Lattice Points in Three-Dimensional Convex Bodies with Points of Gaussian Curvature Zero at the Boundary

 In this article we investigate the number of lattice points in a three-dimensional convex body which contains non-isolated points with Gaussian curvature zero but a finite number of flat points at the boundary. Especially, in case of rational tangential planes in these points we investigate not only the influence of the flat points but also of the other points with Gaussian curvature zero on the estimation of the lattice rest. Received 19 June 2001; in revised form 17 January 2002 RID="a" ID="a" Dedicated to Professor Edmund Hlawka on the occasion of his 85th birthday     

Lattice Points in Planar Convex Domains

We investigate the number of lattice points in planar convex domains. We give estimates of the remainder in the asymptotic representation with numerical constants, which are astonishingly small. We consider convex planar domains whose boundary has nonvanishing curvature throughout. Here the curvature of the curve of boundary plays an important role. Further, we consider the number of lattice points in domains which are bounded by superellipses. These curves have isolated points with curvature zero.     

Lattice Points in Convex Bodies with Planar Points on the Boundary

 An asymptotic formula is proved for the number of lattice points in large threedimensional convex bodies. In contrast to the usual assumption the Gaussian curvature of the boundary may vanish at non-isolated points. It is only assumed that the second fundamental form vanishes at isolated points where the tangent plane is rational and some ellipticity condition holds.     

Lattice Points in Convex Bodies with Planar Points on the Boundary

 An asymptotic formula is proved for the number of lattice points in large threedimensional convex bodies. In contrast to the usual assumption the Gaussian curvature of the boundary may vanish at non-isolated points. It is only assumed that the second fundamental form vanishes at isolated points where the tangent plane is rational and some ellipticity condition holds. Received 25 April 2001     

The lattice discrepancy of certain three-dimensional bodies

Based on a very precise approximation to the lattice discrepancy of a Lamé disc, an asymptotic formula is established for the number of lattice points in the three-dimensional body $$|u_1|^{mk}+\left(|u_2|^k+|u_3|^k\right)^{m}\le x^{mk},$$ for large real x and fixed reals m, k. Particular attention is paid to the boundary points of Gaussian curvature zero.     

Bergman interpolation on finite Riemann surfaces. Part I: asymptotically flat case

The article considers the Bergman space interpolation problem on open Riemann surfaces obtained from a compact Riemann surface by removing a finite number of points. Such a surface is equipped with what we call an asymptotically flat conformal metric, i.e., a complete metric with zero curvature outside a compact subset. Sufficient conditions for interpolation in weighted Bergman spaces over asymptotically flat Riemann surfaces are then established. When the weights have curvature that is quasi-isometric to the asymptotically flat boundary metric, these sufficient conditions are shown to be necessary, unless the surface has at least one cylindrical end, in which case, the necessary conditions are slightly weaker than the sufficient conditions.     

Boundaries of Flat Compact Surfaces in 3-Space

This paper deals with the problem `which knots or links in3-space bound flat (immersed) compact surfaces?' In aforthcoming paper by the author, it is proven that any simple closedspace curve can be deformed until it bounds a flat orientable compact(Seifert) surface. The main results of this paper are that there existknots that do not bound any flat compact surfaces. The lower bound oftotal curvature of a knot bounding an orientable nonnegatively curvedcompact surface can, for varying knot types, be arbitrarily much greaterthan the infimum of curvature needed for the knot to have its knot type.The number of 3-singular points (points of zero curvatureor if not then of zero torsion) on the boundary of a flat immersedcompact surface is greater than or equal to twice the absolute value ofthe Euler characteristic of the surface. A set of necessary and, in aweakened sense, sufficient conditions for a knot or link to be what wecall a generic boundary of a flat immersed compact surface withoutplanar regions is given.     

完备Riemann流形之共轭点

本文证明了具非负曲率完备Ｒｉｅｍａｎｎ测地线为无共轭点测地线的充要条件；并由此证明了若该流形上的截面含有一无共轭点测地线的切向量，则其对应的截曲率为零．     

Volume and Lattice Points of Reflexive Simplices

Using new number-theoretic bounds on the denominators of unit fractions summing up to one, we show that in any dimension d ≥ 4 there is only one d-dimensional reflexive simplex having maximal volume. Moreover, only these reflexive simplices can admit an edge that has the maximal number of lattice points possible for an edge of a reflexive simplex. In general, these simplices are also expected to contain the largest number of lattice points even among all lattice polytopes with only one interior lattice point. Translated in algebro-geometric language, our main theorem yields a sharp upper bound on the anticanonical degree of d-dimensional Q-factorial Gorenstein toric Fano varieties with Picard number one, e.g., of weighted projective spaces with Gorenstein singularities.     

Symplectic curvature tensors

In the paper, one establishes the decomposition of the space of tensors which have the symmetries of the curvature of a torsionless symplectic connection into Sp (n)-irreducible components. This leads to three interesting classes of symplectic connections: flat, Ricci flat, and similar to the Levi-Civita connections of Kähler manifolds with constant holomorphic sectional curvature (we call them connections with reducible curvature). A symplectic manifold with two transversal polarizations has a canonical symplectic connection, and we study properties that are encountered if this canonical connection belongs to the classes mentioned above. For instance, in the reducible case we can compute the Pontrjagin classes, and these will be zero if the polarizations are real, etc. If the polarizations are real and there exist points where they are either singular or nontransversal, one has residues in the sense ofLehmann [L], which should be expected to play an interesting role in symplectic geometry.     

Surfaces in Sol3 Space Foliated by Circles

In this paper we study surfaces foliated by a uniparametric family of circles in the homogeneous space Sol₃. We prove that there do not exist such surfaces with zero mean curvature or with zero Gaussian curvature. We extend this study considering surfaces foliated by geodesics, equidistant lines or horocycles in totally geodesic planes and we classify all such surfaces under the assumption of minimality or flatness.     

Deformability of surfaces of positive curvature

For surfaces of positive Gaussian curvature bounded away from zero the following statement is proved: A piece of a given surface containing a preassigned finite set of points and having a Lyapunov boundary can be deformed with an arbitrary given (as large as we like) bending at these points under the condition that the area of the piece is sufficiently small.Translated from Matematicheskie Zametki, Vol. 19, No. 5, pp. 815–823, May, 1976.I thank V. T. Fomenko for guidance.     

Random complex zeroes, II. Perturbed lattice

We show that the flat chaotic analytic zero points (i.e. zeroes of a random entire function where ζ₀, ζ₁, … are independent standard complex-valued Gaussian variables) can be regarded as a random perturbation of a lattice in the plane. The distribution of the distances between the zeroes and the corresponding lattice points is shift-invariant and has a Gaussian-type decay of the tails. Supported by the Israel Science Foundation of the Israel Academy of Sciences and Humanities.     

On extrinsic symmetric spaces with zero mean curvature in Minkowski space-time

For an extrinsic symmetric space M in Minkowski space-time, we prove that if M is spacelike with zero mean curvature, then it is totally geodesic and if M is timelike with zero mean curvature, then it is totally geodesic or it is a flat hypersurface.     

Ruled surfaces with vanishing second Gaussian curvature

For a surface free of points of vanishing Gaussian curvature in Euclidean space the second Gaussian curvature is defined formally. It is first pointed out that a minimal surface has vanishing second Gaussian curvature but that a surface with vanishing second Gaussian curvature need not be minimal. Ruled surfaces for which a linear combination of the second Gaussian curvature and the mean curvature is constant along the rulings are then studied. In particular the only ruled surface in Euclidean space with vanishing second Gaussian curvature is a piece of a helicoid.     

Counting Lattice Points by Means of the Residue Theorem

We use the residue theorem to derive an expression for the number of lattice points in a dilated n-dimensional tetrahedron with vertices at lattice points on each coordinate axis and the origin. This expression is known as the Ehrhart polynomial. We show that it is a polynomial in t, where t is the integral dilation parameter. We prove the Ehrhart-Macdonald reciprocity law for these tetrahedra, relating the Ehrhart polynomials of the interior and the closure of the tetrahedra. To illustrate our method, we compute the Ehrhart coefficient for codimension 2. Finally, we show how our ideas can be used to compute the Ehrhart polynomial for an arbitrary convex lattice polytope.     

THE DISTRIBUTION OF LATTICE POINTS IN ELLIPTIC ANNULI

We study the distribution of the number of lattice points lyingin thin elliptical annuli. It has been conjectured by Bleherand Lebowitz that if the width of the annuli tends to zero andtheir area tends to infinity, then the distribution of thisnumber, normalized to have zero mean and unit variance, is Gaussian.This has been proved by Hughes and Rudnick for circular annuliwhose width shrinks to zero sufficiently slowly. We prove thisconjecture for ellipses whose aspect ratio is transcendentaland strongly Diophantine, also assuming the width shrinks slowlyto zero.     

Some compact conformally flat manifolds with non-negative scalar curvature

In this paper we study some compact locally conformally flat manifolds with a compatible metric whose scalar curvature is nonnegative, and in particular with nonnegative Ricci curvature. In the last section we study such manifolds of dimension 4 and scalar curvature identically zero.     

Normal forms and invariants for 2-dimensional almost-Riemannian structures

2-Dimensional almost-Riemannian structures are generalized Riemannian structures on surfaces for which a local orthonormal frame is given by a Lie bracket generating pair of vector fields that can become collinear. Generically, there are three types of points: Riemannian points where the two vector fields are linearly independent, Grushin points where the two vector fields are collinear but their Lie bracket is not, and tangency points where the two vector fields and their Lie bracket are collinear and the missing direction is obtained with one more bracket.In this paper we consider the problem of finding normal forms and functional invariants at each type of point. We also require that functional invariants are “complete” in the sense that they permit to recognize locally isometric structures.The problem happens to be equivalent to the one of finding a smooth canonical parameterized curve passing through the point and being transversal to the distribution.For Riemannian points such that the gradient of the Gaussian curvature K is different from zero, we use the level set of K as support of the parameterized curve. For Riemannian points such that the gradient of the curvature vanishes (and under additional generic conditions), we use a curve which is found by looking for crests and valleys of the curvature. For Grushin points we use the set where the vector fields are parallel.Tangency points are the most complicated to deal with. The cut locus from the tangency point is not a good candidate as canonical parameterized curve since it is known to be non-smooth. Thus, we analyse the cut locus from the singular set and we prove that it is not smooth either. A good candidate appears to be a curve which is found by looking for crests and valleys of the Gaussian curvature. We prove that the support of such a curve is uniquely determined and has a canonical parametrization.     

4-dimensional anti-Kähler manifolds and Weyl curvature

On a 4-dimensional anti-Kähler manifold, its zero scalar curvature implies that its Weyl curvature vanishes and vice versa. In particular any 4-dimensional anti-Kähler manifold with zero scalar curvature is flat.