Singularities of maximal surfaces

We show that the singularities of spacelike maximal surfaces in Lorentz–Minkowski 3-space generically consist of cuspidal edges, swallowtails and cuspidal cross caps. The same result holds for spacelike mean curvature one surfaces in de Sitter 3-space. To prove these, we shall give a simple criterion for a given singular point on a surface to be a cuspidal cross cap. Dedicated to Yusuke Sakane on the occasion of his 60th birthday.     

On maximal operators along surfaces

In this paper, we establish the boundedness of the following maximal operator

An efficient algorithm for maximal margin clustering

Maximal margin based frameworks have emerged as a powerful tool for supervised learning. The extension of these ideas to the unsupervised case, however, is problematic since the underlying optimization entails a discrete component. In this paper, we first study the computational complexity of maximal hard margin clustering and show that the hard margin clustering problem can be precisely solved in O(n ^d+2) time where n is the number of the data points and d is the dimensionality of the input data. However, since it is well known that many datasets commonly ‘express’ themselves primarily in far fewer dimensions, our interest is in evaluating if a careful use of dimensionality reduction can lead to practical and effective algorithms. We build upon these observations and propose a new algorithm that gradually increases the number of features used in the separation model in each iteration, and analyze the convergence properties of this scheme. We report on promising numerical experiments based on a ‘truncated’ version of this approach. Our experiments indicate that for a variety of datasets, good solutions equivalent to those from other existing techniques can be obtained in significantly less time.     

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).     

Complex analytic curves and maximal surfaces

Maximal immersions of a surfaceM ² inton-dimensional Lorentz space which are isometric to a fixed holomorphic mapping ofM ² into complex Lorentz space are determined. The main tool is an adaption of Calabi's Rigidity Theorem. Such an adaption is necessary because of the existence of degenerate hyperplanes in complex Lorentz space.Partially supported by a grant from Wellesley College.     

Congruence subgroups and maximal Riemann surfaces

A global maximal Riemann surface is a surface of constant curvature ?1 with the property that the length of its shortest simple closed geodesic is maximal with respect to all surfaces of the corresponding Teichmüller space. I show that the Riemann surfaces that correspond to the principal congruence subgroups of the modular group are global maximal surfaces. This result provides a strong geometrical reason that the Selberg conjecture, which says that these surfaces have no eigenvalues of the Laplacian in the open interval (0, 1/4), is true.     

Skew ray tracing and sensitivity analysis of ellipsoidal optical boundary surfaces

The 4 × 4 homogeneous transformation matrix is one of the most commonly applied mathematical tools in the fields of robotics, mechanisms and computer graphics. Here we extend further this mathematical tool to geometrical optics by addressing the following two topics: (1) skew ray tracing to determine the paths of reflected/refracted skew rays crossing ellipsoidal boundary surfaces; and (2) sensitivity analysis to determine via direct mathematical analysis the differential changes of the incident point and the reflected/refracted vector with respect to changes in the incident light source. The proposed ray tracing and sensitivity analysis are projected as the nucleus of other geometrical optical computations.     

Classification ofm-functions on surfaces

We establish a necessary and sufficient condition of conjugacy ofm=functions on surfaces. Institute of Mathematics, Ukrainian Academy of Sciences, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 8, pp. 1129–1135, August, 1999.     

A family of maximal surfaces in Lorentz-Minkowski three-space

We prove the existence of an infinite family of complete spacelike maximal surfaces with singularities in Lorentz-Minkowski three-space and study their properties. These surfaces are distinguished by their number of handles and have two elliptic catenoidal ends.

     

Parabolicity of maximal surfaces in Lorentzian product spaces

In this paper we establish some parabolicity criteria for maximal surfaces immersed into a Lorentzian product space of the form ${M^2 \times \mathbb {R}_1}$ , where M ² is a connected Riemannian surface with non-negative Gaussian curvature and ${M^2 \times \mathbb {R}_1}$ is endowed with the Lorentzian product metric ${{\langle , \rangle}={\langle , \rangle}_M-dt^2}$ . In particular, and as an application of our main result, we deduce that every maximal graph over a starlike domain ${\Omega \subseteq M}$ is parabolic. This allows us to give an alternative proof of the non-parametric version of the Calabi–Bernstein result for entire maximal graphs in ${M^2 \times \mathbb {R}_1}$ .     

Reimann surfaces with shortest geodesic of maximal length

I describe Riemann surfaces of constant curvature –1 with the property that the length of its shortest simple closed geodesic is maximal with respect to an open neighborhood in the corresponding Teichmüller space. I give examples of such surfaces. In particular, examples are presented which are modelled upon (Euclidean) polyhedra. This problem is a non-Euclidean analogue of the well known best lattice sphere packing problem.Supported by the Schweizerischer Nationalfonds zur Förderung wissenschaftlicher Forschung     

On maximal surfaces in the n-dimensional Lorentz-Minkowski space

Complete maximal surfaces in the n-dimensional Lorentz-Minkowski space are studied from the behaviour of their normal vectors. Moreover, several examples of maximal surfaces are constructed.Research partially supported by DGICYT Grant PS87-0115-C03-02.     

Discrete maximal surfaces with singularities in Minkowski space

We describe discrete maximal surfaces with singularities in 3-dimensional Minkowski space and give a Weierstrass type representation for them. In the smooth case, maximal surfaces (spacelike surfaces with mean curvature identically 0) in Minkowski 3-space generally have certain singularities. We give a criterion that naturally describes the “singular set” for discrete maximal surfaces, including a classification of the various types of singularities that are possible in the discrete case.     

Noncommutative cyclic covers and maximal orders on surfaces

In this paper, we construct various examples of maximal orders on surfaces, including some del Pezzo orders, some ruled orders and some numerically Calabi-Yau orders. The method of construction is a noncommutative version of the cyclic covering trick. These noncommutative cyclic covers are very computable and we give a formula for their ramification data. This often allows us to determine if a maximal order, described via ramification data, can be constructed as a noncommutative cyclic cover. The construction also has applications to Brauer-Severi varieties and, in the quaternion case, we show how to obtain some Brauer-Severi varieties from G-Hilbert schemes of P¹-bundles.