Kähler–Ricci Flow With Small Initial Energy

In this paper, we prove that the K?hler–Ricci flow converges to a K?hler–Einstein metric when E ₁ energy is small. We also prove that E ₁ is bounded from below if and only if the K-energy is bounded from below in the canonical class. The first named author is partially supported by a NSF grant, while the third author was partially supported by a NSF supplement grant.     

Compact deformations of Fuchsian groups

A conformal map Φ on the unit disk is called a compact deformation of a Fuchsian groupG if Φ has a quasiconformal extension to the planeh which conjugatesG to a Kleinian group G′ and the dilatation ofh is compactly supported moduloG. We show that for such deformations δ = dim(∧(G′)) = dim(∧_c(G′)) (if δ ≥1) and the image of ∧_e = ∧ ∧_c is contained in a countable union of rectifiable curves and has zero length iffG is divergence type. The first author is partially supported by NSF Grant DMS 01-03626. The second author is partially supported by NSF grant DMS-99-71311.     

On the number of generators needed for free profinite products of finite groups

We provide lower estimates on the minimal number of generators of the profinite completion of free products of finite groups. In particular, we show that if C ₁, ..., C _n are finite cyclic groups then there exists a finite group G which is generated by isomorphic copies of C ₁, ..., C _n and the minimal number of generators of G is n. The first author’s research is partially supported by NSF grant DMS-0701105. The second author’s research is partially supported by OTKA grant T38059 and the Magyary Zoltán Postdoctoral Fellowship.     

On a graph packing conjecture by Bollobás,Eldridge and Catlin

Two graphs G ₁ and G ₂ of order n pack if there exist injective mappings of their vertex sets into [n], such that the images of the edge sets are disjoint. In 1978, Bollobás and Eldridge, and independently Catlin, conjectured that if (Δ(G ₁) + 1)(Δ(G ₂) + 1) ≤ n + 1, then G ₁ and G ₂ pack. Towards this conjecture, we show that for Δ(G ₁),Δ(G ₂) ≥ 300, if (Δ(G ₁) + 1)(Δ(G ₂) + 1) ≤ 0.6n + 1, then G ₁ and G ₂ pack. This is also an improvement, for large maximum degrees, over the classical result by Sauer and Spencer that G ₁ and G ₂ pack if Δ(G ₁)Δ(G ₂) < 0.5n. This work was supported in part by NSF grant DMS-0400498. The work of the second author was also partly supported by NSF grant DMS-0650784 and grant 05-01-00816 of the Russian Foundation for Basic Research. The work of the third author was supported in part by NSF grant DMS-0652306.     

Sharp trace theorems for null hypersurfaces on Einstein metrics with finite curvature flux

The main objective of the paper is to prove a geometric version of sharp trace and product estimates on null hypersurfaces with finite curvature flux. These estimates play a crucial role to control the geometry of such null hypersurfaces. The paper is based on an invariant version of the classical Littlewood–Paley theory, in a noncommutative setting, defined via heat flow on surfaces. Received: April 2004 Revision: June 2005 Accepted: July 2005 The first author is partially supported by NSF grant DMS-0070696. The second author is a Clay Prize Fellow and is partially supported by NSF grant DMS-01007791.     

On galois descent for Hochschild and cyclic homology

LetG be a finite group acting by automorphisms on an algebraS over some commutative ringk. We show that if the action ofG restricted to the center ofS is Galois in the sense of [C-H-R], thenHH _*(S ^G)≊HH _* (S) ^G. An analogous result holds for cyclic homology, provided the order ofG is invertible ink. The author was supported in part by a grant from the NSF.     

Wiggly sets and limit sets

We show that a compact, connected set which has uniform oscillations at all points and at all scales has dimension strictly larger than 1. We also show that limit sets of certain Kleinian groups have this property. More generally, we show that ifG is a non-elementary, analytically finite Kleinian group, and its limit set Λ(G) is connected, then Λ(G) is either a circle or has dimension strictly bigger than 1. The first author is partially supported by NSF Grant DMS 95-00577 and an Alfred P. Sloan research fellowship. The second author is partially supported by NSF grant DMS-94-23746.     

The space of surface group representations

In this note we prove that the number of irreducible components of Hom (π,G) is the same as π₁(G), where π is a surface group andG is complex semisimple. This is established by studying the flat bundles on Riemann surfaces. The present work is partially supported by NSF grant DMS89-04922     

The moduli space of complete embedded constant mean curvature surfaces

We examine the space of finite topology surfaces in ³ which are complete, properly embedded and have nonzero constant mean curvature. These surfaces are noncompact provided we exclude the case of the round sphere. We prove that the spaceM _k of all such surfaces withk ends (where surfaces are identified if they differ by an isometry of ³) is locally a real analytic variety. When the linearization of the quasilinear elliptic equation specifying mean curvature equal to one has noL ²-nullspace, we prove thatM _k is locally the quotient of a real analytic manifold of dimension 3k–6 by a finite group (i.e. a real analytic orbifold), fork 3. This finite group is the isotropy subgroup of the surface in the group of Euclidean motions. It is of interest to note that the dimension ofM _k is independent of the genus of the underlying punctured Riemann surface to which is conformally equivalent. These results also apply to hypersurfaces of H ⁿ⁺¹ with nonzero constant mean curvature greater than that of a horosphere and whose ends are cylindrically bounded.Research of the first author supported in part by NSF grant # DMS9404278 and an NSF Postdoctoral Fellowship, of the second auther by NSF Young Investigator Award, a Sloan Foundation Postdoctoral Fellowship and NSF grant # DMS9303236, and of the third author by NSF grant # DMS9022140 and an NSF Postdoctoral Fellowship.     

Coloring complexes and arrangements

Steingrimsson’s coloring complex and Jonsson’s unipolar complex are interpreted in terms of hyperplane arrangements. This viewpoint leads to short proofs that all coloring complexes and a large class of unipolar complexes have convex ear decompositions. These convex ear decompositions impose strong new restrictions on the chromatic polynomials of all finite graphs. Similar results are obtained for characteristic polynomials of submatroids of type ℬ _n arrangements. The first author was supported by NSF grant DMS-0500638. The second author was supported by NSF grant DMS-0245623.     

Invariant Radon measures for horocycle flows on Abelian covers

We classify the ergodic invariant Radon measures for horocycle flows on ^d–covers of compact Riemannian surfaces of negative curvature, thus proving a conjecture of M. Babillot and F. Ledrappier. An important tool is a result in the ergodic theory of equivalence relations concerning the reduction of the range of a cocycle by the addition of a coboundary. Dedicated to the memory of M. Babillot     

On closures of orbits and arithmetic of quaternions

ForG=PGL₂(ℚ _p )×PGL₂ ℚ we study the closures of orbits under the maximal split Cartan subgroup ofG in homogeneous spacesΓ\G. We show that if a closure of an orbit contains a closed orbit then the orbit is either dense or closed. We show the relation of this to divisibility properties of integral quaternions and other lattices. Sponsored in part by the Edmund Landau Center for Research in Mathematical Analysis supported by the Minerva Foundation (Germany). Research at MSRI supported by NSF grant DMS8505550.     

Algebraic invariants for right-angled Artin groups

A finite simplicial graph Γ determines a right-angled Artin group G_Γ, with generators corresponding to the vertices of Γ, and with a relation υw=wυ for each pair of adjacent vertices. We compute the lower central series quotients, the Chen quotients, and the (first) resonance variety of G_Γ, directly from the graph Γ. Partially supported by NSF grant DMS-0311142.     

Toeplitz Operators on Analytic Besov Spaces

We characterize complex measures μ on the unit disk for which the Toeplitz operator T _μ is bounded or compact on the analytic Besov spaces B _p with 1 ≤ p < ∞. Research supported in part by NSF grant, DMS 0200587 (first author); and by a NSERC grant (third author).     

The K-admissibility of SL(2, 5)

Let K be a field and let G be a finite group. G is K-admissible if there exists a Galois extension L of K with G=Gal(L/K) such that L is a maximal subfield of a central K-division algebra. We characterize those number fields K such that H is K-admissible where H is any subgroup of SL(2, 5) which contains a S ₂-group. The method also yields refinements and alternate proofs of some known results including the fact that A ₅ is K-admissible for every number field K.Dedicated to Professor Jacques Tits on the occasion of his sixtieth birthdayThe first author was partly supported by NSF fellowship DMS-8601130; the second author was partly supported by NSF grant DMS-8806371.     

Quasiequational Theories of Flat Algebras

We prove that finite flat digraph algebras and, more generally, finite compatible flat algebras satisfying a certain condition are finitely q-based (possess a finite basis for their quasiequations). We also exhibit an example of a twelve-element compatible flat algebra that is not finitely q-based. The first author was partially supported by the grant # 201/02/0594 of the Grant Agency of the Czech Republic, and by the Institutional grant MSM0021620839; the second author was partially supported by the grant No. Tn37877 of the Hungarian National Foundation for Scientific Research (OTHA); the third author was supported by the NSF grant # DMS-9971352.     

Pluricanonical systems on irregular 3-folds of general type

In this paper we prove that if X is an irregular 3-fold with χ(ω _X ) > 0, then |mK _X | is birational for all m ≥ 5.The first author was partially supported by NSC and NCTS of Taiwan. The second author was partially supported by NSF research grant no: 0456363.     

The Horocycle Flow and the Laplacian on Hyperbolic Surfaces of Infinite Genus

Consider a complete hyperbolic surface which can be partitioned into countably many pairs of pants whose boundary components have lengths less than some constant. We show that any infinite ergodic invariant Radon measure for the horocycle flow is either supported on a single horocycle associated with a cusp, or corresponds canonically to an extremal positive eigenfunction of the Laplace–Beltrami operator.     

On capability of finite abelian groups

Baer characterized capable finite abelian groups (a group is capable if it is isomorphic to the group of inner automorphisms of some group) by a condition on the size of the factors in the invariant factor decomposition (the group must be noncyclic and the top two invariant factors must be equal). We provide a different characterization, given in terms of a condition on the lattice of subgroups. Namely, a finite abelian group G is capable if and only if there exists a family {H _i } of subgroups of G with trivial intersection, such that the union generates G and all quotients G/H _i have the same exponent. Other variations of this condition are also provided (for instance, the condition that the union generates G can be replaced by the condition that it is equal to G). The work presented here is partially supported by NSF/DMS-0805932.     

Bounded characteristic functions and models for noncontractive sequences of operators

The Sz.-Nagy-FoiaŞ functional model for completely non-unitary contractions is extended to completely non-coisometric sequences of bounded operatorsT = (T₁,...,T _d) (d finite or infinite) on a Hilbert space, with bounded characteristic functions. For this class of sequences, it is shown that the characteristic function θ_T is a complete unitary invariant. We obtain, as the main result, necessary and sufficient conditions for a bounded multi-analytic operator on Fock spaces to coincide with the characteristic function associated with a completely non-coisometric sequence of bounded operators on a Hilbert space. Research supported in part by a COBASE grant from the National Research Council. The first author was partially supported by a grant from Ministerul Educaţiei Şi Cercetarii. The second author was partially supported by a National Science Foundation grant.