Browsing by Browse by FOR 2008 "010102 Algebraic and Differential Geometry"

We consider two cases of the asymptotically flat scalar-flat Yamabe problem on a non-compact manifold with inner boundary in dimension n ≥ 3. First, following arguments of Cantor and Brill in the compact case, we show that given an asymptotically flat metric g, there is a conformally equivalent asymptotically flat scalar-flat metric that agrees with g on the boundary. We then replace the metric boundary condition with a condition on the mean curvature: given a function f on the boundary that is not too large, we show that there is an asymptotically flat scalar-flat metric, conformally equivalent to g whose boundary mean curvature is given by f. The latter case involves solving an elliptic PDE with critical exponent using the method of sub- and supersolutions. Both results require the usual assumption that the Sobolev quotient is positive.

We consider a closed orientable Riemannian 3-manifold (M,g) and a vector field X with unit norm whose integral curves are geodesics of g. Any such vector field determines naturally a 2-plane bundle contained in the kernel of the contact form of the geodesic flow of g. We study when this 2-plane bundle remains invariant under two natural almost complex structures. We also provide a geometric condition that ensures that X is the Reeb vector field of the 1-form λ obtained by contracting g with X. We apply these results to the case of great circle flows on the 3-sphere with two objectives in mind: one is to recover the result in [4] that a volume preserving great circle flow must be Hopf and the other is to characterize in a similar fashion great circle flows that are conformal relative to the almost complex structure in the kernel of λ given by rotation by π/2 according to the orientation of M.

We study higher dimensional versions of shearfree null-congruences in conformal Lorentzian manifolds. We show that such structures induce a subconformal structure and a partially integrable almost CR-structure on the leaf space and we classify the Lorentzian metrics that induce the same subconformal structure. In the last section we survey some known applications of the correspondence between almost CR-structures and shearfree null-congruences in dimension 4.

We introduce a CR-invariant class of Lorentzian metrics on a circle bundle over a three-dimensional CR structure, which we call FRT metrics. These metrics generalise the Fefferman metric, allowing for more control of the Ricci curvature, but are more special than the shearfree Lorentzian metrics introduced by Robinson and Trautman. Our main result is a criterion for embeddability of three-dimensional CR structures in terms of the Ricci curvature of the FRT metrics in the spirit of the results by Lewandowski et al. (Class Quantum Gravity 7(11):L241–L246, 1990) and also Hill et al. (Indiana Univ Math J 57(7):3131–3176, 2008. https://doi.org/10.1512/iumj.2008.57.3473).

We study O(d)-equivariant biharmonic maps in the critical dimension. A major consequence of our study concerns the corresponding heat flow. More precisely, we prove that blowup occurs in the biharmonic map heat flowfrom B⁴(0, 1) into S⁴. To our knowledge, this was the first example of blowup for the biharmonic map heat flow. Such results have been hard to prove, due to the inapplicability of the maximum principle in the biharmonic case. Furthermore, we classify the possible O(4)-equivariant biharmonic maps from R⁴ into S⁴, and we show that there exists, in contrast to the harmonic map analogue, equivariant biharmonic maps from B⁴(0, 1) into S⁴ that wind around S⁴ as many times as we wish. We believe that the ideas developed herein could be useful in the study of other higher-order parabolic equations.

In this paper a family of curves - Riemannian cubics - in the unit sphere and the real projective plane is investigated. Riemannian cubics naturally arise as solutions to variational problems in Riemannian spaces. It is remarkable to find that an envelope of lines generated by a Riemannian cubic in one space is (nearly) a Riemannian cubic in another space.

There are only some exceptional CR dimensions and codimensions such that the geometries enjoy a discrete classification of the pointwise types of the homogeneous models. The cases of CR dimensions n and codimensions n² are among the very few possibilities of the so-called parabolic geometries. Indeed, the homogeneous model turns out to be PSU(n+1,n)/P with a suitable parabolic subgroup P. We study the geometric properties of such real (2n+n²)-dimensional submanifolds in C n+n² for all n > 1. In particular, we show that the fundamental invariant is of torsion type, we provide its explicit computation, and we discuss an analogy to the Fefferman construction of a circle bundle in the hypersurface type CR geometry.

It is well known from undergraduate complex analysis that holomorphic functions of one complex variable are fully determined by their values at the boundary of a complex domain via the Cauchy integral formula. This is the first instance in which students encounter the general principle of complex analysis in one and several variables that the study of holomorphic objects often reduces to the study of their boundary values. The boundaries of complex domains, having odd topological dimension, cannot be complex objects. This motivated the study of the geometry of real hypersurfaces in complex space. In particular, since all established facts about a particular hypersurface carry over to its image via a biholomorphic mapping in the ambient space, it is important to decide which hypersurfaces are equivalent with respect to such mappings - that is, to solve an equivalence problem for real hypersurfaces in a complex space.

In this paper, we present a generalization of the classical least squares problem on Euclidean spaces, introduced by Lagrange, to more general Riemannian manifolds. Using the variational definition of Riemannian polynomials, we formulate a higher-order variational problem on a manifold equipped with a Riemannian metric, which depends on a smoothing parameter and gives rise to what we call smoothing geometric splines. These are curves with a certain degree of smoothness that best fit a given set of points at given instants of time and reduce to Riemannian polynomials when restricted to each subinterval. We show that the Riemannian mean of the given points is achieved as a limiting process of the above. Also, when the Riemannian manifold is an Euclidean space, our approach generates, in the limit, the unique polynomial curve which is the solution of the classical least squares problem. These results support our belief that the approach presented in this paper is the natural generalization of the classical least squares problem to Riemannian manifolds.

We introduce the notion of involutive Kodaira-Spencer deformations of the regular part X0 of a normal surface singularity, which form a subspace of the analytic cohomology H1(X0, T1,0X0). Examples of involutive deformations for which the Stein completion does not embed in a complex Euclidean space of stable dimension are in fact well-known. Under the assumption that X0 admits a Kähler metric with L2-curvature, we show that unstable deformations are avoided if the holomorphic functions which determine an embedding of the central fibre are correspondingly deformed into functions which can be uniformly bounded on compact subsets.

We define the property of involutivity for deformations of complex structure on a manifold X, with particular reference to the regular part of a normal surface. Our main result is a sufficient condition for involutivity in terms of a "Ə-Cartan formula", previously examined in [3] in the more special context of cone singularities. By way of examples we show that some involutive deformations of the regular part determine a subspace, if not the entire versal space, of flat deformations of normal surface singularities, while others may determine Stein surfaces which lie outside the versal space of flat deformations of a given normal surface.

The third Japanese-Australian Workshop on Real and Complex Singularities (JARCS SYDNEY 2009) was held at the University of Sydney, Australia, during the period 15-18 September 2009. There were 33 participants, mostly from Japan and Australia. The workshop covered a variety of topics in singularity theory and brought together experts, early career researchers, and doctoral students from Australia, France and Japan. This volume contains research papers in real and complex singularities, algebraic geometry and three introductory Lectures on Ominimal structures. It is our hope that this volume reflects the lively research atmosphere of this conference.

This article follows recent work of Miyajima on the complex-analytic approach to deformations of the regular part (i.e. the punctured smooth neighbourhood) of isolated singularities. Attention has previously focused on stably-embeddable infinitesimal deformations as those which correspond to standard algebraic deformations of the germ of a variety, and which also lead to convergent series solutions of the Kodaira-Spencer integrability equation. The emphasis of the present paper, however, is on the subspaces Zₒ of first cohomology classes containing infinitesimal deformations with vanishing Kodaira-Spencer bracket, and Wₒ, consisting more broadly of deformations for which the bracket represents the trivial second cohomology class. Deformations representing classes in Zₒ are automatically integrable, regardless of their analytic behaviour near the singular point. Classes in Wₒ are those for which only the first formal obstruction to integrability is overcome. After some preliminary results on cohomology, the main theorem of this paper gives a partial description of the analytic geometry of Zₒ and Wₒ for affine cones of arbitrary dimension.

We extend the results of [11] on embedded CR manifolds of CR dimension and codimension two to abstract partially integrable almost CR manifolds. We prove that points on such manifolds fall into three different classes, two of which (the hyperbolic and the elliptic points) always make up open seats. We prove that manifolds consisting entirely of hyperbolic (respectively elliptic) points admit canonical Cartan connections. More precisely, these structures are shown to be exactly the normal parabolic geometries of types (PSU(2,1) x PSU(2,1),B x B), respectively (PSL(3,C),B), where B indicates a Borel subgroup. We then show how general tools for parabolic geometries can be used to obtain geometric interpretations of the torsion part of the harmonic components of the curvature of the Cartan connection in the elliptic case.

The First Australian-Japanese Workshop on Real and Complex Singularities (JARCS SYDNEY 2005) was held at the University of Sydney, Australia, during the period 5-8 September, 2005. There were 35 participants, mostly from Japan and Australia. The present volume contains mainly the texts of most of the invited talks delivered at the workshop. The Australian-Japanese cooperation in singularity theory has quite a long history, starting in the late 70s, with two main protagonists, Tzee-Char Kuo in Australia and Takuo Fukuda in Japan. Since then we have witnessed a fruitful and intensive collaboration in the field of singularities. This volume contains expository articles on several topics in singularity theory, and research papers in real and complex singularities topology of stable maps, algebraic geometry, differential geometry and dynamical systems. The text is enhanced by beautiful figures and illustrations, many of which appear fully-coloured in the attached CD-ROM.

Let (Mˆ, g) be a compact, oriented Riemannian three-manifold corresponding to the metric point-completion M ∪{P₀} of a manifold M, and let ξ denote a geodesible Killing unit vector field on Mˆ such that the Ricci curvature function Ricg (ξ ) > 0 everywhere, and is constant outside a compact subset K ⊂⊂ M. Suppose further that (E, ∇, ϕ) supply the essential data of a monopole field on M, smooth outside isolated singularities all contained in K. The main theorem of this article provides a sufficient condition for smooth extension of (E, ∇, ϕ) across P₀, in terms of the Higgs potential Φ, defined in a punctured neighbourhood of P₀ by ∇ξΦ − 2i[ϕ, Φ] = ϕ . The sufficiency condition is expressed by a system of equations on the same neighbourhood, which can be effectively simplified in the case that Mˆ is a regular Sasaki manifold, such as the round S³.

We give a classification of Sasakian manifolds that are CR-equivalent to hyperquadrics by describing their exact parameter space. For "half" of the parameter space, we find an explicit representation by defining equations. This problem is related to the problem of finding pseudo-Kähler potentials with prescribed Ricci curvature.

We study rolling of one connected submanifold upon another connected submanifold, both isometrically embedded into one and the same Riemannian manifold. We generalise the definition of rolling wellknown from the literature. By this new definition, the Euclidean group of motions is replaced by the connected component of the Lie group of isometries of the embedding manifold. We show that rolling in this more general situation is again unique. We prove a theorem that enables us to learn how to roll non-Euclidean manifolds that result from deformations of Euclidean submanifolds from the knowledge of the kinematic equations of rolling the Euclidean submanifolds. Taking into account that the ellipsoid is a deformed sphere, we apply the above mentioned theorem and the kinematic equations for the rolling sphere to derive the kinematic equations for rolling the ellipsoid. This example might serve as a motivation to roll other manifolds as well.

We study rolling of one submanifold upon another submanifold, both isometrically embedded in a Riemannian manifold.We generalise the definition of rolling in Sharpe (1997). In this new definition, the Euclidean group of motions is replaced by the Lie group of orientation preserving isometries. We show that rolling in this general situation is unique. We prove a theorem that enables us to learn how to roll non-Euclidean manifolds that result from deformations of Euclidean submanifolds from the knowledge of the kinematic equations of rolling these Euclidean submanifolds. Taking into account that the ellipsoid is a deformed sphere, we apply the above mentioned theorem and the kinematic equations for the rolling sphere to derive the kinematic equations for rolling the ellipsoid. This example serves as a motivation to roll other manifolds.

We introduce a CR-invariant class of Lorentzian metrics on a circle bundle over a 3-dimensional CR-structure, which we call FRT-metrics. These metrics generalise the Fefferman metric, allowing for more control of the Ricci curvature, but are more special than the shearfree Lorentzian metrics introduced by Robinson and Trautman. Our main result is a criterion for embeddability of 3-dimensional CR-structures in terms of the Ricci curvature of the FRT-metrics in the spirit of the results by Lewandowski et al. in [37] and also Hill et al. in [25]. We also study higher dimensional versions of shearfree null congruences in conformal Lorentzian manifolds. We show that such structures induce a subconformal structure and a partially integrable almost CR structure on the leaf space and we classify the Lorentzian metrics that induce the same subconformal structure.

We describe the automorphisms of a singular multicontact structure, that is a generalisation of the Martinet distribution. Such a structure is interpreted as a para-CR structure on a hypersurface M of a direct product space R²+ x R²-. We introduce the notion of a finite type singularity analogous to CR geometry and, along the way, we prove extension results for para-CR functions and mappings on embedded para-CR manifolds into the ambient space.

The first part of this paper describes in detail the action of small gauge transformations in heterotic supergravity. We show a convenient gauge fixing is 'holomorphic gauge' together with a condition on the holomorphic top form. This gauge fixing, combined with supersymmetry and the Bianchi identity, allows us to determine a set of non-linear PDEs for the terms in the Hodge decomposition. Although solving these in general is highly non-trivial, we give a prescription for their solution perturbatively in α‵ and apply this to the moduli space metric. The second part of this paper relates small gauge transformations to a choice of connection on the moduli space. We show holomorphic gauge is related to a choice of holomorphic structure and Lee form on a 'universal bundle'. Connections on the moduli space have field strengths that appear in the second order deformation theory and we point out it is generically the case that higher order deformations do not commute.

The fourth Japanese-Australian Workshop on Real and Complex Singularities (JARCS4 KOBE 2011) was held at the Kobe Satellite of Hyogo University of Teacher Education during the period 22-25 November, 2011. There were 31 participants from Australia and Japan. The Australian and Japanese Singularity groups have built up strong research relationship in the past three decades. For instance, the blow-analytic theory introduced by Tzee-Char Kuo in Australia has been intensively developed in Japan. In addition, a lot of joint research works of both countries have been established in several topics related to real and complex singularities. The present volume mainly consists of the texts of the invited talks of the workshop. Some of them are joint works of Australians and Japanese. This volume contains original articles on real and complex singularities, topology of differentiable maps, openings of differentiable map-germs, the relationship between free divisors and holonomic systems, effective computation method of invariants of singularities, the application of singularity theory to differential geometry, the deformation theory of CR structures and differential equation with singular points. In these articles some new notions important for characterizations of singularities are introduced, and several new results are presented. New approaches to classical topics and new computation methods of singularities are also presented.

In this paper, we present the general analytic solution to the zero curvature equation for rigid three-dimensional CR-manifolds. The solutions are uniquely determined by one function and four real parameters.