Best Approximations of Convex Compact Sets by Balls in the Hausdorff Metric

We deduce an upper bound for the Hausdorff distance between a nonempty bounded set and the set of all closed balls in a strictly convex straight geodesic space X of nonnegative curvature. We prove that the set $\chi \left[ {\rm M} \right]$ of centers of closed balls approximating a convex compact set in the Hausdorff metric in the best possible way is nonempty _X [M] and is contained in M. Some other properties of $\chi \left[ {\rm M} \right]$ also are investigated.     

Finite Topological Type and Volume Growth

We show that a complete noncompact n-dimensional Riemannian manifold Mwith Ricci curvature Ric _M –(n – 1) and conjugateradius conj _M c > 0 has finite topological type, provided that the volume growth of geodesic balls in M is not very far from that of the balls in an n-dimensional hyperbolic space H ⁿ (–1)of sectional curvature –1. We also show that a complete open Riemannian manifold M with nonnegative intermediate Ricci curvature and quadratic curvature decay has finite topological typeif the volume of geodesic balls of M around the base point grows slowly.     

Formulas for best extrapolation

We considern-point Lagrange-Hermite extrapolation forf(x), x>1, based uponf(x _i ),i=1(1)n, –1x _i 1, including non-distinct pointsx _i in confluent formulas involving derivatives. The problem is to find the pointsx _i that minimize the factor in the remainderP _n (x)f ⁽ⁿ⁾()/n, –1<<x subject to the condition|P _n (x)|M, –1x1,2^–n+1M2 ⁿ . The solution is significant only when a single set of pointsx _i suffices for everyx>1. The problem is here completely solved forn=1(1)4. Forn>4 it may be conjectured that there is a single minimal , 0 rn, whererr(M) is a non-decreasing function ofM, P _n (–1)=(–1) ⁿ M, and for 0rn–2, thej-th extremumP _n (x _{e, j} )=(–1) ^n–j M,j=1(1)n–r–1 (except forM=M _r ,r=1(1)n–1, whenj=1(1)n–r).     

Some Applications of Critical Point Theory of Distance Functions on Riemannian Manifolds

In this paper, we use the theory of critical points of distance functions to study the rigidity and topology of Riemannian manifolds with sectional curvature bounded below. We prove that an n-dimensional complete connected Riemannian manifold M with sectional curvature K _M 1 is isometric to an n-dimensional Euclidean unit sphere if M has conjugate radius bigger than /2 and contains a geodesic loop of length 2. We also prove that if M is an n(3)-dimensional complete connected Riemannian manifold with K _M 1 and radius bigger than /2, then any closed connected totally geodesic submanifold of dimension not less than two of M is homeomorphic to a sphere.     

The formation of black holes in spherically symmetric gravitational collapse

We consider the spherically symmetric, asymptotically flat Einstein–Vlasov system. We find explicit conditions on the initial data, with ADM mass M, such that the resulting spacetime has the following properties: there is a family of radially outgoing null geodesics where the area radius r along each geodesic is bounded by 2M, the timelike lines \({r=c\in [0,2M]}\) are incomplete, and for r > 2M the metric converges asymptotically to the Schwarzschild metric with mass M. The initial data that we construct guarantee the formation of a black hole in the evolution. We give examples of such initial data with the additional property that the solutions exist for all r ≥ 0 and all Schwarzschild time, i.e., we obtain global existence in Schwarzschild coordinates in situations where the initial data are not small. Some of our results are also established for the Einstein equations coupled to a general matter model characterized by conditions on the matter quantities.     

Mathematical expectations of functions of sums of a random number of independent terms

Conditions are found which must be imposed on a function g(x) in order that M g(₁+₂+ + _v < if M g(_i) < and M g(v) < ,, ₁, ₂, , _n, ... being non-negative and independent, being integral, and {_i} being identically distributed. The result is applied to the theory of branching processes.Translated from Matematicheskie Zametki, Vol. 3, No. 4, pp. 387–394, April, 1968.     

A note on the illumination of convex bodies

LetKE ^d be a convex body and letl _r(K) denote the minimum number ofr-dimensional affine subspaces ofE ^d lying outsideK with which it is possible to illuminateK, where 0rd–1. We give a new proof of the theorem thatl _r(K)(d+1)/(r+1) with equality for smoothK.The work was supported by Hung. Nat. Found. for Sci . Research No. 326-0213 and 326-0113.     

The space of geodesics

Let M be a manifold with linear connection . The space G(M) of all geodesics of M may be given a topological structure and may be realized as a quotient space of the reduced tangent bundle of M. The space G(M) is a T ₁ space iff the image of each geodesic is a closed subset of M. It is Hausdorff iff each tangentially convergent sequence of geodesics converges in the Hausdorff limit sense to the limit geodesic. If M has no conjugate points and G(M) is Hausdorff, then M is geodesically connected.Supported in part by NSF grant DMS-8803511.     

Orlik-Solomon Algebras and Tutte Polynomials

The OS algebra A of a matroid M is a graded algebra related to the Whitney homology of the lattice of flats of M. In case M is the underlying matroid of a hyperplane arrangement A in ^r , A is isomorphic to the cohomology algebra of the complement ^r A. Few examples are known of pairs of arrangements with non-isomorphic matroids but isomorphic OS algebras. In all known examples, the Tutte polynomials are identical, and the complements are homotopy equivalent but not homeomorphic.We construct, for any given simple matroid M ₀, a pair of infinite families of matroids M _n and M _n , n 1, each containing M ₀ as a submatroid, in which corresponding pairs have isomorphic OS algebras. If the seed matroid M ₀ is connected, then M _n and M _n have different Tutte polynomials. As a consequence of the construction, we obtain, for any m, m different matroids with isomorphic OS algebras. Suppose one is given a pair of central complex hyperplane arrangements A ₀ and A ₁ . Let S denote the arrangement consisting of the hyperplane {0} in ₁ . We define the parallel connection P(A ₀, A ₁), an arrangement realizing the parallel connection of the underlying matroids, and show that the direct sums A ₀ A ₁ and S P (A ₀, A ₁) have diffeomorphic complements.     

Densely two-valued metric projections in uniformly convex Banach spaces

For every uniformly convex Banach spaceX with dimX2 there is a residual setU in the Hausdorff metric spaceB(X) of bounded and closed sets inX such that the metric projection generated by a set fromU is two-valued and upper semicontinuous on a dense and everywhere continual subset ofX. For any two closed and separated subsetsM ₁ andM ₂ ofX the points on the equidistant hypersurface which have best approximations both inM ₁ andM ₂ form a dense G set in the induced topology.The author is partially supported by the National Fund for Scientific Research at the Bulgarian Ministry of Science and Education under contract MM 408/94.     

Approximation by Fourier sums of classes of functions with several bounded derivatives

An ordered estimate is obtained for the approximation by Fourier sums, in the metric ofd=(d ₁ , ...,d _n ), 1<dj<,j=1, ...,_n of classes of periodic functions of several variables with zero means with respect to all their arguments, having m mixed derivatives of order a¹..., a^m., aⁱ rⁿ. which are bounded in the metrics ofp ⁱ =p ₁ ⁱ , ..., p _n ⁱ , i

j ⁱ <,i=i, ...,n, j=1, ...,n by the constants ₁, ., _m, respectively.Translated from Matematicheskie Zametki, Vol. 23, No. 2, pp. 197–212, February, 1978.     

A Semigroup Approach to Harmonic Maps

We present a semigroup approach to harmonic maps between metric spaces. Our basic assumption on the target space (N,d) is that it admits a barycenter contraction, i.e. a contracting map which assigns to each probability measure q on N a point b(q) in N. This includes all metric spaces with globally nonpositive curvature in the sense of Alexandrov as well as all metric spaces with globally nonpositive curvature in the sense of Busemann. It also includes all Banach spaces.The analytic input comes from the domain space (M,) where we assume that we are given a Markov semigroup (p_t)_t>0. Typical examples come from elliptic or parabolic second-order operators on Rⁿ, from Lévy type operators, from Laplacians on manifolds or on metric measure spaces and from convolution operators on groups. In contrast to the work of Korevaar and Schoen (1993, 1997), Jost (1994, 1997), Eells and Fuglede (2001) our semigroups are not required to be symmetric.The linear semigroup acting, e.g., on the space of bounded measurable functions u:MR gives rise to a nonlinear semigroup (P_t^*)_t acting on certain classes of measurable maps f:MN. We will show that contraction and smoothing properties of the linear semigroup (p_t)_t can be extended to the nonlinear semigroup (P_t^*)_t, for instance, L_p–L_q smoothing, hypercontractivity, and exponentially fast convergence to equilibrium. Among others, we state existence and uniqueness of the solution to the Dirichlet problem for harmonic maps between metric spaces. Moreover, for this solution we prove Lipschitz continuity in the interior and Hölder continuity at the boundary.Our approach also yields a new interpretation of curvature assumptions which are usually required to deduce regularity results for the harmonic map flow: lower Ricci curvature bounds on the domain space are equivalent to estimates of the L₁-Wasserstein distance between the distribution of two Brownian motions in terms of the distance of their starting points; nonpositive sectional curvature on the target space is equivalent to the fact that the L₁-Wasserstein distance of two distributions always dominates the distance of their barycenters.Dedicated to the memory of Professor Dr. Heinz Bauer     

On weakly p-harmonic maps to a closed hemisphere

We consider weakly p-harmonic maps (p2) from a compact connected Riemannian manifold M^m(m2) to the the standard sphere Sⁿ with values in the closed hemisphere Sⁿ₊ = {x Sⁿ : x_n+1 0 } (n 2). We first prove that if u=(u₁,...,u_n+1):MSⁿ is a weakly p-harmonic map satisfying u_n+1(x)>0 a.e. on M, then it is a minimizing p-harmonic map. Next, we give a necessary and sufficient condition for the boundary data : M Sⁿ₊ to achieve uniqueness; and when this condition fails, we are able to describe the set of all minimizers. When M is without boundary, we obtain a Liouville type Theorem for weakly p-harmonic maps.Mathematics Subject Classification (2000): 58E20; 35J70     

m-Systems and Partial m-Systemsof Polar Spaces

LetP be a finite classical polar space of rankr, withr 2. A partialm-systemM ofP, with 0 m r - 1, is any set (₁), ₂,..., _k ofk ( 0) totally singularm-spaces ofP such that no maximal totally singular space containing _i has a point in common with (_{1 2} ... _k) — _i,i = 1, 2,...,k. In a previous paper an upper bound for ¦M¦ was obtained (Theorem 1). If ¦M¦ = , thenM is called anm-system ofP. Form = 0 them-systems are the ovoids ofP; form =r - 1 them-systems are the spreads ofP. In this paper we improve in many cases the upper bound for the number of elements of a partialm-system, thus proving the nonexistence of several classes ofm-systems.Dedicated to Hanfried Lenz on the occasion of his 80th birthday     

Einstein Metrics with Nonpositive Sectional Curvature on Extensions of Lie Algebras of Heisenberg Type

The purpose of this article is to study some simply connected Lie groups with left invariant Einstein metric, negative Einstein constant and nonpositive sectional curvature. These Lie groups are classified if their associated metric Lie algebra s is of Iwasawa type and s = An₁n₂...n_r, where all n_iare Lie algebras of Heisenberg type with [[n_i,n_j] = {0} for ij. The most important ideas of the article are based on a construction method for Einstein spaces introduced by Wolter in 1991. By this method some new examples of Einstein spaces with nonpositive curvature are constructed. In another part of the article it is shown that Damek-Ricci spaces have negative sectional curvature if and only if they are symmetric spaces.     

The Symmetry of 2-Flat Homogeneous Spaces

A Riemannian manifold M is called 2-flat homogeneous if every geodesic is contained in some 2-flat , and if the group of isometries of M acts transitively on the set of pairs (p, ) with p . By a 2-flat we mean a closed, connected, flat, totally geodesic, 2-dimensional submanifold of M. It is proved in the paper that 2-flat homogeneous spaces are symmetric.     

A Distance-Regular Graph with Strongly Closed Subgraphs

Let be a distance-regular graph of diameter d, valency k and r := maxi | (c _i,b _i) = (c ₁,b ₁). Let q be an integer with r + 1 q d – 1.In this paper we prove the following results: Theorem 1 Suppose for any pair of vertices at distance q there exists a strongly closed subgraph of diameter q containing them. Then for any integer i with 1 i q and for any pair of vertices at distance i there exists a strongly closed subgraph of diameter i containing them. Theorem 2 If r 2, then c _2r+3 1.As a corollary of Theorem 2 we have d k ²(r + 1) if r 2.     

The Moduli Space of Riemann Surfaces of Large Genus

Let ${\mathcal{M}_{g,\epsilon}}$ be the ${\epsilon}$ -thick part of the moduli space ${\mathcal{M}_g}$ of closed genus g surfaces. In this article, we show that the number of balls of radius r needed to cover ${\mathcal{M}_{g,\epsilon}}$ is bounded below by ${(c_1g)^{2g}}$ and bounded above by ${(c_2g)^{2g}}$ , where the constants c ₁, c ₂depend only on ${\epsilon}$ and r, and in particular not on g. Using this counting result we prove that there are Riemann surfaces of arbitrarily large injectivity radius that are not close (in the Teichmüller metric) to a finite cover of a fixed closed Riemann surface. This result illustrates the sharpness of the Ehrenpreis conjecture.     

The multiple-point schemes of a finite curvilinear map of codimension one

LetX andY be smooth varieties of dimensionsn−1 andn over an arbitrary algebraically closed field,f: X→Y a finite map that is birational onto its image. Suppose thatf is curvilinear; that is, for allxεX, the Jacobian ϱf(x) has rank at leastn−2. Forr≥1, consider the subschemeN _r ofY defined by the (r−1)th Fitting ideal of the -module , and setM _r ∶=f ⁻¹ N _r . In this setting—in fact, in a more general setting—we prove the following statements, which show thatM _r andN _r behave like reasonable schemes of source and targetr-fold points off. If each component ofM _r , or equivalently ofN _r , has the minimal possible dimensionn−r, thenM _r andN _r are Cohen-Macaulay, and their fundamental cycles satisfy the relation,f _*[M _r ]=r[N _r ]. Now, suppose that each component ofM _s , or ofN _s , has dimensionn−s fors=1,...,r+1. Then the blowup Bl(N _r ,N _r+1 ) is equal to the Hilbert scheme Hilb _f ^r and the blowup Bl(M _r ,M _r+1 ) is equal to the universal subscheme Univ _f ^r of Hilb _f ^r × _Y X; moreover, Hilb _f ^r and Univ _f ^r are Gorenstein. In addition, the structure maph:Hilb _f ^r →Y is finite and birational onto its image; and its conductor is equal to the ideal ofN _r+1 inN _r , and is locally self-linked. Reciprocally, is equal to . Moreover,h _* [h ⁻¹ N _r+1 ]=(r+1)[N _r+1 ]. Similar assertions hold for the structure maph ₁: Univ _f ^r →X ifr≥2. Supported in part by NSF grant 9106444-DMS. Supported in part by NSA grant MDA904-92-3007, and at MIT 21–30 May 1989 by Sloan Foundation grant 88-10-1. Supported in part by NSF grant DMS-9305832.     

Curve shortening in a Riemannian manifold

In this paper, we study the curve shortening flow in a general Riemannian manifold. We have many results for the global behavior of the flow. In particular, we show the following results: let M be a compact Riemannian manifold. (1) If the curve shortening flow exists for infinite time, and , then for every n > 0, . Furthermore, the limiting curve exists and is a closed geodesic in M. (2) In M × S ¹, if γ₀ is a ramp, then we have a global flow which converges to a closed geodesic in C ^∞ norm. As an application, we prove the theorem of Lyusternik and Fet.