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Given a configuration of pebbles on the vertices of a graph, a pebbling move is defined by removing two pebbles from some vertex and placing one pebble on an adjacent vertex. The cover pebbling number of a graph, gamma(G), is the smallest number of pebbles such that through a sequence of pebbling moves, a pebble can eventually be placed on every vertex simultaneously, no matter how the pebbles are initially distributed. The cover pebbling number for complete multipartite graphs and wheel graphs is determined. We also prove a sharp bound for gamma(G) given the diameter and number of vertices of G. | Cover pebbling numbers and bounds for certain families of graphs | 14,100 |
In this paper, we give a simple combinatorial explanation of a formula of A. Postnikov relating bicolored rooted trees to bicolored binary trees. We also present generalized formulas for the number of labeled k-ary trees, rooted labeled trees, and labeled plane trees. | A combinatorial proof of Postnikov's identity and a generalized
enumeration of labeled trees | 14,101 |
The Ehrhart polynomial of an integral convex polytope counts the number of lattice points in dilates of the polytope. In math.CO/0402148, the authors conjectured that for any cyclic polytope with integral parameters, the Ehrhart polynomial of it is equal to its volume plus the Ehrhart polynomial of its lower envelope and proved the case when the dimension d = 2. In our article, we prove the conjecture for any dimension. | Ehrhart polynomials of cyclic polytopes | 14,102 |
Given a graph G and a configuration C of pebbles on the vertices of G, a pebbling step removes two pebbles from one vertex and places one pebble on an adjacent vertex. The cover pebbling number g=g(G) is the minimum number so that every configuration of g pebbles has the property that, after some sequence of pebbling steps, every vertex has a pebble on it. We prove that the cover pebbling number of the d-dimensional hypercube Q^d equals 3^d. | Cover Pebbling Hypercubes | 14,103 |
We use a greedy probabilistic method to prove that for every $\epsilon > 0$, every $m\times n$ Latin rectangle on $n$ symbols has an orthogonal mate, where $m=(1-\epsilon)n$. That is, we show the existence of a second Latin rectangle such that no pair of the $mn$ cells receives the same pair of symbols in the two rectangles. | Orthogonal latin rectangles | 14,104 |
Applying results from partial difference sets, quadratic forms, and recent results of Brouwer and Van Dam, we construct the first known amorphic association scheme with negative Latin square type graphs and whose underlying set is a non-elementary abelian 2-group. We give a simple proof of a result of Hamilton that generalizes Brouwer's result. We use multiple distinct quadratic forms to construct amorphic association schemes with a large number of classes. | Amorphic association schemes with negative Latin square type graphs | 14,105 |
The Grundy number of an impartial game G is the size of the unique Nim heap equal to G. We introduce a new variant of Nim, Restricted Nim, which restricts the number of stones a player may remove from a heap in terms of the size of the heap. Certain classes of Restricted Nim are found to produce sequences of Grundy numbers with a self-similar fractal structure. Extending work of C. Kimberling, we obtain new characterizations of these "fractal sequences" and give a bijection between these sequences and certain upper-triangular arrays. As a special case we obtain the game of Serial Nim, in which the Nim heaps are ordered from left to right, and players can move only in the leftmost nonempty heap. | Fractal Sequences and Restricted Nim | 14,106 |
A rather simple natural outer derivation of the graded Lie algebra of all vector valued differential forms with the Fr\"olicher-Nijenhuis bracket turns out to be a differential and gives rise to a cohomology of the manifold, which is functorial under local diffeomorphisms. This cohomology is determined as the direct product of the de Rham cohomology space and the graded Lie algebra of "traceless" vector valued differential forms, equipped with a new natural differential concomitant as graded Lie bracket. We find two graded Lie algebra structures on the space of differential forms. Some consequences and related results are also discussed. | A cohomology for vector valued differential forms | 14,107 |
Shoen and Uhlenbeck showed that ``tangent maps'' can be defined at singular points of energy minimizing maps. Unfortunately these are not unique, even for generic boundary conditions. Examples are discussed which have isolated singularities with a continuum of distinct tangent maps. | Nonunique tangent maps at isolated singularities of harmonic maps | 14,108 |
In the main part of this paper a connection is just a fiber projection onto a (not necessarily integrable) distribution or sub vector bundle of the tangent bundle. Here curvature is computed via the Froelicher-Nijenhuis bracket, and it is complemented by cocurvature and the Bianchi identity still holds. In this situation we determine the graded Lie algebra of all graded derivations over the horizontal projection of a connection and we determine their commutation relations. Finally, for a principal connection on a principal bundle and the induced connection on an associated bundle we show how one may pass from one to the other. The final results relate derivations on vector bundle valued forms and derivations over the horizontal projection of the algebra of forms on the principal bundle with values in the standard vector space. | Graded derivations of the algebra of differential forms associated with
a connection | 14,109 |
We study the action of the diffeomorphism group $\Diff(M)$ on the space of proper immersions $\Imm_{\text{prop}}(M,N)$ by composition from the right. We show that smooth transversal slices exist through each orbit, that the quotient space is Hausdorff and is stratified into smooth manifolds, one for each conjugacy class of isotropy groups. | The action of the diffeomorphism group on the space of immersions | 14,110 |
It is shown that a strong system of vector fields on a fiber bundle in the sense of [Modugno, M. Systems of connections and invariant lagrangians. In: Differential geometric methods in theoretical physics, Proc. 15th Int. Conf., DGM, Clausthal/FRG 1986, 518-534 World Scientific Publishing Co. (1987)] is induced from a principal fiber bundle if and only if each vertical vector field of the system is complete. | The relation between systems and associated bundles | 14,111 |
We use an extension of Sunada's theorem to construct a nonisometric pair of isospectral simply connected domains in the Euclidean plane, thus answering negatively Kac's question, ``can one hear the shape of a drum?'' In order to construct simply connected examples, we exploit the observation that an orbifold whose underlying space is a simply connected manifold with boundary need not be simply connected as an orbifold. | One cannot hear the shape of a drum | 14,112 |
Let $G\subset GL(V)$ be a linear Lie group with Lie algebra $\frak g$ and let $A(\frak g)^G$ be the subalgebra of $G$-invariant elements of the associative supercommutative algebra $A(\frak g)= S(\frak g^*)\otimes \La(V^*)$. To any $G$-structure $\pi:P\to M$ with a connection $\omega$ we associate a homomorphism $\mu_\omega:A(\frak g)^G\to \Omega(M)$. The differential forms $\mu_\omega(f)$ for $f\in A(\frak g)^G$ which are associated to the $G$-structure $\pi$ can be used to construct Lagrangians. If $\omega$ has no torsion the differential forms $\mu_\omega(f)$ are closed and define characteristic classes of a $G$-structure. The induced homomorphism $\mu'_\omega:A(\g)^G\to H^*(M)$ does not depend on the choice of the torsionfree connection $\omega$ and it is the natural generalization of the Chern Weil homomorphism. | Characteristic classes for $G$-structures | 14,113 |
An action of a Lie algebra $\frak g$ on a manifold $M$ is just a Lie algebra homomorphism $\zeta:\frak g\to \frak X(M)$. We define orbits for such an action. In general the space of orbits $M/\frak g$ is not a manifold and even has a bad topology. Nevertheless for a $\frak g$-manifold with equidimensional orbits we treat such notions as connection, curvature, covariant differentiation, Bianchi identity, parallel transport, basic differential forms, basic cohomology, and characteristic classes, which generalize the corresponding notions for principal $G$-bundles. As one of the applications, we derive a sufficient condition for the projection $M\to M/\frak g$ to be a bundle associated to a principal bundle. | Differential geometry of $\frak g$-manifolds | 14,114 |
This note announces a general construction of characteristic currents for singular connections on a vector bundle. It develops, in particular, a Chern-Weil-Simons theory for smooth bundle maps $\alpha : E \rightarrow F$ which, for smooth connections on $E$ and $F$, establishes formulas of the type $$ \phi \ = \ \text{\rm Res}_{\phi}\Sigma_{\alpha} + dT. $$ Here $\phi$ is a standard charactersitic form, $\text{Res}_{\phi}$ is an associated smooth ``residue'' form computed canonically in terms of curvature, $\Sigma_{\alpha}$ is a rectifiable current depending only on the singular structure of $\alpha$, and $T$ is a canonical, functorial transgression form with coefficients in $\loc$. The theory encompasses such classical topics as: Poincar\'e-Lelong Theory, Bott-Chern Theory, Chern-Weil Theory, and formulas of Hopf. Applications include:\ \ a new proof of the Riemann-Roch Theorem for vector bundles over algebraic curves, a $C^{\infty}$-generalization of the Poincar\'e-Lelong Formula, universal formulas for the Thom class as an equivariant characteristic form (i.e., canonical formulas for a de Rham representative of the Thom class of a bundle with connection), and a Differentiable Riemann-Roch-Grothendieck Theorem at the level of forms and currents. A variety of formulas relating geometry and characteristic classes are deduced as direct consequences of the theory. | A theory of characteristic currents associated with a singular
connection | 14,115 |
We develop an asymptotic expansion of the spectral measures on a degenerating family of hyperbolic Riemann surfaces of finite volume. As an application of our results, we study the asymptotic behavior of weighted counting functions, which, if $M$ is compact, is defined for $w \geq 0$ and $T > 0$ by $$N_{M,w}(T) = \sum\limits_{\lambda_n \leq T}(T-\lambda_n)^w $$ where $\{\lambda_n\}$ is the set of eigenvalues of the Laplacian which acts on the space of smooth functions on $M$. If $M$ is non-compact, then the weighted counting function is defined via the inverse Laplace transform. Now let $M_{\ell}$ denote a degenerating family of compact or non-compact hyperbolic Riemann surfaces of finite volume which converges to the non-compact hyperbolic surface $M_{0}$. As an example of our results, we have the following theorem: There is an explicitly defined function $G_{\ell,w}(T)$ which depends solely on $\ell$, $w$, and $T$ such that for $w > 3/2$ and $T>0$, we have $$N_{M_{\ell},w}(T) = G_{\ell,w}(T) +N_{M_{0},w}(T) +o(1)$$ for $\ell \to 0$. We also consider the setting when $w < 3/2$, and we obtain a new proof of the continuity of small eigenvalues on degenerating hyperbolic Riemann surfaces of finite volume. | On the Asymptotic Behavior of Counting Functions Associated to
Degenerating Hyperbolic Riemann Surfaces | 14,116 |
We present a new construction for obtaining pairs of higher-step isospectral Riemannian nilmanifolds and compare several resulting new examples. In particular, we present new examples of manifolds that are isospectral on functions, but not isospectral on one-forms. | A New Construction of Isospectral Riemannian Nilmanifolds with Examples | 14,117 |
We define geometric zeta functions for locally symmetric spaces as generalizations of the zeta functions of Ruelle and Selberg. As a special value at zero we obtain the Reidemeister torsion of the manifold. For hermitian spaces these zeta functions have as special value the quotient of the holomorphic torsion of Ray and Singer and the holomorphic $L^2$-torsion, where the latter is defined via the $L^2$-theory of Atiyah. For higher fundamental rank twisted torsion numbers appear. | Geometric Zeta Functions, $L^2$-Theory, and Compact Shimura Manifolds | 14,118 |
Two Riemannian manifolds are said to have $C^k$-conjugate geodesic flows if there exist an $C^k$ diffeomorphism between their unit tangent bundles which intertwines the geodesic flows. We obtain a number of rigidity results for the geodesic flows on compact 2-step Riemannian nilmanifolds: For generic 2-step nilmanifolds the geodesic flow is $C^2$ rigid. For special classes of 2-step nilmanifolds, we show that the geodesic flow is $C^0$ or $C^2$ rigid. In particular, there exist continuous families of 2-step nilmanifolds whose Laplacians are isospectral but whose geodesic flows are not $C^0$ conjugate. | Geodesic Conjugacy in two-step nilmanifolds | 14,119 |
We give an arithmetic criterion which is sufficient to imply the discreteness of various two-generator subgroups of $PSL(2,{\bold C})$. We then examine certain two-generator groups which arise as extremals in various geometric problems in the theory of Kleinian groups, in particular those encountered in efforts to determine the smallest co-volume, the Margulis constant and the minimal distance between elliptic axes. We establish the discreteness and arithmeticity of a number of these extremal groups, the associated minimal volume arithmetic group in the commensurability class and we study whether or not the axis of a generator is simple. | Arithmeticity, Discreteness and Volume | 14,120 |
The Margulis constant for Kleinian groups is the smallest constant $c$ such that for each discrete group $G$ and each point $x$ in the upper half space ${\bold H}^3$, the group generated by the elements in $G$ which move $x$ less than distance c is elementary. We take a first step towards determining this constant by proving that if $\langle f,g \rangle$ is nonelementary and discrete with $f$ parabolic or elliptic of order $n \geq 3$, then every point $x$ in ${\bold H}^3$ is moved at least distance $c$ by $f$ or $g$ where $c=.1829\ldots$. This bound is sharp. | On the Margulis constant for Kleinian groups, I curvature | 14,121 |
We show that strictly abnormal geodesics arise in graded nilpotent Lie groups. We construct such a group, for which some Carnot geodesics are strictly abnormal; in fact, they are not normal in any subgroup. In the step-2 case we also prove that these geodesics are always smooth. Our main technique is based on the equations for the normal and abnormal curves, that we derive (for any Lie group) explicitly in terms of the structure constants. | A note on Carnot geodesics in nilpotent Lie groups | 14,122 |
We survey what is known about minimal surfaces in $\bold R^3 $ that are complete, embedded, and have finite total curvature. The only classically known examples of such surfaces were the plane and the catenoid. The discovery by Costa, early in the last decade, of a new example that proved to be embedded sparked a great deal of research in this area. Many new examples have been found, even families of them, as will be described below. The central question has been transformed from whether or not there are any examples except surfaces of rotation to one of understanding the structure of the space of examples. | Complete embedded minimal surfaces of finite total curvature | 14,123 |
Grafting is a surgery on Riemann surfaces introduced by Thurston which connects hyperbolic geometry and the theory of projective structures on surfaces. We will discuss the space of projective structures in terms of the Thurston's geometric parametrization given by grafting. From this approach we will prove that on any compact Riemann surface with genus greater than $1$ there exist infinitely many projective structures with Fuchsian holonomy representations. In course of the proof it will turn out that grafting is closely related to harmonic maps between surfaces. | Grafting, harmonic maps, and projective structures on surfaces | 14,124 |
The subject of this paper is the relationship among the marked length spectrum, the length spectrum, the Laplace spectrum on functions, and the Laplace spectrum on forms on Riemannian nilmanifolds. In particular, we show that for a large class of three-step nilmanifolds, if a pair of nilmanifolds in this class has the same marked length spectrum, they necessarily share the same Laplace spectrum on functions. In contrast, we present the first example of a pair of isospectral Riemannian manifolds with the same marked length spectrum but not the same spectrum on one-forms. Outside of the standard spheres vs. the Zoll spheres, which are not even isospectral, this is the only example of a pair of Riemannian manifolds with the same marked length spectrum, but not the same spectrum on forms. This partially extends and partially contrasts the work of Eberlein, who showed that on two-step nilmanifolds, the same marked length spectrum implies the same Laplace spectrum both on functions and on forms. | The Marked Length Spectrum Versus the Laplace Spectrum on Forms on
Riemannian Nilmanifolds | 14,125 |
The authors study the Hodge theory of the exterior differential operator $d$ acting on $q$-forms on a smoothly bounded domain in $\RR^{N+1}$, and on the half space $\rnp$. The novelty is that the topology used is not an $L^2$ topology but a Sobolev topology. This strikingly alters the problem as compared to the classical setup. It gives rise to a boundary-value problem belonging to a class of problems first introduced by Vi\v{s}ik and Eskin, and by Boutet de Monvel. | Hodge theory in the Sobolev topology for the de Rham complex | 14,126 |
Motivated by the physical concept of special geometry two mathematical constructions are studied, which relate real hypersurfaces to tube domains and complex Lagrangean cones respectively. Me\-thods are developed for the classification of homogeneous Riemannian hypersurfaces and for the classification of linear transitive reductive algebraic group actions on pseudo Riemannian hypersurfaces. The theory is applied to the case of cubic hypersurfaces, which is the case most relevant to special geometry, obtaining the solution of the two classification problems and the description of the corresponding homogeneous special K\"ahler manifolds. | Homogeneous Special Geometry | 14,127 |
The paper sketches a recent progress and formulates several open problems in studying equivariant quasiconformal and quasisymmetric homeomorphisms in negatively curved spaces as well as geometry and topology of noncompact geometrically finite negatively curved manifolds and their boundaries at infinity having Carnot--Carath\'eodory structures. Especially, the most interesting are complex hyperbolic manifolds with Cauchy--Riemannian structure at infinity, which occupy a distinguished niche and whose properties make them surprisingly different from real hyperbolic ones. | Quasiconformality and geometrical finiteness in Carnot--Carathéodory
and negatively curved spaces | 14,128 |
We prove the existence of a complete, embedded, singly periodic minimal surface, whose quotient by vertical translations has genus one and two ends. The existence of this surface was announced in our paper in {\it Bulletin of the AMS}, 29(1):77--84, 1993. Its ends in the quotient are asymptotic to one full turn of the helicoid, and, like the helicoid, it contains a vertical line. Modulo vertical translations, it has two parallel horizontal lines crossing the vertical axis. The nontrivial symmetries of the surface, modulo vertical translations, consist of: $180^\circ$ rotation about the vertical line; $180^\circ$ rotation about the horizontal lines (the same symmetry); and their composition. | The Singly Periodic Genus-One Helicoid | 14,129 |
An intrinsic definition in terms of conformal capacity is proposed for the conformal type of a Carnot--Carath\'eodory space (parabolic or hyperbolic). Geometric criteria of conformal type are presented. They are closely related to the asymptotic geometry of the space at infinity and expressed in terms of the isoperimetric function and the growth of the area of geodesic spheres. In particular, it is proved that a sub-Riemannian manifold admits a conformal change of metric that makes it into a complete manifold of finite volume if and only if the manifold is of conformally parabolic type. Further applications are discussed, such as the relation between local and global invertibility properties of quasiconformal immersions (the global homeomorphism theorem). | Asymptotic geometry and conformal types of Carnot--Carathéodory spaces | 14,130 |
The geodesic flow on the tangent bundle is the flow of a certain vector field which is called the spray $S:TM\to TTM$. The flow lines of the vector field $\ka_{TM}\o TS:TTM\to TTTM$ project to the Jacobi fields on $TM$. This could be called the Jacobi flow. | The Jacobi flow | 14,131 |
Let K be an algebraically closed field endowed with a complete non-archimedean norm with valuation ring R. Let f:Y -> X be a map of K-affinoid varieties. In this paper we study the analytic structure of the image f(Y) in X; such an image is a typical example of a subanalytic set. We show that the subanalytic sets are precisely the D-semianalytic sets, where D is the truncated division function first introduced by Denef and van den Dries. This result is most conveniently stated as a Quantifier Elimination result for the valuation ring R in an analytic expansion of the language of valued fields. | Flattening and subanalytic sets in rigid analytic geometry | 14,132 |
We prove the existence of Verdier stratifications for sets definable in any o-minimal structure on (R, +, .). It is also shown that the Verdier condition (w) implies the Whitney condition (b) in o-minimal structures on (R, +, .). As a consequence the Whitney Stratification Theorem holds. The existence of (wf)-stratification of functions definable in polynomially bounded o-minimal structures is presented. | Verdier stratifications and [wf]-stratification in o-minimal structures | 14,133 |
Let g:X -> Y be a smooth (i.e. C^\infty differentiable) map between two smooth manifolds. In analogy with the case of complex polynomial functions, we say that y_0 in Y is a typical value of g if there exists an open neighbourhood U of y_0 in Y, such that the restriction g:g^{-1}(U) -> U is a C^\infty trivial fibration. If y_0 in Y is not a typical value of g, then y_0 is called an atypical value of g. We denote by B_g the bifurcation set of g, i.e. the set of atypical values of g. In the case of a complex polynomial function f:C^n -> C it is known that B_f is a finite set. It was previously proved that the bifurcation sets of real polynomial functions are also finite. The aim of this note is to show that the bifurcation set B_f of a smooth definable function f:R^n -> R is finite . | On the bifurcation sets of functions definable in o-minimal structures | 14,134 |
Let K be an algebraically closed field of characteristic zero, endowed with a complete nonarchimedean norm. Let X be a K-rigid analytic variety and \Sigma a semianalytic subset of X. Then the closure of \Sigma in X with respect to the canonical topology is again semianalytic. The proof uses Embedded Resolution of Singularities. | Closure of rigid semianalytic sets | 14,135 |
Suppose curves are moving by curvature in a plane, but one embeds the plane in $R^3$ and looks at the plane from an angle. Then circles shrinking to a round point would appear to be ellipses shrinking to an ``elliptical point,'' and the surface energy would appear to be anisotropic as would the mobility. The result of this paper is that if one uses the apparent surface energy and the apparent mobility, then the motion by weighted curvature with mobility in the apparent plane is the same as motion by curvature in the original plane but then viewed from the angle. This result applies not only to the isotropic case but to arbitrary surface energy functions and mobilities in the plane, to surfaces in 3-space, and (in the case that the surface energy function is twice differentiable) to the case of motion viewed through distorted lenses (i.e., diffeomorphisms) as well. This result is to be contrasted with an earlier result which states that for area-preserving affine transformations of the plane where the energy and mobility are NOT also transformed, motion by curvature to the power 1/3 (rather than 1) is invariant. | Motion by weighted mean curvature is affine invariant | 14,136 |
We consider a capillary drop that contacts several planar bounding walls so as to produce singularities (vertices) in the boundary of its free surface. It is shown under various conditions that when the number of vertices is less than or equal to three, then the free surface must be a portion of a sphere. These results extend the classical theorem of H. Hopf on constant mean curvature immersions of the sphere. The conclusion of sphericity cannot be extended to more than three vertices, as we show by examples. | Vertex theorems for capillary drops on support planes | 14,137 |
Using the `Riemann Problem with zeros' method, Ward has constructed exact solutions to a (2+1)-dimensional integrable Chiral Model, which exhibit solitons with nontrivial scattering. We give a correspondence between what we conjecture to be all pure soliton solutions and certain holomorphic vector bundles on a compact surface. | Ward's solitions | 14,138 |
We classify simple singularities of functions on space curves. We show that their bifurcation sets have properties very similar to those of functions on smooth manifolds and complete intersections [1,2]: the k(pi, 1)-theorem for the bifurcations diagram of functions is true, and both this diagram and the discriminant are Saito's free divisors. | Functions on space curves | 14,139 |
The ends of a complete embedded minimal surface of {\em finite total curvature} are well understood (every such end is asymptotic to a catenoid or to a plane). We give a similar characterization for a large class of ends of {\em infinite total curvature}, showing that each such end is asymptotic to a helicoid. The result applies, in particular, to the genus one helicoid and implies that it is embedded outside of a compact set in ${\mathbb R}^3$. | Embedded minimal ends asymptotic to the helicoid | 14,140 |
The Chekanov theorem generalizes the classic Lyusternik-Shnirel'man and Morse theorems concerning critical points of a smooth function on a closed manifold. A Legendrian submanifold \Lambda of space of 1-jets of the functions on a manifold M defines a multi-valued function whose graph is the projection of \Lambda in J^0 M = M x R. The Chekanov theorem asserts that if \Lambda is homotopic to the 1-jet of a smooth function in the class of embedded Legendrian manifolds, then such a graph of a multi-valued function must have a lot of points (their number is determined by the topology of M) at which the tangent plane to the graph is parallel to M \times 0. In the present paper a similar estimate is proved for a wider class of Legendrian manifolds. We consider Legendrian manifolds homotopic (in the class of embedded Legendrian manifolds) to Legendrian manifolds specified by generating families. | Generalization of the Chekanov theorem: diameters of immersed manifolds
and wave fronts | 14,141 |
In this paper we introduce a new approach to variational problems on the space Riem(M^n) of Riemannian structures (i.e. isometry classes of Riemannan metrics) on any fixed compact manifold M^n of dimension n >= 5. This approach often enables one to replace the considered variational problem on Riem(M^n) (or on some subset of Riem(M^n)) by the same problem but on spaces Riem(N^n) for every manifold N^n from a class of compact manifolds of the same dimension and with the same homology as M^n but with the following two useful properties: (1) If \nu is any Riemannian structure on any manifold N^n from this class such that Ric_(N^n,\nu) >= -(n-1), then the volume of (N^n,\nu) is greater than one; and (2) Manifolds from this class do not admit Riemannian metrics of non-negative scalar curvature. As a first application we prove a theorem which can be informally explained as follows: Let M be any compact connected smooth manifold of dimension greater than four, M et(M) be the space of isometry classes of compact metric spaces homeomorphic to M endowed with the Gromov-Hausdorff topology, Riem_1(M) in M et(M ) be the space of Riemannian structures on M such that the absolute values of sectional curvature do not exceed one, and R_1(M) denote the closure of Riem_1(M) in M et(M ). Then diameter regarded as a functional on R_1(M) has infinitely many "very deep" local minima. | Variational problems for Riemannian functionals and arithmetic groups | 14,142 |
Regular Lie groups are infinite dimensional Lie groups with the property that smooth curves in the Lie algebra integrate to smooth curves in the group in a smooth way (an `evolution operator' exists). Up to now all known smooth Lie groups are regular. We show in this paper that regular Lie groups allow to push surprisingly far the geometry of principal bundles: parallel transport exists and flat connections integrate to horizontal foliations as in finite dimensions. As consequences we obtain that Lie algebra homomorphisms intergrate to Lie group homomorphisms, if the source group is simply connected and the image group is regular. | Regular infinite dimensional Lie groups | 14,143 |
The Weierstrass representation for spheres in $\R^3$ and, in particular, effective construction of immersions from data of spectral theory origin is discussed. These data are related to Dirac operators on a plane and on an infinite cylinder and these operators are just representations of Dirac operators acting in spinor bundles over the two-sphere which is naturally obtained as a completion of a plane or of a cylinder. Spheres described in terms of Dirac operators with one-dimensional potentials on a cylinder are completely studied and, in particular, for them a lower estimate of the Willmore functional in terms of the dimension of the kernel of the corresponding Dirac operator on a two-sphere is obtained. It is conjectured that this estimate is valid for all Dirac operators on spheres and some reasonings for this conjecture are discussed. In Appendix a criterion distinguishing Weierstrass representations, of universal coverings of compact surfaces of higher genera, converted into immersions of compact surfaces is given. | The Weierstrass representation of spheres in $R^3$, the Willmore
numbers, and soliton spheres | 14,144 |
Lie bialgebra structures are reviewed and investigated in terms of the double Lie algebra, of Manin- and Gau{\ss}-decompositions. The standard R-matrix in a Manin decomposition then gives rise to several Poisson structures on the correponding double group, which is investigated in great detail. | Poisson structures on double Lie groups | 14,145 |
We consider a natural Riemannian metric on the infinite dimensional manifold of all embeddings from a manifold into a Riemannian manifold, and derive its geodesic equation in the case $\Emb(\Bbb R,\Bbb R)$ which turns out to be Burgers' equation. Then we derive the geodesic equation, the curvature, and the Jacobi equation of a right invariant Riemannian metric on an infinite dimensional Lie group, which we apply to $\Diff(\Bbb R)$, $\Diff(S^1)$, and the Virasoro-Bott group. Many of these results are well known, the emphasis is on conciseness and clarity. | Curvature of the Virasoro-Bott group | 14,146 |
One computes the cohomology of the projective embedding of sl(m+1,R) acting on the differential operators on densities on R^m of various weights. This cohomology is non vanishing only for some special critical values of the weights. This allows us first to explain some strange feature pointed out by Gargoubi in his classification if the module of differential operators on the line. This also allows us to provide a so called sl(m+1,R) invariant symbol for differential operators acting on densities of non critical weights. | On the cohomology of sl(m+1,R) acting on differential operators and
sl(m+1,R)-equivariant symbol | 14,147 |
The aim of this paper is to describe a class of conservative systems on $S^2$ possessing an integral cubic in momenta. We prove that this class of systems consists off the case of Goryachev-Chaplygin, the one-parameter family of systems which has been found by the author in the previous paper (dg-ga/9711005) and a new two-parameter family of conservative systems on $S^2$ possessing an integral cubic in momenta. | On the case of Goryachev-Chaplygin and new examples of integrable
conservative systems on S^2 | 14,148 |
We introduce a canonical outer vector field on a Poisson manifold, also due independently to A. Weinstein. We view it as a global section of the sheaf of Poisson vector fields modulo the subsheaf of hamiltonian vector fields. We study this outer derivation mostly in the case of holomorphic Poisson manifolds. | The outer derivation of a complex Poisson manifold | 14,149 |
A closed spin K\"ahler manifold of positive scalar curvature with smallest possible first eigenvalue of the Dirac operator is characterized by holomorphic spinors. It is shown that on any spin K\"ahler-Einstein manifold each holomorphic spinor is a finite sum of eigenspinors of the square of the Dirac operator. Vanishing theorems for holomorphic spinors are proved. | Holomorphic spinors and the Dirac equation | 14,150 |
In this paper we announce a gluing theorem for conformal structures with anti-self-dual (ASD) Weyl tensor that applies in geometrical situations that are more general than those considered by previous authors. By adapting a method proposed by Floer, sufficient conditions are given for the existence of ASD conformal structures on `generalized connected sums' of non-compact ASD 4-manifolds with cylindrical ends. The gluing theorem applies in particular to give results about connected sums of ASD orbifolds along (isolated) singular points. | Gluing theorems for anti-self-dual metrics | 14,151 |
Mathai-Quillen forms are used to give an integral formula for the Lefschetz number of a smooth map of a closed manifold. Applied to the identity map, this formula reduces to the Chern-Gauss-Bonnet theorem. The formula is computed explicitly for constant curvature metrics. There is in fact a one-parameter family of integral expressions. As the parameter goes to infinity, a topological version of the heat equation proof of the Lefschetz fixed submanifold formula is obtained. As the parameter goes to zero and under a transversality assumption, a lower bound for the number of points mapped into their cut locus is obtained. For diffeomorphisms with Lefschetz number unequal to the Euler characteristic, this number is infinite for most metrics, in particular for metrics of non-positive curvature. | Mathai-Quillen forms and Lefschetz theory | 14,152 |
Recently, a set of tools has been developed with the purpose of the study of Quantum Gravity. Until now, there have been very few attempts to put these tools into a rigorous mathematical framework. This is the case, for example, of the so called path bundle of a manifold. It is well known that this topological principal bundle plays the role of a universal bundle for the reconstruction of principal bundles and their connections. The path bundle is canonically endowed with a parallel transport and associated with it important types of derivatives have been considered by several authors: the Mandelstam derivative, the connection derivative and the Loop derivative. In the present article we shall give a unified viewpoint for all these derivatives by developing a differentiable calculus on the path bundle. In particular we shall show that the loop derivative is the curvature of a canonically defined one form that we shall called the universal connection one form. | The loop derivative as a curvature | 14,153 |
Using the higher analytic torsion form of Bismut and Lott we construct a characteristic class for smooth sphere bundles. We calculate this class in the case where the sphere bundle comes from a complex vector bundle. Related to these characteristic classes we define nontrivial continuous group cohomology classes of the diffeomorphism group of odd dimensional spheres. | Higher analytic torsion of sphere bundles and continuous cohomology of
$Diff(S^{2n-1})$ | 14,154 |
We extend our earlier work in [TZ1], where an analytic approach to the Guillemin-Sternberg conjecture [GS] was developed, to cases where the Spin$^c$-complex under consideration is allowed to be further twisted by certain natural exterior power bundles. The main result is a weighted quantization formula in the presence of commuting Hamiltonian actions. The corresponding Morse type inequalities in holomorphic situations are also established. | Symplectic reduction and a weighted multiplicity formula for twisted
Spin$^c$-Dirac operators | 14,155 |
We give a normal form for families of 3-dimensional Poisson structures. This allows us to classify singularities with nonzero 1-jet and typical bifurcations. The Appendix contains corollaries on classification of families of integrable 1-forms on $R^3 | Singularities and bifurcations of 3-dimensional Poisson structure | 14,156 |
A Clifford algebra model for M"obius geometry is presented. The notion of Ribaucour pairs of orthogonal systems in arbitrary dimensions is introduced, and the structure equations for adapted frames are derived. These equations are discretized and the geometry of the occuring discrete nets and sphere congruences is discussed in a conformal setting. This way, the notions of ``discrete Ribaucour congruences'' and ``discrete Ribaucour pairs of orthogonal systems'' are obtained --- the latter as a generalization of discrete orthogonal systems in Euclidean space. The relation of a Cauchy problem for discrete orthogonal nets and a permutability theorem for the Ribaucour transformation of smooth orthogonal systems is discussed. | Orthogonal nets and Clifford algebras | 14,157 |
Let T be the standard torus of revolution in R^3 with radii b and 1, 0<b<1. Let \alpha be a (p,q) torus curve on T. We show that there are points of zero curvature on \alpha for only one value of the variable radius of T, b=p^2/(p^2+q^2). The curve \alpha has non-vanishing curvature for all other values of b. Moreover, for this value of b, there are exactly q points of zero curvature on \alpha. | Torus Curves With Vanishing Curvature | 14,158 |
Geometric problems are usually formulated by means of (exterior) differential systems. In this theory, one enriches the system by adding algebraic and differential constraints, and then looks for regular solutions. Here we adopt a dual approach, which consists to enrich a plane field, as this is often practised in control theory, by adding brackets of the vector fields tangent to it, and then, look for singular solutions of the obtained distribution. We apply this to the isometry problem of rigid geometric structures. | On Gromov's theory of rigid transformation groups: A dual approach | 14,159 |
This paper shows that there are symplectic four-manifolds M with the following property: a single isotopy class of smooth embedded two-spheres in M contains infinitely many Lagrangian submanifolds, no two of which are isotopic as Lagrangian submanifolds. The examples are constructed using a special class of symplectic automorphisms ("generalized Dehn twists"). The proof uses Floer homology. Revised version: one footnote removed, one reference added | Lagrangian two-spheres can be symplectically knotted | 14,160 |
Let M be the cotangent bundle of S^2, with the standard symplectic structure. By adapting an argument of Gromov we determine the weak homotopy type of the group S of those symplectic automorphisms of M which are trivial at infinity. It turns out that S is weakly homotopy equivalent to \Z. \pi_0(S) is generated by the class of the standard "generalized Dehn twist". As a consequence, we show that there are different connected components of S which lie in the same connected component of the corresponding group of diffeomorphisms. | Symplectic automorphisms of T^*S^2 | 14,161 |
Let M be the product of \C P^m and \C P^n, with the standard integral symplectic form. We prove that the inclusion map from the group of symplectic automorphisms of M to its diffeomorphism group is not surjective on homotopy groups. More precisely, it is not surjective on \pi_j for all odd j \leq \max\{2m-1,2n-1\}. This is a weak higher-dimensional analogue of Gromov's results for \C P^1 \times \C P^1. The proof uses parametrized Gromov-Witten invariants in a new (?) way. We also give some information about the symplectic automorphism groups of M with differently weighted product symplectic structures. | On the group of symplectic automorphisms of $\C P^m \times \C P^n$ | 14,162 |
In this letter we investigate some aspects of the noncommutative differential geometry based on derivations of the algebra of endomorphisms of an oriented complex hermitian vector bundle. We relate it, in a natural way, to the geometry of the underlying principal bundle and compute the cohomology of its complex of noncommutative differential forms. | On the Noncommutative Geometry of the Endomorphism Algebra of a Vector
Bundle | 14,163 |
Let $M= G/\Gamma$ be a compact nilmanifold endowed with an invariant complex structure. We prove that, on an open set of any connected component of the moduli space ${\cal C} ({\frak g})$ of invariant complex structures on $M$, the Dolbeault cohomology of $M$ is isomorphic to the one of the differential bigraded algebra associated to the complexification $\cg^\C$ of the Lie algebra of $G$. To obtain this result, we first prove the above isomorphism for compact nilmanifolds endowed with a rational invariant complex structure. This is done using a descending series associated to the complex structure and the Borel spectral sequences for the corresponding set of holomorphic fibrations. Then we apply the theory of Kodaira-Spencer for deformations of complex structures. | Dolbeault Cohomology of compact Nilmanifolds | 14,164 |
We formulate a more conceptual interpretation of the Cappell-Lee-Miller glueing/splitting theorem using the new language of asymptotic maps and asymptotic exactness. Additionally, we present an asymptotic description of the Mayer-Vietoris sequence naturally associated to the Cech cohomology of the sheaf of local solutions of a Dirac type operator. We discuss applications to eigenvalue estimates, approximation of obstruction bundles and glueing of determinant line bundles frequently arising in gauge theoretic problems. The operators involved in all these results need not be translation invariant. | On the Cappell-Lee-Miller glueing theorem | 14,165 |
We determine the Fukaya Floer homology of the three-manifold which is the product of a Riemann surface of genus $g\geq 1$ times the circle. This sets up the groundwork for finding the structure of the Donaldson invariants of four-manifolds not of simple type in the future. We give the following applications: 1) We show that every four-manifold with $b^+>1$ is of finite type. 2) Some results relevant to Donaldson invariants of connected sums along surfaces. 3) We find the invariants of the product of two Riemann surfaces both of genus greater or equal than one. | Fukaya Floer homology of $Σ\times S^1$ and applications | 14,166 |
The following version of a conjecture of Fischer-Colbrie and Schoen is proved: If M is a complete Riemannian 3-manifold with nonnegative scalar curvature which contains a two-sided torus S which is of least area in its isotopy class then M is flat. This follows from a local version derived in the paper. | Rigidity of area minimizing tori in 3-manifolds of nonnegative scalar
curvature | 14,167 |
We prove that the ring $\Aff{\R}{M}$ of all polynomials defined on a real algebraic variety $M\subset\R^n$ is dense in the Hilbert space $L^2(M,e^{-|x|^2}\de\mu)$, where $\de\mu$ denotes the volume form of $M$ and $\de\nu=e^{-|x|^2}\de\mu$ the Gaussian measure on $M$. | The Gaussian Measure On Algebraic Varieties | 14,168 |
Let X be a Hermitian locally symmetric space. We prove that every Chern class of X has a canonical lift to the cohomology of the Baily- Borel-Satake compactification X* of X and that the resulting Chern numbers satisfy the Hirzebruch proportionality formula with respect to the compact dual X^ of X. The same result holds for any automorphic vector bundle over X in place of the tangent bundle. As a consequence there is a surjection of the subalgebra of H*(X*) generated by these lifted classes onto H*(X^). The method of proof is to construct fiberwise flat connections on these bundles near the singular strata of X*, where one then finds de Rham representatives of the Chern classes which are pulled back from the strata. | Chern classes of modular varieties | 14,169 |
This paper constructs a Hodge theory of noncompact topologically tame manifolds $M$. The main result is an isomorphism between the de Rham cohomology with compact supports of $M$ and the kernel of the Hodge--Witten--Bismut Laplacian $\lap_\mu$ associated to a measure $d\mu$ which has sufficiently rapid growth at infinity on $M$. This follows from the construction of a space of forms associated to $\lap_\mu$ which satisfy an ``extension by zero'' property. The ``extension by zero'' property is proved for manifolds with cylindrical ends possessing gaussian growth measures. | Hodge theory and cohomology with compact supports | 14,170 |
It is demonstrated that the stationary Veselov-Novikov (VN) and the stationary modified Veselov-Novikov (mVN) equations describe one and the same class of surfaces in projective differential geometry: the so-called isothermally asymptotic surfaces, examples of which include arbitrary quadrics and cubics, quartics of Kummer, projective transforms of affine spheres and rotation surfaces. The stationary mVN equation arises in the Wilczynski approach and plays the role of the projective "Gauss-Codazzi" equations, while the stationary VN equation follows from the Lelieuvre representation of surfaces in 3-space. This implies an explicit Backlund transformation between the stationary VN and mVN equations which is an analog of the Miura transformation between their (1+1)-dimensional limits. | Stationary Veselov-Novikov equation and isothermally asymptotic surfaces
in projective differential geometry | 14,171 |
An explicit construction of surfaces with flat normal bundle in the Euclidean space (unit hypersphere) in terms of solutions of certain linear system is proposed. In the case of 3-space our formulae can be viewed as the direct Lie sphere analog of the generalized Weierstrass representation of surfaces in conformal geometry or the Lelieuvre representation of surfaces in the affine space. An explicit parametrization of Ribaucour congruences of spheres by three solutions of the linear system is obtained. In view of the classical Lie correspondence between Ribaucour congruences and surfaces with flat normal bundle in the Lie quadric this gives an explicit representation of surfaces with flat normal bundle in the 4-dimensional space form of the Lorentzian signature. Direct projective analog of this construction is the known parametrization of W-congruences by three solutions of the Moutard equation. Under the Pl\"ucker embedding W-congruences give rise to surfaces with flat normal bundle in the Pl\"ucker quadric. Integrable evolutions of surfaces with flat normal bundle and parallels with the theory of nonlocal Hamiltonian operators of hydrodynamic type are discussed in the conclusion. | Surfaces with flat normal bundle: an explicit construction | 14,172 |
New universal invariant operators are introduced in a class of geometries which include the quaternionic structures and their generalisations as well as 4-dimensional conformal (spin) geometries. It is shown that, in a broad sense, all invariants and invariant operators arise from these universal operators and that they may be used to reduce all invariants problems to corresponding algebraic problems involving homomorphisms between modules of certain parabolic subgroups of Lie groups. Explicit application of the operators is illustrated by the construction of all non-standard operators between exterior forms on a large class of the geometries which includes the quaternionic structures. | Invariant local twistor calculus for quaternionic structures and related
geometries | 14,173 |
It is shown that the geometry of locally homogeneous multisymplectic manifolds (that is, smooth manifolds equipped with a closed nondegenerate form of degree > 1, which is locally homogeneous of degree k with respect to a local Euler field) is characterized by their automorphisms. Thus, locally homogeneous multisymplectic manifolds extend the family of classical geometries possessing a similar property: symplectic, volume and contact. The proof of the first result relies on the characterization of invariant differential forms with respect to the graded Lie algebra of infinitesimal automorphisms, and on the study of the local properties of Hamiltonian vector fields on locally multisymplectic manifolds. In particular it is proved that the group of multisymplectic diffeomorphisms acts (strongly locally) transitively on the manifold. It is also shown that the graded Lie algebra of infinitesimal automorphisms of a locally homogeneous multisymplectic manifold characterizes their multisymplectic diffeomorphisms. | Invariant Forms and Automorphisms of Locally Homogeneous Multisymplectic
Manifolds | 14,174 |
We compute eta invariants of various Dirac type operators on circle bundles over Riemann surfaces via two approaches: an adiabatic approach based on the results of Bismut-Cheeger-Dai and a direct elementary one. These results, coupled with some delicate spectral flow computations are then used to determine the virtual dimensions of Seiberg-Witten finite energy moduli spaces on any 4-manifold bounding unions of circle bundles. This belated paper should be regarded as the analytical backbone of dg-ga/9711006. There, we indicated only what changes are needed to extend the methods of the present paper to Seifert fibrations and we focused only to topological and number theoretic aspects related to Froyshov invariants | Eta invariants of Dirac operators on Circle bundles over Riemann
surfaces and virtual dimensions of finite energy Seiberg-Witten moduli spaces | 14,175 |
An equifocal submanifold M of a symmetric space N of compact type induces a foliation with singular leaves on N. In this paper we will show how to reconstruct the equifocal foliation starting from one of the singular leaves, the so-called focal manifolds. To be more concrete: The equifocal submanifold is equal to a partial tube B around the focal manifold and we will show how to construct B in this paper. Moreover, we will find a geometrical characterization of focal manifolds. | Equifocal families in symmetric spaces of compact type | 14,176 |
The L^2-torsion is an invariant defined for compact L^2-acyclic manifolds of determinant class, for example odd dimensional hyperbolic manifolds. It was introduced by John Lott and Varghese Mathai and computed for hyperbolic manifolds in low dimensions. In this paper we show that the L^2-torsion of hyperbolic manifolds of arbitrary odd dimension does not vanish. This was conjectured by J. Lott and W. Lueck. Some concrete values are computed and an estimate of their growth with the dimension is given. | L2-torsion of hyperbolic manifolds | 14,177 |
In the present paper we study locally semiflat (we also call them semiintegrable) almost Grassmann structures. We establish necessary and sufficient conditions for an almost Grassmann structure to be alpha- or beta-semiintegrable. These conditions are expressed in terms of the fundamental tensors of almost Grassmann structures. Since we are not able to prove the existence of locally semiflat almost Grassmann structures in the general case, we give many examples of alpha- and beta-semiintegrable structures, mostly four-dimensional. For all examples we find systems of differential equations of the families of integral submanifolds V_alpha and V_beta of the distributions Delta_alpha and Delta_beta of plane elements associated with an almost Grassmann structure. For some examples we were able to integrate these systems and find closed form equations of submanifolds V_alpha and V_beta. | Semiintegrable almost Grassmann structures | 14,178 |
The main results on the theory of conformal and almost Grassmann structures are presented. The common properties of these structures and also the differences between them are outlined. In particular, the structure groups of these structures and their differential prolongations are found. A complete system of geometric objects of the almost Grassmann structure totally defining its geometric structure is determined. The vanishing of these objects determines a locally Grassmann manifold. It is proved that the integrable almost Grassmann structures are locally Grassmann. The criteria of semiintegrability of almost Grassmann structures is proved in invariant form. | Conformal and Grassmann structures | 14,179 |
The differential geometry of almost Grassmann structures defined on a differentiable manifold of dimension n = pq by a fibration of Segre cones SC (p, q) is studied. The peculiarities in the structure of almost Grassmann structures for the cases p=q=2; p = 2, q > 2 (or p > 2, q = 2), and p > 2, q > 2 are clarified. The fundamental geometric objects of these structures up to fourth order are derived. The conditions under which an almost Grassmann structure is locally flat or locally semiflat are found for all cases indicated above. | On the theory of almost Grassmann structures | 14,180 |
For a hypersurface V of a conformal space, we introduce a conformal differential invariant I = h^2/g, where g and h are the first and the second fundamental forms of V connected by the apolarity condition. This invariant is called the conformal quadratic element of V. The solution of the problem of conformal rigidity is presented in the framework of conformal differential geometry and connected with the conformal quadratic element of V. The main theorem states: Let n \geq 4 and V and V' be two nonisotropic hypersurfaces without umbilical points in a conformal space C^n or a pseudoconformal space C^n_q of signature (p, q), p = n - q. Suppose that there is a one-to-one correspondence f: V ---> V' between points of these hypersurfaces, and in the corresponding points of V and V' the following condition holds: I' = f_* I, where f_*: T (V) ---> T (V) is a mapping induced by the correspondence f. Then the hypersurfaces V and V' are conformally equivalent. | A conformal differential invariant and the conformal rigidity of
hypersurfaces | 14,181 |
The authors study singular points of lightlike hypersurfaces of the de Sitter space S^{n+1}_1 and the geometry of hypersurfaces and use them for construction of an invariant normalization and an invariant affine connection of lightlike hypersurfaces. | Singular points of lightlike hypersurfaces of the de Sitter space | 14,182 |
In this paper we will prove new extrinsic upper bounds for the eigenvalues of the Dirac operator on an isometrically immersed surface $M^2 \hookrightarrow {\Bbb R}^3$ as well as intrinsic bounds for 2-dimensional compact manifolds of genus zero and genus one. Moreover, we compare the different estimates of the eigenvalue of the Dirac operator for special families of metrics. | Upper bounds for the first eigenvalue of the Dirac operator on surfaces | 14,183 |
There are three types of hypersurfaces in a pseudoconformal space C^n_1 of Lorentzian signature: spacelike, timelike, and lightlike. These three types of hypersurfaces are considered in parallel. Spacelike hypersurfaces are endowed with a proper conformal structure, and timelike hypersurfaces are endowed with a conformal structure of Lorentzian type. Geometry of these two types of hypersurfaces can be studied in a manner that is similar to that for hypersurfaces of a proper conformal space. Lightlike hypersurfaces are endowed with a degenerate conformal structure. This is the reason that their investigation has special features. It is proved that under the Darboux mapping such hypersurfaces are transferred into tangentially degenerate (n-1)-dimensional submanifolds of rank n-2 located on the Darboux hyperquadric. The isotropic congruences of the space C^n_1 that are closely connected with lightlike hypersurfaces and their Darboux mapping are also considered. | On geometry of hypersurfaces of a pseudoconformal space of Lorentzian
signature | 14,184 |
On the Grassmann manifold G (m, n) of m-dimensional subspaces of an n-dimensional projective space P^n, a certain supplementary construction called the normalization is considered. By means of this normalization, one can construct the structure of a Riemannian or semi-Riemannian manifold or an affine connection on G(m, n). | On a normalization of a Grassmann manifold | 14,185 |
We develop Teichmuller theoretical methods to construct new minimal surfaces in $\BE^3$ by adding handles and planar ends to existing minimal surfaces in $\BE^3$. We exhibit this method on an interesting class of minimal surfaces which are likely to be embedded, and have a low degree Gau\ss map for their genus; the (Weierstrass data) period problem for these surfaces is of arbitrary dimension. In particular, we exhibit a two-parameter family of complete minimal surfaces in the Euclidean three-space $\BE^3$ which generalize the breakthrough minimal surface of C. Costa; these new surfaces are embedded (at least) outside a compact set, and are indexed (roughly) by the number of ends they have and their genus. They have at most eight self-symmetries despite being of arbitrarily large genus, and are interesting for a number of reasons. Moreover, our methods also extend to prove that some natural candidate classes of surfaces cannot be realized as minimal surfaces in $\BE^3$. As a result of both aspects of this work, we obtain a classification of a family of surfaces as either realizable or unrealizable as minimal surfaces. | Teichmuller theory and handle addition for minimal surfaces | 14,186 |
Asymptotic subcone of an unbounded metric space is another metric space, capturing the structure of the original space at infinity. In this paper we define a functional metric space S which is an asymptotic subcone of the hyperbolic plane. This space is a real tree branching at every its point. Moreover, it is a homogeneous metric space such that any real tree with countably many vertices can be isometrically embedded into it. This implies that every such tree is also an asymptotic subcone of the hyperbolic plane. | On the asymptotic geometry of the hyperbolic plane | 14,187 |
We present the solution of a longstanding internal problem of noncommutative geometry, namely the computation of the index of a transversally elliptic operator on an arbitrary foliation. The new and crucial ingredient is a certain Hopf algebra associated to the transverse frame bundle. Its cyclic cohomology is defined and shown to be canonically isomorphic to the Gelfand-Fuks cohomology. | Hopf algebras, cyclic cohomology and the transverse index theory | 14,188 |
Ray Singer torsion is a numerical invariant associated with a compact Riemannian manifold equipped with a flat bundle and a Hermitian structure on this bundle. In this note we show how one can remove the dependence on the Riemannian metric and on the Hermitian structure with the help of a base point and of an Euler structure, in order to obtain a topological invariant. A numerical invariant for an Euler structure and additional data is also constructed. | Removing Metric Anomalies from Ray Singer Torsion | 14,189 |
Witten- Helffer-Sj\"ostrand theory is a considerable addition to the De Rham- Hodge theory for Riemannian manifolds and can serve as a general tool to prove results about comparison of numerical invariants associated to compact manifolds analytically, i.e. by using a Riemannian metric, or combinatorially, i.e by using a triangulation. In this presentation a triangulation, or a partition of a smooth manifold in cells, will be viewed in a more analytic spirit, being provided by the stable manifolds of the gradient of a nice Morse function. WHS theory was recently used both for providing new proofs for known but difficult results in topology, as well as new results and a positive solution for an important conjecture about $L_2-$torsion, cf [BFKM]. This presentation is a short version of a one quarter course I have given during the spring of 1997 at OSU. | Lectures on Witten Helffer Sjöstrand Theory | 14,190 |
It is proved that the geometry of lightlike hypersurfaces of the de Sitter space S^{n+1}_1 is directly connected with the geometry of hypersurfaces of the conformal space C^n. This connection is applied for a construction of an invariant normalization and an invariant affine connection of lightlike hypersurfaces as well as for studying singularities of lightlike hypersurfaces. | The geometry of lightlike hypersurfaces of the de Sitter space | 14,191 |
The geometry of canal hypersurfaces of an n-dimensional conformal space C^n is studied. Such hypersurfaces are envelopes of r-parameter families of hyperspheres, 1 \leq r \leq n-2. In the present paper the conditions that characterize canal hypersurfaces, and which were known earlier, are made more precise. The main attention is given to the study of the Darboux maps of canal hypersurfaces in the de Sitter space M_1^{n+1} and the projective space P^{n+1}. To canal hypersurfaces there correspond r-dimensional spacelike tangentially nondegenerate submanifolds in M_1^{n+1} and tangentially degenerate hypersurfaces of rank r in P^{n+1}. In this connection the problem of existence of singular points on canal hypersurfaces is considered. | The Darboux mapping of canal hypersurfaces | 14,192 |
We consider the asymptotic behaviour of positive solutions u of the conformal scalar curvature equation, \Delta u + n(n-2)/4 u^{(n+2)(n-2) = 0, in the neighbourhood of isolated singularities in the standard Euclidean ball. Although asymptotic radial symmetry for such solutions was proved some time ago, by Caffarelli, Gidas and Spruck, we present a much simpler and more geometric derivation of this fact. We also discuss a refinement, showing that any such solution is asymptotic to one of the deformed radial singular solutions. Finally we give some applications of these refined asymptotics, first to computing the global Pohozaev invariants of solutions on the sphere with isolated singularities, and then to the regularity of the moduli space of all such solutions. | Refined asymptotics for constant scalar curvature metrics with isolated
singularities | 14,193 |
We construct a new class of complete constant mean curvature surfaces in R^3. These are geometrically different than the surfaces constructed by Kapouleas' gluing technique. These are obtained by piecing together half-Delaunay surfaces to the truncations of minimal k-noids. The gluing techniques are new: the surfaces are obtained by matching Cauchy data from the infinite dimensional family of exact solutions of the problem on each of the component pieces. These surfaces are also proved to be nondegenerate in the CMC moduli space. | Constant mean curvature surfaces with Delaunay ends | 14,194 |
Let X be a closed Riemannian manifold and let H\hookrightarrow X be an embedded hypersurface. Let X=X_+ \cup_H X_- be a decomposition of X into two manifolds with boundary, with X_+ \cap X_- = H. In this expository article, surgery -- or gluing -- formul\ae for several geometric and spectral invariants associated to a Dirac-type operator \eth_X on X are presented. Considered in detail are: the index of \eth_X, the index bundle and the determinant bundle associated to a family of such operators, the eta invariant and the analytic torsion. In each case the precise form of the surgery theorems, as well as the different techniques used to prove them, are surveyed. | Dirac operators, heat kernels and microlocal analysis Part II: analytic
surgery | 14,195 |
An embedded cubic graph consisting of segments of geodesics such that the angles at any vertex are equal to $2\pi/3$ is a closed local minimal net. This net is regular if all segments of geodesics are equal. The problem of classification of closed local minimal nets on surfaces of constant negative curvature has been formulated in the context of the famous Plateau problem in the one-dimensional case. In this paper we prove an asymptotic for $\sharp (W^r(g))$ as $g\to +\infty$ where $g$ is genus and $W^r(g)$ is the set of the regular single-face minimal nets on surfaces of curvature -1. Then we construct some examples of $f$-face regular nets, $f>1$. | Regular minimal nets on surfaces of constant negative curvature | 14,196 |
The coherent states are viewed as a powerful tool in differential geometry. It is shown that some objects in differential geometry can be expressed using quantities which appear in the construction of the coherent states. The following subjects are discussed via the coherent states: the geodesics, the conjugate locus and the cut locus; the divisors; the Calabi's diastasis and its domain of definition; the Euler-Poincar\'e characteristic of the manifold, the number of Borel-Morse cells, Kodaira embeding theorem.... | Coherent states and geometry | 14,197 |
For homogeneous simply connected Hodge manifolds it is proved that the set of coherent vectors orthogonal to a given one is the divisor responsible for the homogeneous holomorphic line bundle of the coherent vectors. In particular, for naturally reductive spaces, the divisor is the cut locus. | Coherent states, line bundles and divisors | 14,198 |
Generalizations of the classical affine Lelieuvre formula to surfaces in projective three-dimensional space and to hypersurfaces in multi- dimensional projective space are given. A discrete version of the projective Lelieuvre formula is presented too. | Projective generalizations of Lelieuvre's formula | 14,199 |
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