The Grothendieck conjecture for affine curves

We prove that the isomorphism class of an affine hyperbolic curve defined over a field finitely generated over Q is completely determined by its arithmetic fundamental group. We also prove a similar result for an affine curve defined over a finite field.     

The view-obstruction problem for 3-dimensional spheres

LetC be ann-dimensional sphere with diameter 1 and center at the origin inE ⁿ . The view-obstruction problem forn-dimensional spheres is to determine a constant ν(n) to be the lower bound of those α for which any half-lineL, given byx _i =a _i t (i=1,2,...,n) where parametert≥0 anda _i (i=1,2,...,n) are positive real numbers, intersects

Isotropic affine spheres

In this paper, we study affine spheres which are isotropic and we obtain a complete classification. In particular, we show that all such affine spheres are hyperbolic affine spheres, isometric with SL(3 , R ) / SO(3), SL(3 , C ) / SU(3), SU*(6) / Sp(3) or E6 (-26) /F4 .     

A characterization of affine spheres in R 4

A surface immersed in R ⁴ is called a proper affine sphere if the position vector belongs to the affine normal plane. We classify proper affine spheres with ?^?? g ^??=0 whose affine mean curvature vector has constant length. Moreover, we find some concrete examples of affine spheres which are not affine umbilical.     

Harmonic spheres conjecture

We discuss the harmonic spheres conjecture that the space of harmonic maps of the Riemann sphere into the loop space of a compact Lie group G are related to the moduli space of Yang-Mills G-fields on the four-dimensional Euclidean space.     

View-obstruction problem for 3-dimensional spheres

The view-obstruction problem for ann-dimensional closed convex bodyC containing the origin in its interior has been formulated by T. W. Cusick in 1973. The problem is to determine the constantK(C) defined to be the infimum of those >0 such that for any rayL starting from the origin and lying in the positive quadrant there are non-negative integersm ₁,...,m _n for whichL intersects C+(m ₁+1/2,...,m _n +1/2). Here a conjecture of T. W. Cusick for the 3-dimensional spheres with centre 0 is proved. In fact an infinite sequence of isolated minima is obtained.Dedicated to Professor R. P. Bambah on the Occasion of his Sixtieth Birthday     

Convex affine domains and Markus conjecture

In this paper we show that any properly convex quasi-homogeneous affine domain with irreducible projective automorphism group is projectively equivalent to a homogeneous affine domain. Then as an application we answer positively the Markus conjecture when the manifold M is in a certain class of convex affine manifolds.Mathematics Subject Classification (1991): 51M10, 57S25This work was supported by Korea Research Foundation Grant (KRF-2002-070-C00010).in final form: 6 October 2003     

Flat affine spheres in R 3

Nondegenerate affine surfaces in R ³ which are affine spheres and have flat affine metrics are classified. Those spheres which are proper are shown to be equivalent to open subsets of the surface defined by xyz=1 or the surface defined by (x ²+y ²)z=1.Partially supported by NSF Grant DMS 8802664.Partially supported by NSERC Operating Grant A2501.     

The Cauchy problem for indefinite improper affine spheres and their Hessian equation

We give a conformal representation for indefinite improper affine spheres which solve the Cauchy problem for their Hessian equation. As consequences, we can characterize their geodesics and obtain a generalized symmetry principle. Then, we classify the helicoidal indefinite improper affine spheres and find a new family with geodesically complete non-flat affine metric. Moreover, we present interesting examples with singular curves and isolated singularities.     

Left-symmetric algebras and homogeneous improper affine spheres

The nonzero level sets in n-dimensional flat affine space of a translationally homogeneous function are improper affine spheres if and only if the Hessian determinant of the function is equal to a nonzero constant multiple of the nth power of the function. The exponentials of the characteristic polynomials of certain left-symmetric algebras yield examples of such functions whose level sets are analogues of the generalized Cayley hypersurface of Eastwood–Ezhov. There are found purely algebraic conditions sufficient for the characteristic polynomial of the left-symmetric algebra to have the desired properties. Precisely, it suffices that the algebra has triangularizable left multiplication operators and the trace of the right multiplication is a Koszul form for which right multiplication by the dual idempotent is projection along its kernel, which equals the derived Lie subalgebra of the left-symmetric algebra.     

On the generalized lower bound conjecture for polytopes and spheres

In 1971, McMullen and Walkup posed the following conjecture, which is called the generalized lower bound conjecture: If P is a simplicial d-polytope then its h-vector (h ₀, h ₁, …, h _d ) satisfies $ {h_0}\leq {h_1}\leq \ldots \leq {h_{{\left\lfloor {{d \left/ {2} \right.}} \right\rfloor }}} $ . Moreover, if h _r?1 = h _r for some $ r\leq \frac{1}{2}d $ then P can be triangulated without introducing simplices of dimension ≤d ? r. The first part of the conjecture was solved by Stanley in 1980 using the hard Lefschetz theorem for projective toric varieties. In this paper, we give a proof of the remaining part of the conjecture. In addition, we generalize this result to a certain class of simplicial spheres, namely those admitting the weak Lefschetz property.     

Calabi conjecture on hyperbolic affine hyperspheres (2)

Research supported by the Alexander von Humboldt-Stiftung, the National Natural Science Foundation of China, the Science Foundation of the Educational Committee of China     

Inflection points and affine vertices of closed curves on 2-dimensional affine flat tori

This paper deals with a geometric problem on inflection points and affine vertices for closed curves in an affine flat torus. We show that the least number of inflection points lying on a closed curve that is not homotopic to zero is 2 if the torus is affinely equivalent to a euclidean torus and 0 otherwise. We consider also the number of affine vertices on a strictly convex closed curve on a flat torus. An explicit example of a closed curve with six affine vertices is given.     

The classification of 3-dimensional Lorentzian affine hypersurfaces with parallel cubic form

We study Lorentzian affine hypersurfaces in Rn+1 with parallel cubic form with respect to the Levi-Civita connection of the affine metric. As main result, a complete classification of such non-degenerate affine hypersurfaces in R⁴ is given.