Complex<Emphasis Type="BoldItalic">n</Emphasis>-dimensional manifolds with a real<Emphasis Type="BoldItalic">n</Emphasis><Superscript>2</Superscript>-dimensional automorphism group

The main theorem of this article is a characterization of non compact simply connected complete Kobayashi hyperbolic complex manifold of dimension n≽ 2 with real n ²-dimensional holomorphic automorphism group. Together with the earlier work [11, 12] and [13] of Isaev and Krantz, this yields a complete classification of the simply-connected, complete Kobayashi hyperbolic manifolds with dim_ℝ Aut (M) ≽ (dim_ℂ M)².     

1 -convex manifolds are p- k?hler

We study here K?hler-type properties of 1-convex manifolds, using the duality between forms and compactly supported currents, and some properties of the Aeppli groups of (q-convex manifolds. We prove that, when the exceptional setS of the l-convex manifoldX has dimensionk, X is p-K?hler for everyp > k, and isk-K?hler if and only if “the fundamental class” ofS does not vanish. There are classical examples whereX is notk-K?hler even with a smoothS, but we prove that this cannot happen if2k ≥n = dimX, nor for suitable neighborhoods of S; in particular,X is always balanced (i.e.,(n - 1)-Kahler). Partially supported by MIUR research funds.     

Codimension of immersions with parallel pluri-mean curvature

Immersions with parallel pluri-mean curvature into euclidean n-space generalize constant mean curvature immersions of surfaces to Kähler manifolds of complex dimension m. Examples are the standard embeddings of Kähler symmetric spaces into the Lie algebra of its transvection group. We give a lower bound for the codimension of arbitrary ppmc immersions. In particular we show that M is locally symmetric if the codimension is minimal.     

Flat nearly Kähler manifolds

We classify flat strict nearly Kähler manifolds with (necessarily) indefinite metric. Any such manifold is locally the product of a flat pseudo-Kähler factor of maximal dimension and a strict flat nearly Kähler manifold of split signature (2m, 2m) with m ≥ 3. Moreover, the geometry of the second factor is encoded in a complex three-form $\zeta \in \Lambda^3 (\mathbb{C}^m)^*We classify flat strict nearly K?hler manifolds with (necessarily) indefinite metric. Any such manifold is locally the product of a flat pseudo-K?hler factor of maximal dimension and a strict flat nearly K?hler manifold of split signature (2m, 2m) with m ≥ 3. Moreover, the geometry of the second factor is encoded in a complex three-form . The first nontrivial example occurs in dimension 4m = 12.      

Suites d’Applications Méromorphes Multivaluées et Courants Laminaires

Let F_n: X₁ → X₂ be a sequence of (multivalued) meromorphic maps between compact Kähler manifolds. We study the asymptotic distribution of preimages of points by F_n and, for multivalued self-maps of a compact Riemann surface, the asymptotic distribution of repelling fixed points. Let (Z_n) be a sequence of holomorphic images of ?^s in a projective manifold. We prove that the currents, defined by integration on Z_n, properly normalized, converge to currents which satisfy some laminarity property. We also show this laminarity property for the Green currents, of suitable bidimensions, associated to a regular polynomial automorphism of ?^k or an automorphism of a projective manifold.     

Automorphisms of Relatively Free Nilpotent Lie Algebras

Let L be a relatively free nilpotent Lie algebra over ? of rank n and class c, with n ≥ 2; freely generated by a set 𝒵. Give L the structure of a group, denoted by R, by means of the Baker–Campbell–Hausdorff formula. Let G be the subgroup of R generated by the set 𝒵 and N _Aut(L)(G) the normalizer in Aut(L) of the set G. We prove that the automorphism group of L is generated by GL _n (?) and N _Aut(L)(G). Let H be a subgroup of finite index in Aut(G) generated by the tame automorphisms and a finite subset X of IA-automorphisms with cardinal s. We construct a set Y consisting of s + 1 IA-automorphisms of L such that Aut(L) is generated by GL _n (?) and Y. We apply this particular method to construct generating sets for the automorphism groups of certain relatively free nilpotent Lie algebras.     

Cohomology of compact hyperkähler manifolds and its applications

We announce the structure theorem for theH ²(M)-generated part of cohomology of a compact hyperkähler manifold. This computation uses an action of the Lie algebra so(4,n–2) wheren=dimH ²(M) on the total cohomology space ofM. We also prove that every two points of the connected component of the moduli space of holomorphically symplectic manifolds can be connected with so-called twistor lines — projective lines holomorphically embedded in the moduli space and corresponding to the hyperkähler structures. This has interesting implications for the geometry of compact hyperkähler manifolds and of holomorphic vector bundles over such manifolds.     

On toric locally conformally Kähler manifolds

We study compact toric strict locally conformally Kähler manifolds. We show that the Kodaira dimension of the underlying complex manifold is \(-\infty \), and that the only compact complex surfaces admitting toric strict locally conformally Kähler metrics are the diagonal Hopf surfaces. We also show that every toric Vaisman manifold has lcK rank 1 and is isomorphic to the mapping torus of an automorphism of a toric compact Sasakian manifold.     

On the -equation¶over pseudoconvex Kähler manifolds

The Picard variety Pic⁰(? ⁿ ) of a complex n-dimensional torus? ⁿ is the group of all holomorphic equivalence classes of topologically trivial holomorphic (principal) line bundles on ? ⁿ . The total space of a topologically trivial holomorphic (principal) line bundle on a compact K?hler manifold is weakly pseudoconvex. Thus we can regard Pic⁰(? ⁿ ) as a family of weakly pseudoconvex K?hler manifolds. We consider a problem whether the Kodaira's -Lemma holds on a total space of holomorphic line bundle belonging to Pic⁰(? ⁿ ). We get a criterion for this problem using a dynamical system of translations on Pic⁰(? ⁿ ). We also discuss the problem whether the -Lemma holds on strongly pseudoconvex K?hler manifolds or not. Using the result of ColColţoiu, we find a 1-convex complete K?hler manifold on which the -Lemma does not hold. Received: 11 June 1999 / Revised version: 22 November 1999     

Tetravalent half‐edge‐transitive graphs and non‐normal Cayley graphs

Let X be a vertex‐transitive graph, that is, the automorphism group Aut(X) of X is transitive on the vertex set of X. The graph X is said to be symmetric if Aut(X) is transitive on the arc set of X. suppose that Aut(X) has two orbits of the same length on the arc set of X. Then X is said to be half‐arc‐transitive or half‐edge‐transitive if Aut(X) has one or two orbits on the edge set of X, respectively. Stabilizers of symmetric and half‐arc‐transitive graphs have been investigated by many authors. For example, see Tutte [Canad J Math 11 (1959), 621–624] and Conder and Maru?i? [J Combin Theory Ser B 88 (2003), 67–76]. It is trivial to construct connected tetravalent symmetric graphs with arbitrarily large stabilizers, and by Maru?i? [Discrete Math 299 (2005), 180–193], connected tetravalent half‐arc‐transitive graphs can have arbitrarily large stabilizers. In this article, we show that connected tetravalent half‐edge‐transitive graphs can also have arbitrarily large stabilizers. A Cayley graph Cay(G, S) on a group G is said to be normal if the right regular representation R(G) of G is normal in Aut(Cay(G, S)). There are only a few known examples of connected tetravalent non‐normal Cayley graphs on non‐abelian simple groups. In this article, we give a sufficient condition for non‐normal Cayley graphs and by using the condition, infinitely many connected tetravalent non‐normal Cayley graphs are constructed. As an application, all connected tetravalent non‐normal Cayley graphs on the alternating group A₆ are determined. © 2011 Wiley Periodicals, Inc. J Graph Theory     

Lip Automorphism Germ and Lip Automorphism

Let X be a metric space, ε^n(X) be the standard trivial Lip n-bundle over X, and Φ be a Lip automorphism germ of ε^n(X). This paper proves that there is a Lip automorphism Φ‘ of ε^n(X) such that the germ of Φ‘ is Φ.     

A structure theorem for boundary-transitive graphs with infinitely many ends

We prove a structure theorem for locally finite connected graphsX with infinitely many ends admitting a non-compact group of automorphisms which is transitive in its action on the space of ends, Ω _X . For such a graphX, there is a uniquely determined biregular treeT (with both valencies finite), a continuous representationφ : Aut(X) → Aut(T) with compact kernel, an equivariant homeomorphism λ : Ω _X → Ω _T , and an equivariant map τ : Vert(X) → Vert(T) with finite fibers. Boundary-transitive trees are described, and some methods of constructing boundary-transitive graphs are discussed, as well as some examples.     

CR Hypersurfaces with a Contracting Automorphism

Let M be a real analytic CR hypersurface in ℂ ⁿ⁺¹ admitting no varieties of positive dimension. We show first that every contracting local CR automorphism of M is linearizable. As a consequence, we show that such M admitting a contracting local CR automorphism is holomorphically equivalent to a weighted homogeneous hypersurface. Finally, we apply these results to prove that a bounded domain in ℂ ⁿ⁺¹ with a real analytic boundary admitting an automorphism contracting at a boundary point must admit a Lie subgroup of real dimension at least two in its automorphism group. Research of the first named author is partially supported by The Grant R01-2005-000-10771-0 of The Korea Science and Engineering Foundation.     

Complex Monge-Ampère equations and totally real submanifolds

We study the Dirichlet problem for complex Monge-Ampère equations in Hermitian manifolds with general (non-pseudoconvex) boundary. Our main result (Theorem 1.1) extends the classical theorem of Caffarelli, Kohn, Nirenberg and Spruck in Cn. We also consider the equation on compact manifolds without boundary, attempting to generalize Yau's theorems in the Kähler case. As applications of the main result we study some connections between the homogeneous complex Monge-Ampère (HCMA) equation and totally real submanifolds, and a special Dirichlet problem for the HCMA equation related to Donaldson's conjecture on geodesics in the space of Kähler metrics.     

Nilpotency of unstable K-theory

In the preceding papers [H. Hamanaka, A. Kono, On [X,U(n)], when dimX is 2n, J. Math. Kyoto Univ. 43 (2) (2003) 333-348; H. Hamanaka, On [X,U(n)], when dimX is 2n+1, J. Math. Kyoto Univ. 44 (3) (2004) 655-667; H. Hamanaka, Adams e-invariant, Toda bracket and [X,U(n)], J. Math. Kyoto Univ. 43 (4) (2003) 815-828], the group structure of the homotopy set [X,U(n)] with the pointwise multiplication is studied, where X is a finite CW-complex and U(n) is the unitary group. It is seen that nil[X,U(n)]=2 for some X with its dimension 2n, and, when dimX=2n+1 and n is even, [X,U(n)] is expressed as the two stage central extension of an Abelian group, i.e., nil[X,U(n)]?3.In this paper, we consider the nilpotency class of [X,U(n)], especially, for given k, the maximum of the nil[X,U(n)] under the condition dimX?2n+k is estimated and determined for k=0,1,2.     

The Automorphism Groups of Finite Type G-Structures on Orbifolds

The automorphism group of a G-structure of finite type and order k on a smooth n-dimensional orbifold is proved to be a Lie group of dimension n+dim(g+g ₁+...+g _k-1), where g _i is the ith prolongation of the Lie algebra g of a given group G. This generalizes the corresponding result by Ehresmann for finite type G-structures on manifolds. The presence of orbifold points is shown to sharply decrease the dimension of the automorphism group of proper orbifolds. Estimates are established for the dimension of the isometry group and the dimension of the group of conformal transformations of Riemannian orbifolds, depending on the types of orbifold points.     

Counting points in hypercubes and convolution measure algebras

It is shown that ifA andB are non-empty subsets of {0, 1} ⁿ (for somenεN) then |A+B|≧(|A||B|)^α where α=(1/2) log₂ 3 here and in what follows. In particular if |A|=2 ^n-1 then |A+A|≧3 ^n-1 which anwers a question of Brown and Moran. It is also shown that if |A| = 2 ^n-1 then |A+A|=3 ^n-1 if and only if the points ofA lie on a hyperplane inn-dimensions. Necessary and sufficient conditions are also given for |A +B|=(|A||B|)^α. The above results imply the following improvement of a result of Talagrand [7]: ifX andY are compact subsets ofK (the Cantor set) withm(X),m(Y)>0 then λ(X+Y)≧2(m(X)m(Y))^α wherem is the usual measure onK and λ is Lebesgue measure. This also answers a question of Moran (in more precise terms) showing thatm is not concentrated on any proper Raikov system.     

Hyperbolic manifolds with high-dimensional automorphism group

We survey our recent classification results for Kobayashi-hyperbolic n-dimensional manifolds with holomorphic automorphism group of dimension at least n ² − 1 for n ≥ 2.