature. In the paper an isoperimetricinequality for extrinsic balls in minimal submanifolds of ariemannian manifold N with sectional curvatures bounded fromabove by a non-positive constant is found. As a corollary ofthis result an alternative proof is obtained of the comparisonfor the volume of extrinsic balls stated by the preceding authors,but now the equality case is characterized when the upper boundfor the sectional curvatures of the ambient manifold is strictlynegative. Finally, when the sectional curvatures of N are boundedfrom above for any constant (positive or negative), it is provedthat the -isoperimetric quotient of the extrinsic balls is boundedfrom below by the mean curvature of the geodesic spheres inthe m-dimensional real space forms.     

Current graphs and bi-embeddings

Let N(γ,γ′) denote the size of the smallest complete graph which cannot be edge-partitioned into two parts embeddable in closed orientable surfaces of genera γ,γ′ respectively. The theory of current graphs is used to determine the values of N(γ,γ′) in certain cases. Some related block designs are discussed.     

A Survey of the Coupon Collector’s Problem with Random Sample Sizes

This paper surveys the coupon collector’s waiting time problem with random sample sizes and equally likely balls. Consider an urn containing m red balls. For each draw, a random number of balls are removed from the urn. The group of removed balls is painted white and returned to the urn. Several approaches to addressing this problem are discussed, including a Markov chain approach to compute the distribution and expected value of the number of draws required for the urn to contain j white balls given that it currently contains i white balls. As a special case, E[N], the expected number of draws until all the balls are white given that all are currently red is also obtained.      

How many samples does it take to see all the balls in an urn?

Let an urn containN balls, numbered from 1 toN. A random number of balls are drawn without replacements from the urn, their numbers are noted and the balls are then returned to the urn. This is done repeatedly, the sample sizes being independent identically distributed. Letv be the number of samples needed to see all the balls. A simple approximation forEv and the asymptotic distribution ofv asN are obtained.Translated fromMatematicheskie Zametki, Vol. 64, No. 1, pp. 58–63, July, 1998.This research was supported by the Russian Foundation for Basic Research under grant No. 97-01-00387.     

Comparison of Exit Moment Spectra for Extrinsic Metric Balls

We prove explicit upper and lower bounds for the L ¹-moment spectra for the Brownian motion exit time from extrinsic metric balls of submanifolds P ^m in ambient Riemannian spaces N ⁿ . We assume that P and N both have controlled radial curvatures (mean curvature and sectional curvature, respectively) as viewed from a pole in N. The bounds for the exit moment spectra are given in terms of the corresponding spectra for geodesic metric balls in suitably warped product model spaces. The bounds are sharp in the sense that equalities are obtained in characteristic cases. As a corollary we also obtain new intrinsic comparison results for the exit time spectra for metric balls in the ambient manifolds N ⁿ themselves.     

Mazur Sets in Normed Spaces

In this paper we investigate different questions concerning Mazur sets in normed spaces, which point out the close connections between geometric functional analysis and discrete geometry. Motivated by a result of Chen and Lin, we study the relationship between Mazur disks and weak* denting points of the dual unit ball. We prove that the only Mazur sets of the spaces l₁ⁿ are points and closed balls. Finally, a new stability property for the family of all sets which are intersections of closed balls is found.     

Extrinsic isoperimetry and compactification of minimal surfaces in Euclidean and hyperbolic spaces

We study the topology of (properly) immersed complete minimal surfaces P ² in Hyperbolic and Euclidean spaces which have finite total extrinsic curvature, using some isoperimetric inequalities satisfied by the extrinsic balls in these surfaces (see [10]). We present an alternative and unified proof of the Chern-Osserman inequality satisfied by these minimal surfaces (in ? ⁿ and in ? ⁿ (b)), based in the isoperimetric analysis mentioned above. Finally, we show a Chern-Osserman-type equality attained by complete minimal surfaces in the Hyperbolic space with finite total extrinsic curvature.     

Branched immersions and braids

Branch points of a real 2-surface Σ in a 4-manifold M generalize branch points of complex curves in complex surfaces: for example, they can occur as singularities of minimal surfaces. We investigate such a branch point p when Σ is topologically embedded. It defines a link L(p), the components of which are closed braids with the same axis up to orientation. If Σ is closed without boundary, the contribution of p to the degree of the normal bundle of Σ in M can be computed on the link L(p), in terms of the algebraic crossing numbers of its components and of their linking numbers with one another.      

Mixed strategy solutions forN-person quadratic games

Mixed strategy -equilibrium points are given forN-person games with cost functions consisting of quadratic, bilinear, and linear terms and strategy spaces consisting of closed balls in Hilbert spaces. The results are applied to linear-quadratic differential games with no information and quadratic integral constraints on the control functions.This work was supported by a Commonwealth of Australia, Postgraduate Research Award.     

Finiteness of the number of ends of minimal submanifolds in euclidean space

We prove a version of the well-known Denjoy-Ahlfors theorem about the number of asymptotic values of an entire function for properly immersed minimal surfaces of arbitrary codimension in ℝ ^N . The finiteness of the number of ends is proved for minimal submanifolds with finite projective volume. We show, as a corollary, that a minimal surface of codimensionn meeting anyn-plane passing through the origin in at mostk points has no morec(n, N)k ends.     

Coalescing times for IID random variables with applications to population biology

We consider a coalescing particle model where particles move in discrete time. At each time period, each remaining ball is independently put in one of n bins according to a probability distribution p = (p₁, …, p_n), and all balls put into the same bin merge into a single ball. Starting with k balls, we are interested in the properties of E[N( p , k)], the expected time until all balls merge into one. We derive both upper and lower bounds for E[N( p , k)], some asymptotic results, and show that P{N( p , k) > t}, and thus E[N( p , k)], is a Schur concave function of p . Applications to population biology are noted. © 2003 Wiley Periodicals, Inc. Random Struct. Alg., 23: 155–166, 2003