Non-radial clustered spike solutions for semilinear elliptic problems on s<Superscript>N</Superscript>

We consider the superlinear elliptic equation on Sⁿ

Regularity of volume-minimizing flows on 3-manifolds

In Johnson and Smith (Indiana Univ Math J 44:45–85, 1995; Ann Global Anal Geometry 30:239–287, 2006; Proceedings of the VII International Colloquium on Differential Geometry, 1994, World Scientific, pp. 81–98), the authors characterized the singular set (discontinuities of the graph) of a volume-minimizing rectifiable section of a fiber bundle, showing that, except under certain circumstances, there exists a volume-minimizing rectifiable section with the singular set lying over a codimension-3 set in the base space. In particular, it was shown that for 2-sphere bundles over 3-manifolds, a minimizer exists with a discrete set of singular points. In this article, we show that for a 2-sphere bundle over a compact 3-manifold, such a singular point cannot exist. As a corollary, for any compact 3-manifold, there is a C ¹ volume-minimizing one-dimensional foliation. In addition, this same analysis is used to show that the examples, due to Pedersen (Trans Am Math Soc 336:69–78, 1993), of potentially volume-minimizing rectifiable sections (rectifiable foliations) of the unit tangent bundle to S ²ⁿ⁺¹ are not, in fact, volume minimizing.      

A general approach to index theorems for holomorphic maps and foliations

Let M be a smooth complex manifold, and S(⊂ M) be a compact irreducible subvariety with dim _C S > 0. Let be given either a holomorphic map f : M → M with f _|S  = id _S , f ≠ id _M , or a holomorphic foliation on M: we describe an approach that can be applied to both map and foliation in order to obtain index theorems. Partially supported by GNSAGA, Centro de Giorgi, M.U.R.S.T.     

Hypersurfaces of restricted type in Minkowski space

A submanifold M _n ^r of Minkowski space is said to be of restricted type if its shape operator with respect to the mean curvature vector is the restriction of a fixed linear transformation of to the tangent space of M _n ^r at every point of M _n ^r . In this paper we completely classify hypersurfaces of restricted type in . More precisely, we prove that a hypersurface of is of restricted type if and only if it is either a minimal hypersurface, or an open part of one of the following hypersurfaces: S ^k × , S _k ¹ × , H ^k × , S _n ¹ , H ⁿ , with 1kn–1, or an open part of a cylinder on a plane curve of restricted type.This work was done when the first and fourth authors were visiting Michigan State University.Aangesteld Navorser N.F.W.O., Belgium.     

A splitting theorem for open nonnegatively curved manifolds

We prove that an open nonnegatively curved manifold Mⁿ with soul S^k splits isometrically as S × ^2–k if (and only if) the holonomy group of the normal bundle NS is trivial.Supported by the Heinrich Hertz foundation     

Non-linear Grassmannians as coadjoint orbits

For a given manifold M we consider the non-linear Grassmann manifold Gr _n (M) of n–dimensional submanifolds in M. A closed (n+2)–form on M gives rise to a closed 2–form on Gr _n (M). If the original form was integral, the 2–form will be the curvature of a principal S ¹ –bundle over Gr _n (M). Using this S ¹ –bundle one obtains central extensions for certain groups of diffeomorphisms of M. We can realize Gr _m–2 (M) as coadjoint orbits of the extended group of exact volume preserving diffeomorphisms and the symplectic Grassmannians SGr _2k (M) as coadjoint orbits in the group of Hamiltonian diffeomorphisms. Mathematics Subject Classification (2000):58B20Both authors are supported by the Fonds zur Förderung der wissenschaftlichen Forschung (Austrian Science Fund), project number P14195-MAT     

Minimal sets of Cartan foliations

A foliation that admits a Cartan geometry as its transversal structure is called a Cartan foliation. We prove that on a manifold M with a complete Cartan foliation ?, there exists one more foliation (M, \(\mathcal{O}\)), which is generally singular and is called an aureole foliation; moreover, the foliations ? and \(\mathcal{O}\) have common minimal sets. By using an aureole foliation, we prove that for complete Cartan foliations of the type ?/? with a compactly embedded Lie subalgebra ? in ?, the closure of each leaf forms a minimal set such that the restriction of the foliation onto this set is a transversally locally homogeneous Riemannian foliation. We describe the structure of complete transversally similar foliations (M, ?). We prove that for such foliations, there exists a unique minimal set ?, and ? is contained in the closure of any leaf. If the foliation (M, ?) is proper, then ? is a unique closed leaf of this foliation.     

Commutative Algebras of Toeplitz Operators on the Reinhardt Domains

Let D be a bounded logarithmically convex complete Reinhardt domain in centered at the origin. Generalizing a result for the one-dimensional case of the unit disk, we prove that the C ^*-algebra generated by Toeplitz operators with bounded measurable separately radial symbols (i.e., symbols depending only on is commutative. We show that the natural action of the n-dimensional torus defines (on a certain open full measure subset of D) a foliation which carries a transverse Riemannian structure having distinguished geometric features. Its leaves are equidistant with respect to the Bergman metric, and the orthogonal complement to the tangent bundle of such leaves is integrable to a totally geodesic foliation. Furthermore, these two foliations are proved to be Lagrangian. We specify then the obtained results for the unit ball.     

Examples of scalar-flat hypersurfaces in $${\mathbb{R}^{n+1}}$$

Given a hypersurface M of null scalar curvature in the unit sphere , n ≥ 4, such that its second fundamental form has rank greater than 2, we construct a singular scalar-flat hypersurface in as a normal graph over a truncated cone generated by M. Furthermore, this graph is 1-stable if the cone is strictly 1-stable.     

Extremal <Emphasis Type="Italic">G</Emphasis>-invariant eigenvalues of the Laplacian of <Emphasis Type="Italic">G</Emphasis>-invariant metrics

The study of extremal properties of the spectrum often involves restricting the metrics under consideration. Motivated by the work of Abreu and Freitas in the case of the sphere S ² endowed with S ¹-invariant metrics, we consider the subsequence of the spectrum of a Riemannian manifold M which corresponds to metrics and functions invariant under the action of a compact Lie group G. If G has dimension at least 1, we show that the functional λ _k ^G admits no extremal metric under volume-preserving G-invariant deformations. If, moreover, M has dimension at least three, then the functional is unbounded when restricted to any conformal class of G-invariant metrics of fixed volume. As a special case of this, we can consider the standard O(n)-action on S ⁿ ; however, if we also require the metric to be induced by an embedding of S ⁿ in , we get an optimal upper bound on .      

Inverse-preserving Linear Maps Between Spaces of Matrices over Fields

Suppose F is a field different from F2, the field with two elements. Let Mn（F） and Sn（F） be the space of n × n full matrices and the space of n ×n symmetric matrices over F, respectively. For any G1, G2 ∈ {Sn（F）, Mn（F）}, we say that a linear map f from G1 to G2 is inverse-preserving if f（X）^-1 = f（X^-1） for every invertible X ∈ G1. Let L （G1, G2） denote the set of all inverse-preserving linear maps from G1 to G2. In this paper the sets .L（Sn（F）,Mn（F））, L（Sn（F）,Sn（F））, L （Mn（F）,Mn（F）） and L（Mn （F）, Sn （F）） are characterized.     

Para-tt*-bundles on the tangent bundle of an almost para-complex manifold

In this paper we study para-tt ^*-bundles (TM, D, S) on the tangent bundle of an almost para-complex manifold (M, τ). We characterise those para-tt ^*-bundles with ${\nabla=D + S}In this paper we study para-tt ^*-bundles (TM, D, S) on the tangent bundle of an almost para-complex manifold (M, τ). We characterise those para-tt ^*-bundles with induced by the one-parameter family of connections given by and prove a uniqueness result for solutions with a para-complex connection D. Flat nearly para-K?hler manifolds and special para-complex manifolds are shown to be such solutions. We analyse which of these solutions admit metric or symplectic para-tt ^*-bundles. Moreover, we give a generalisation of the notion of a para-pluriharmonic map to maps from almost para-complex manifolds (M, τ) into pseudo-Riemannian manifolds and associate to the above metric and symplectic para-tt ^*-bundles generalised para-pluriharmonic maps into , respectively, into SO ₀(n,n)/U ^π(C ⁿ ), where U ^π(C ⁿ ) is the para-complex analogue of the unitary group.      

Foliations with One-Sided Branching

We generalize the main results from the author's paper in Geom. Topol. 4 (2000), 457–515 and from Thurston's eprint math.GT/9712268 to taut foliations with one-sided branching. First constructed by Meigniez, these foliations occupy an intermediate position between -covered foliations and arbitrary taut foliations of 3-manifolds. We show that for a taut foliation with one-sided branching of an atoroidal 3-manifold M, one can construct a pair of genuine laminations ^± of M transverse to with solid torus complementary regions which bind every leaf of in a geodesic lamination. These laminations come from a universal circle, a refinement of the universal circles proposed by Thurston (unpublished), which maps monotonely and ₁(M)-equivariantly to each of the circles at infinity of the leaves of , and is minimal with respect to this property. This circle is intimately bound up with the extrinsic geometry of the leaves of . In particular, let denote the pulled-back foliation of the universal cover, and co-orient so that the leaf space branches in the negative direction. Then for any pair of leaves of with , the leaf is asymptotic to in a dense set of directions at infinity. This is a macroscopic version of an infinitesimal result from Thurston and gives much more drastic control over the topology and geometry of , than is achieved by him. The pair of laminations ^± can be used to produce a pseudo-Anosov flow transverse to which is regulating in the nonbranching direction. Rigidity results for ^± in the -covered case are extended to the case of one-sided branching. In particular, an -covered foliation can only be deformed to a foliation with one-sided branching along one of the two laminations canonically associated to the -coveredfoliation constructed in Geom. Topol. 4 (2000), 457–515, and these laminations become exactly the laminations ^± for the new branched foliation. Other corollaries include that the ambient manifold is -hyperbolic in the sense of Gromov, and that a self-homeomorphism of this manifold homotopic to the identity is isotopic to the identity.     

A Ricci inequality for hypersurfaces in the sphere

Let Mⁿ be a complete Riemannian manifold immersed isometrically in the unity Euclidean sphere In [9], B. Smyth proved that if Mⁿ, n ≧ 3, has sectional curvature K and Ricci curvature Ric, with inf K > −∞, then sup Ric ≧ (n − 2) unless the universal covering of Mⁿ is homeomorphic to Rn or homeomorphic to an odd-dimensional sphere. In this paper, we improve the result of Smyth. Moreover, we obtain the classification of complete hypersurfaces of with nonnegative sectional curvature.Received: 11 November 2003     

On the solution of singular linear systems of algebraic equations by semiiterative methods

Summary For a square matrixT ^n,n , where (I–T) is possibly singular, we investigate the solution of the linear fixed point problemx=T x+c by applying semiiterative methods (SIM's) to the basic iterationx ₀ ⁿ ,x _k T c _k–1+c(k1). Such problems arise if one splits the coefficient matrix of a linear systemA x=b of algebraic equations according toA=M–N (M nonsingular) which leads tox=M ^–1 N x+M ^–1 bT x+c. Even ifx=T x+c is consistent there are cases where the basic iteration fails to converge, namely ifT possesses eigenvalues 1 with ||1, or if =1 is an eigenvalue ofT with nonlinear elementary divisors. In these cases — and also ifx=T x+c is incompatible — we derive necessary and sufficient conditions implying that a SIM tends to a vector which can be described in terms of the Drazin inverse of (I–T). We further give conditions under which is a solution or a least squares solution of (I–T)x=c.Research supported in part by the Alexander von Humboldt-Stiftung     

Simultaneous uniformization for the leaves of projective foliations by curves

In this paper we prove that, given a holomorphic foliation by curves on P ⁿ , of degree 2, whose singularities have nondegenerate linear part, then there exists a hermitian metricg on P ⁿ -S (S=singular set) which is complete and induces strictly negative Gaussian curvature on the leaves of the foliation (Theorem B). This implies, in particular, that all leaves of the foliation are uniformized by the unit disc and that the set of uniformizations of the leaves is paracompact (Theorem A). We obtain also some consequences concerning the non existence of vanishing cycles in the sense of Novikov, the equivalence of the existence of a parabolic element in the group of deck transformations of the leaf and of a separatrix in the leaf, etc...     

Polytopal and nonpolytopal spheres an algorithmic approach

The convexity theory for oriented matroids, first developed by Las Vergnas [17], provides the framework for a new computational approach to the Steinitz problem [13]. We describe an algorithm which, for a given combinatorial (d − 2)-sphereS withn vertices, determines the setC _d,n(S) of rankd oriented matroids withn points and face latticeS. SinceS is polytopal if and only if there is a realizableM εC _d,n(S), this method together with the coordinatizability test for oriented matroids in [10] yields a decision procedure for the polytopality of a large class of spheres. As main new result we prove that there exist 431 combinatorial types of neighborly 5-polytopes with 10 vertices by establishing coordinates for 98 “doubted polytopes” in the classification of Altshuler [1]. We show that for alln ≧k + 5 ≧8 there exist simplicialk-spheres withn vertices which are non-polytopal due to the simple fact that they fail to be matroid spheres. On the other hand, we show that the 3-sphereM ₉₆₃ ⁹ with 9 vertices in [2] is the smallest non-polytopal matroid sphere, and non-polytopal matroidk-spheres withn vertices exist for alln ≧k + 6 ≧ 9.     

Maximum-cover source location problems with objective edge-connectivity three

Given a graph G = (V, E), a set of vertices covers a vertex if the edge-connectivity between S and v is at least a given number k. Vertices in S are called sources. The maximum-cover source location problem, which is a variation of the source location problem, is to find a source set S with a given size at most p, maximizing the sum of the weight of vertices covered by S. In this paper, we show a polynomial-time algorithm for this problem in the case of k = 3 for a given undirected graph with a vertex weight function and an edge capacity function. Moreover, we show that this problem is NP-hard even if vertex weights and edge capacities are both uniform for general k.     

A finiteness theorem for transitive foliations and flat vector bundles

In this paper we prove a finiteness theorem for the spectral sequence (E _i(∇), (d _∇)_i) associated to a transitive foliation on a compact manifoldM, and to a flat vector bundleE overM with flat connection ∇. we also compute some examples of homogeneous Lie foliations on compact connected homogeneous spaces.     

Constant mean curvature hypersurfaces with two principal curvatures in a sphere

In this paper we consider a compact oriented hypersurface M ⁿ with constant mean curvature H and two distinct principal curvatures λ and μ with multiplicities (n − m) and m, respectively, immersed in the unit sphere S ⁿ⁺¹. Denote by the trace free part of the second fundamental form of M ⁿ , and Φ be the square of the length of . We obtain two integral formulas by using Φ and the polynomial . Assume that B _H,m is the square of the positive root of P _H,m (x) = 0. We show that if M ⁿ is a compact oriented hypersurface immersed in the sphere S ⁿ⁺¹ with constant mean curvatures H having two distinct principal curvatures λ and μ then either or . In particular, M ⁿ is the hypersurface .