Uniqueness and non-existence theorems for conformally invariant equations

By using the equivalent integral form for the Q-curvature equation, we generalize the well-known non-existence results on two-dimensional Gaussian curvature equation to all dimensional Q-curvature equation. Somehow, we introduce a new approach to Q-curvature equation which is higher order and even pseudo-differential equation. As a by-product, we do classify the solutions for Q=1 solutions on Sⁿ as well as on Rⁿ with necessary growth rate assumption.     

Schouten curvature functions on locally conformally flat Riemannian manifolds

Consider a compact Riemannian manifold (M, g) with metric g and dimension n ≥ 3. The Schouten tensor A _g associated with g is a symmetric (0, 2)-tensor field describing the non-conformally-invariant part of the curvature tensor of g. In this paper, we consider the elementary symmetric functions {σ _k (A _g ), 1 ≤ k ≤ n} of the eigenvalues of A _g with respect to g; we call σ _k (A _g ) the k-th Schouten curvature function. We give an isometric classification for compact locally conformally flat manifolds which satisfy the conditions: A _g is semi-positive definite and σ _k (A _g ) is a nonzero constant for some k ∈ {2, ... , n}. If k = 2, we obtain a classification result under the weaker conditions that σ₂(A _g ) is a non-negative constant and (M ⁿ , g) has nonnegative Ricci curvature. The corresponding result for the case k = 1 is well known. We also give an isometric classification for complete locally conformally flat manifolds with constant scalar curvature and non-negative Ricci curvature. Udo Simon: Partially supported by Chinese-German cooperation projects, DFG PI 158/4-4 and PI 158/4-5, and NSFC.     

Locally homogeneous Riemannian manifolds and Cartan triples

Ann-dimensional Cartan triple is a triple (g, , ) consisting of a Lie subalgebra g of so(n) (endowed with the Killing form), a linear map : ⁿ g and a bilinear antisymmetric map ²( ⁿ , g), which together satisfy (1.25)–(1.28) of Section 1. LetM _n be the set ofmaximal n-dimensional Cartan triples, and letA _n be thenatural action of the orthogonal group O(n) onM _n (Section 3). One shows that there is a bijective mapping from the set of local isometry classes ofn-dimensional locally homogeneous Riemannian manifolds to the set of orbits ofA _n (Theorem 3.1(a)). Under this bijection, the classes of homogeneous Riemannian manifolds correspond to orbits ofclosed Cartan triples.     

Clifford structures on Riemannian manifolds

We introduce the notion of even Clifford structures on Riemannian manifolds, which for rank r=2 and r=3 reduce to almost Hermitian and quaternion-Hermitian structures respectively. We give the complete classification of manifolds carrying parallel rank r even Clifford structures: Kähler, quaternion-Kähler and Riemannian products of quaternion-Kähler manifolds for r=2,3 and 4 respectively, several classes of 8-dimensional manifolds (for 5?r?8), families of real, complex and quaternionic Grassmannians (for r=8,6 and 5 respectively), and Rosenfeld?s elliptic projective planes OP², (C⊗O)P², (H⊗O)P² and (O⊗O)P², which are symmetric spaces associated to the exceptional simple Lie groups F₄, E₆, E₇ and E₈ (for r=9,10,12 and 16 respectively). As an application, we classify all Riemannian manifolds whose metric is bundle-like along the curvature constancy distribution, generalizing well-known results in Sasakian and 3-Sasakian geometry.     

On conformally flat contact metric manifolds

In the present paper we classify the conformally flat contact metric manifolds of dimension satisfying . We prove that these manifolds are Sasakian of constant curvature 1.     

Conformal holonomy of C-spaces, Ricci-flat, and Lorentzian manifolds

The main result of this paper is that a Lorentzian manifold is locally conformally equivalent to a manifold with recurrent lightlike vector field and totally isotropic Ricci tensor if and only if its conformal tractor holonomy admits a 2-dimensional totally isotropic invariant subspace. Furthermore, for semi-Riemannian manifolds of arbitrary signature we prove that the conformal holonomy algebra of a C-space is a Berger algebra. For Ricci-flat spaces we show how the conformal holonomy can be obtained by the holonomy of the ambient metric and get results for Riemannian manifolds and plane waves.     

Classification of Riemannian manifolds in nonlinear potential theory

The classification theory of Riemann surfaces is generalized to Riemanniann-manifolds in the conformally invariant case. This leads to the study of the existence ofA-harmonic functions of typen with various properties and to an extension of the definition of the classical notions with inclusionsO _G O _HP O _HB O _HD . In the classical case the properness of the inclusions were proved rather late, in the 50's by Ahlfors and Tôki. Our main objective is to show that such inclusions are proper also in the generalized case.This research was supported in part by grants from the Academy of Finland and the U.S. National Science Foundation (NSF DMS 9003438).     

On three-dimensional conformally flat quasi-Sasakian manifolds

LetM be a 3-dimensional quasi-Sasakian manifold. On such a manifold, the so-called structure function is defined. With the help of this function, we find necessary and sufficient conditions forM to be conformally flat. Next it is proved that ifM is additionally conformally flat with = const., then (a)M is locally a product ofR and a 2-dimensional Kählerian space of constant Gauss curvature (the cosymplectic case), or (b)M is of constant positive curvature (the non cosymplectic case; here the quasi-Sasakian structure is homothetic to a Sasakian structure). An example of a 3-dimensional quasi-Sasakian structure being conformally flat with nonconstant structure function is also described. For conformally flat quasi-Sasakian manifolds of higher dimensions see [O₁]     

Immersions of cohomogeneity one Riemannian manifolds

This work deals with positively curved compact Riemannian manifolds which are acted on by a closed Lie group of isometries whose principal orbits have codimension one and are isotropy irreducible homogeneous spaces. For such manifolds we can show that their universal covering manifold may be isometrically immersed as a hypersurface of revolution in an euclidean space.     

Rough isometry and Dirichlet finite harmonic functions on Riemannian manifolds

We prove that the dimension of harmonic functions with finite Dirichlet integral is invariant under rough isometries between Riemannian manifolds satisfying the local conditions, expounded below. This result directly generalizes those of Kanai, of Grigor'yan, and of Holopainen. We also prove that the dimension of harmonic functions with finite Dirichlet integral is preserved under rough isometries between a Riemannian manifold satisfying the same local conditions and a graph of bounded degree; and between graphs of bounded degree. These results generalize those of Holopainen and Soardi, and of Soardi, respectively. Received: 23 July 1998 / Revised version: 10 February 1999     

Pseudo Riemannian manifolds whose generalized Jacobi operator has constant characteristic polynomial

Letg be a non-degenerate innerproduct of signature (p,q) onR ^m . LetGr _r,s (g) be the Grassmanian of planes so the restriction of g to is non-degenerate and has signature (r, s). IfR is an algebraic curvature tensor onR ^m , we define a generalized Jacobi operator onGr _r,s (g) and study when the characteristic polynomial of this operator is constant.Dedicated to Professor Helmut Karzel on his 70th birthdayResearch partially supported by the NSF (USA).Research partially supported by the NFSI (Bulgaria).     

ParaHermitian and paraquaternionic manifolds

A set of canonical paraHermitian connections on an almost paraHermitian manifold is defined. ParaHermitian version of the Apostolov-Gauduchon generalization of the Goldberg-Sachs theorem in General Relativity is given. It is proved that the Nijenhuis tensor of a Nearly paraKähler manifolds is parallel with respect to the canonical connection. Salamon's twistor construction on quaternionic manifold is adapted to the paraquaternionic case. A hyper-paracomplex structure is constructed on Kodaira-Thurston (properly elliptic) surfaces as well as on the Inoe surfaces modeled on . A locally conformally flat hyper-paraKähler (hypersymplectic) structure with parallel Lee form on Kodaira-Thurston surfaces is obtained. Anti-self-dual non-Weyl flat neutral metric on Inoe surfaces modeled on is presented. An example of anti-self-dual neutral metric which is not locally conformally hyper-paraKähler is constructed.     

L q -Estimates of spherical functions and an invariant mean-value property

We find someL ^q -estimates for the spherical functions on Cartan domains. As an application we prove that if the rank of the Cartan domainD is greater than one, then for any 1<-q<, the invariant mean-value property forL ^q -function onD does not imply harmonicity (the converse is known to be true even in the context of general non-compact Riemannian symmetric spacesG/K).     

Geometric connections and geometric Dirac operators on contact manifolds

We construct some natural metric connections on metric contact manifolds compatible with the contact structure and characterized by the Dirac operators they determine. In the case of CR manifolds these are invariants of a fixed pseudo-hermitian structure, and one of them coincides with the Tanaka-Webster connection.     

Orlicz capacities and Hausdorff measures on metric spaces

In the setting of doubling metric measure spaces with a 1-Poincaré inequality, we show that sets of Orlicz Φ-capacity zero have generalized Hausdorff h-measure zero provided thatwhere Θ⁻¹ is the inverse of the function Θ(t)=Φ(t)/t, and s is the “upper dimension” of the metric measure space. This condition is a generalization of a well known condition in Rⁿ. For spaces satisfying the weaker q-Poincaré inequality, we obtain a similar but slightly more restrictive condition. Several examples are also provided.     

A note on dyadic Hausdorff capacities

In this paper, the authors give a counterexample to show that the classical dyadic Hausdorff capacity on Rn when n?2 and 0<d?n−1 is not a capacity in the sense of Choquet. A variant of the classical dyadic Hausdorff capacity, , is then introduced and is further proved to be a capacity in the sense of Choquet.