Conformal energy in four dimension

The conformal energy for 4-manifolds using the Paneitz operator is introduced in this article. The conformal invariance of the energy functional allows us to find a sharp lower bound in terms of the conformal volume. We also demonstrate certain obstruction to existence of minimal immersions to spheres using the fourth order curvature invariance associated to the operator. Received: 17 April 1999 / Revised version: 23 March 2002 / Published online: 5 September 2002     

Conformal dimension of the antenna set

We show that the self-similar set known as the ``antenna set' has the property that (where the infimum is over all quasiconformal mappings of the plane), but that this infimum is not attained by any quasiconformal map; indeed, is not attained for any quasisymmetric map into any metric space.

     

Conformal dimension and canonical splittings of hyperbolic groups

We prove a general criterion for a metric space to have conformal dimension one. The conditions are stated in terms of the existence of enough local cut points in the space. We then apply this criterion to the boundaries of hyperbolic groups and show an interesting relationship between conformal dimension and some canonical splittings of the group.     

Conformal type and isoperimetric dimension of Riemannian manifolds

The notion of conformal isoperimetric dimension is introduced. For Riemannian manifolds, connections between its conformal isoperimetric dimension and its conformal type are established. Translated fromMatematicheskie Zametki, Vol. 63, No. 3, pp. 379–385, March, 1998. The authors are greatly indebted to N. A. Zorich for her help in preparing the computer version of this paper. This research was partially supported by the Russian Foundation for Basic Research under grants No. 96-01-01218 and No. 96-01-00901.     

Conformal measures and Hausdorff dimension for infinitely renormalizable quadratic polynomials

We show in this paper that iff is a quadratic infinitely many times renormalizable polynomial of sufficient high combinatorial type, then: HD (J(f))= inf{: -conformal measure for f} We use Lyubich's construction of the principal nest ([Lyu97]) in order to prove this result.Partially supported by CNP q-Brazil grant # 300534/96-5     

平面上一类自相似集的共形维数

本文证明了平面上一类自相似集的共形维数为1.此外还证明了这些自相似集与任何Hausdorff维数为1的度量空间都不是拟对称等价的.这表明,对于这些自相似集而言,共形维数定义中的下确界不能达到.     

Resistance dimension,random walk dimension and fractal dimension

In this paper, we generalize a theorem due to Telcs concerning random walks on infinite graphs, which describes the relation of random walk dimension, fractal dimension and resistance dimension. Moreover, we obtain a reasonable upper bound and lower bound on the hitting time in terms of resistance for some nice graphs. In fact, the conditions given in this paper are weaker than those obtained by A. Telcs.Partly supported by National Natural Science Foundation and State Educational Committee of China.     

Finitistic dimension and restricted injective dimension

We study the relations between finitistic dimensions and restricted injective dimensions. Let R be a ring and T a left R-module with A = End _R T. If _R T is selforthogonal, then we show that rid(T _A ) ? findim(A _A ) ? findim( _R T) + rid(T _A ). Moreover, if R is a left noetherian ring and T is a finitely generated left R-module with finite injective dimension, then rid(T _A ) ? findim(A _A ) ? fin.inj.dim( _R R)+rid(T _A ). Also we show by an example that the restricted injective dimensions of a module may be strictly smaller than the Gorenstein injective dimension.     

Conformal uniformization and packings

A new short proof is given for Brandt and Harrington’s theorem about conformal uniformizations of planar finitely connected domains as domains with boundary components of specified shapes. This method of proof generalizes to periodic domains. Letting the uniformized domains degenerate in a controlled manner, we deduce the existence of packings of specified shapes and with specified combinatorics. The shapes can be arbitrary smooth disks specified up to homothety, for example. The combinatorics of the packing is described by the contacts graph, which can be specified to be any finite planar graph whose vertices correspond to the shapes. This is in the spirit of Koebe’s proof of the Circle Packing Theorem as a consequence of his uniformization by circle domains. The author thankfully acknowledges support of NSF grant DMS-9112150.     

Conformal mapping and ellipses

We answer a question raised by M. Chuaqui, P. Duren, and B. Osgood by showing that a conformal mapping of a simply connected domain cannot take two circles onto two proper ellipses.

     

Conformal autosimilarity

We prove that the conformally autosimilar closed subsets of the Euclidean sphere are exactly the limit sets of convex cocompact Kleinian groups. We also prove other similar results.     

Conformal cochains

In this paper we define holomorphic cochains and an associated period matrix for triangulated closed topological surfaces. We use the combinatorial Hodge star operator introduced in the author's paper of 2007, which depends on the choice of an inner product on the simplicial 1-cochains.

We prove that for a triangulated Riemannian 2-manifold (or a Riemann surface), and a particularly nice choice of inner product, the combinatorial period matrix converges to the (conformal) Riemann period matrix as the mesh of the triangulation tends to zero.

     

Conformal spaces

A conformal space is a non-singular metric vector space to which has been adjoined a null-cone of points at infinity. We define a conformal space in terms of a higher dimensional coordinate space, and then state and prove a fundamental theorem of conformal geometry.     

Conformal and semi-conformal biharmonic maps

We show that a conformal mapping between Riemannian manifolds of the same dimension n ≥ 3 is biharmonic if and only if the gradient of its dilation satisfies a certain second-order elliptic partial differential equation. On an Einstein manifold solutions can be generated from isoparametric functions. We characterise those semi-conformal submersions that are biharmonic in terms of their dilation and the fibre mean curvature vector field.