Narrow Results

We prove the compactness of the set of solutions to the CR Yamabe problem on a compact strictly pseudoconvex CR manifold of dimension three whose blow-up manifolds at every point have positive p-mass. As a corollary, we deduce that compactness holds for CR-embeddable manifolds which are not CR-equivalent to $S^3$. The theorem is proved by blow-up analysis.

We give lower bounds for the eigenvalues of the submanifold Dirac operator in terms of intrinsic and extrinsic curvature expressions. We also show that the limiting cases give rise to a class of spinor fields generalizing that of Killing spinors. We conclude by translating these results in terms of intrinsic twisted Dirac operators.

Given any two Einstein (pseudo-)metrics, with scalar curvatures suitably related, we give an explicit construction of a Poincaré–Einstein (pseudo-)metric with conformal infinity the conformal class of the product of the initial metrics. We show that these metrics are equivalent to ambient metrics for the given conformal structure. The ambient metrics have holonomy that agrees with the conformal holonomy. In the generic case the ambient metric arises directly as a product of the metric cones over the original Einstein spaces. In general the conformal infinity of the Poincaré metric we construct is not Einstein, and so this describes a class of non-conformally Einstein metrics for which the (Fefferman–Graham) obstruction tensor vanishes.

In the conformal (Möbius) geometry of submanifolds, using algebraic invariants of the shape operator to construct conformal invariants is a frequently used method. But it does not apply to $3$-dim Lorentzian hypersurfaces of the last type, the minimal polynomial of whose shape operator has a triple root. In this paper, using the obstruction of some distribution to be integrable, a new method to construct conformal invariants is introduced. Using this method, a complete conformal invariant system is constructed for generic conformally flat Lorentzian hypersurfaces of the last type. We find that such kind of hypersurfaces allows an infinite parameter family of non-equivalent deformations, which implies they are more abundant than the other three types. For non-generic conformally flat Lorentzian hypersurfaces of the last type, the isometric geometry is studied and a fundamental theorem is obtained in this paper.

We construct and give a complete classification of all the differential symmetry breaking operators ${𝔻}_{\lambda ,\nu }^m:C^{\infty }(S^3,{{𝒱}}_{\lambda }^3)\to C^{\infty }(S^2,{{ℒ}}_{m,\nu })$, between the spaces of smooth sections of a vector bundle of rank $3$ over the $3$-sphere ${{𝒱}}_{\lambda }^3\to S^3$, and a line bundle over the $2$-sphere ${{ℒ}}_{m,\nu }\to S^2$. In particular, we give necessary and sufficient conditions on the tuple of parameters $(\lambda ,\nu ,m)$ for which these operators exist.

A locally conformally product (LCP) structure on compact manifold $M$ is a conformal structure $c$ together with a closed, non-exact and non-flat Weyl connection $D$ with reducible holonomy. Equivalently, an LCP structure on $M$ is defined by a reducible, non-flat, incomplete Riemannian metric $h_D$ on the universal cover $\tilde{M}$ of $M$, with respect to which the fundamental group ${\pi }_1(M)$ acts by similarities. It was recently proved by Kourganoff that in this case $(\tilde{M},h_D)$ is isometric to the Riemannian product of the flat space ${ℝ}^q$ and an incomplete irreducible Riemannian manifold $(N,g_N)$. In this paper, we show that for every LCP manifold $(M,c,D)$, there exists a metric $g\in c$ such that the Lee form of $D$ with respect to $g$ vanishes on vectors tangent to the distribution on $M$ defined by the flat factor ${ℝ}^q$, and use this fact in order to construct new LCP structures from a given one by taking products. We also establish links between LCP manifolds and number field theory, and use them in order to construct large classes of examples, containing all previously known examples of LCP manifolds constructed by Matveev–Nikolayevsky, Kourganoff and Oeljeklaus–Toma (OT-manifolds).

We construct a model of a universe filled with a perfect fluid with isotropic particle pressure. The anisotropic plane symmetric Kasner spacetime is used as a seed metric and through a conformal mapping a perfect fluid is generated. The model is inhomogeneous, irrotational, shearing and accelerating. For negative time, the universe is expanding, while for positive time, it is collapsing. With the aid of graphical three-dimensional plots, it is established that the density and pressure hypersurfaces are smooth and singularity free. Additionally, the sound-speed index is computed and the fluid obeys the causality criterion. The non-static exact solution found may prove useful in the study of gravitational waves.

In this paper we prove that any complete conformal gradient soliton with nonnegative Ricci tensor is either isometric to a direct product ℝ × N^n-1, or globally conformally equivalent to the Euclidean space ℝⁿ or to the round sphere 𝕊ⁿ. In particular, we show that any complete, noncompact, gradient Yamabe-type soliton with positive Ricci tensor is rotationally symmetric, whenever the potential function is nonconstant.

The classical Pohozaev identity constrains potential solutions of certain semilinear PDE boundary value problems. The Kazdan–Warner identity is a similar necessary condition important for the Nirenberg problem of conformally prescribing scalar curvature on the sphere. For dimensions n ≥ 3 both identities are captured and extended by a single identity, due to Schoen in 1988. In each of the three cases the identity requires and involves an infinitesimal conformal symmetry. For structures with such a conformal vector field, we develop a very wide, and essentially complete, extension of this picture. Any conformally variational natural scalar invariant is shown to satisfy a Kazdan–Warner type identity, and a similar result holds for scalars that are the trace of a locally conserved 2-tensor. Scalars of the latter type are also seen to satisfy a Pohozaev–Schoen type identity on manifolds with boundary, and there are further extensions. These phenomena are explained and unified through the study of total and conformal variational theory, and in particular the gauge invariances of the functionals concerned. Our generalization of the Pohozaev–Schoen identity is shown to be a complement to a standard conservation law from physics and general relativity.

Let $m\ge 2$ be an integer. For any open domain $\mathrm{\Omega }\subset {ℝ}^{2m}$, non-positive function $\phi \in C^{\infty }(\mathrm{\Omega })$ such that ${\mathrm{\Delta }}^m\phi \equiv 0$, and bounded sequence $(V_k)\subset L^{\infty }(\mathrm{\Omega })$ we prove the existence of a sequence of functions $(u_k)\subset C^{2m-1}(\mathrm{\Omega })$ solving the Liouville equation of order $2m$

We develop a general regulated volume expansion for the volume of a manifold with boundary whose measure is suitably singular along a separating hypersurface. The expansion is shown to have a regulator independent anomaly term and a renormalized volume term given by the primitive of an associated anomaly operator. These results apply to a wide range of structures. We detail applications in the setting of measures derived from a conformally singular metric. In particular, we show that the anomaly generates invariant ($Q$-curvature, transgression)-type pairs for hypersurfaces with boundary. For the special case of anomalies coming from the volume enclosed by a minimal hypersurface ending on the boundary of a Poincaré–Einstein structure, this result recovers Branson’s $Q$-curvature and corresponding transgression. When the singular metric solves a boundary version of the constant scalar curvature Yamabe problem, the anomaly gives generalized Willmore energy functionals for hypersurfaces with boundary. Our approach yields computational algorithms for all the above quantities, and we give explicit results for surfaces embedded in 3-manifolds.

Let $(M,𝜃)$ be a compact, connected, strictly pseudo-convex CR manifold. In this paper, we give some properties of the CR Yamabe Operator $L_{𝜃}$. We present an upper bound for the Second CR Yamabe Invariant, when the First CR Yamabe Invariant is negative, and the existence of a minimizer for the Second CR Yamabe Invariant, under some conditions.

Conformal geodesics are distinguished curves on a conformal manifold, loosely analogous to geodesics of Riemannian geometry. One definition of them is as solutions to a third-order differential equation determined by the conformal structure. There is an alternative description via the tractor calculus. In this article, we give a third description using ideas from holography. A conformal $n$-manifold $X$ can be seen (formally at least) as the asymptotic boundary of a Poincaré–Einstein $(n+1)$-manifold $Y$. We show that any curve $\gamma$ in $X$ has a uniquely determined extension to a surface ${\mathrm{\Sigma }}_{\gamma }$ in $Y$, which we call the ambient surface of$\gamma$. This surface meets the boundary $X$ in right angles along $\gamma$ and is singled out by the requirement that it be a critical point of renormalized area. The conformal geometry of $\gamma$ is encoded in the Riemannian geometry of ${\mathrm{\Sigma }}_{\gamma }$. In particular, $\gamma$ is a conformal geodesic precisely when ${\mathrm{\Sigma }}_{\gamma }$ is asymptotically totally geodesic, i.e. its second fundamental form vanishes to one order higher than expected.

We also relate this construction to tractors and the ambient metric construction of Fefferman and Graham. In the $(n+2)$-dimensional ambient manifold, the ambient surface is a graph over the bundle of scales. The tractor calculus then identifies with the usual tensor calculus along this surface. This gives an alternative compact proof of our holographic characterization of conformal geodesics.

We define a conformally invariant action ${𝒮}$ on gauge connections on a closed pseudo-Riemannian manifold $M$ of dimension 6. At leading order this is quadratic in the gauge connection. The Euler–Lagrange equations of ${𝒮}$, with respect to variation of the gauge connection, provide a higher-order conformally invariant analogue of the (source-free) Yang–Mills equations. For any gauge connection $A$ on $M$, we define ${𝒮}(A)$ by first defining a Lagrangian density associated to $A$. This is not conformally invariant but has a conformal transformation analogous to a $Q$-curvature. Integrating this density provides the conformally invariant action.

In the special case that we apply ${𝒮}$ to the conformal Cartan-tractor connection, the functional gradient recovers the natural conformal curvature invariant called the Fefferman–Graham obstruction tensor. So in this case, the Euler–Lagrange equations are exactly the “obstruction-flat” condition for 6-manifolds. This extends known results for 4-dimensional pseudo-Riemannian manifolds where the Bach tensor is recovered in the Yang–Mills equations of the Cartan-tractor connection.

Over forty years ago, Paneitz, and independently Fradkin and Tseytlin, discovered a fourth-order conformally invariant differential operator, intrinsically defined on a conformal manifold, mapping scalars to scalars. This operator is a special case of the so-termed extrinsic Paneitz operator defined in the case when the conformal manifold is itself a conformally embedded hypersurface. In particular, this encodes the obstruction to smoothly solving the five-dimensional scalar Laplace equation, and suitable higher dimensional analogs, on conformally compact structures with constant scalar curvature. Moreover, the extrinsic Paneitz operator can act on tensors of general type by dint of being defined on tractor bundles. Motivated by a host of applications, we explicitly compute the extrinsic Paneitz operator. We apply this formula to obtain: an extrinsically-coupled $Q$-curvature for embedded four-manifolds, the anomaly in renormalized volumes for conformally compact five-manifolds with negative constant scalar curvature, Willmore energies for embedded four-manifolds, the local obstruction to smoothly solving the five-dimensional singular Yamabe problem, and new extrinsically-coupled fourth- and sixth-order operators for embedded surfaces and four-manifolds, respectively.

Since the 1935 proposal by Einstein, Podolsky and Rosen the riddle of nonlocality, today demonstrated by the violation of Bell's inequalities within innumerable experiments, has been a cause of concern and confusion within the debate over the foundations of quantum mechanics. The present paper tackles the problem by a nonrelativistic approach based on conformal differential geometry applied to the solution of the dynamical problem of two entangled spin 1/2 particles. It is found that the quantum nonlocality may be understood on the basis of a conformal quantum geometrodynamics acting necessarily on the full "configuration space" of the entangled particles. At the end, the violation of the Bell inequalities is demonstrated without making recourse to the common nonlocality paradigm.

After an overview of variational principles for discrete gravity, and on the basis of the approach to conformal transformations in a simplicial PL setting proposed by Luo and Glickenstein, we present at a heuristic level an improved scheme for addressing the gravitational (Euclidean) path integral and geometrodynamics.

The aim of this paper is to introduce and justify a possible generalization of the classic Bach field equations on a four-dimensional smooth manifold $M$ in the presence of field $\phi$, given by a smooth map with source $M$ and target another Riemannian manifold. Those equations are characterized by the vanishing of a two times covariant, symmetric, traceless and conformally invariant tensor field, called $\phi$-Bach tensor, that in absence of the field $\phi$ reduces to the classic Bach tensor, and by the vanishing another tensor related to the bi-energy of $\phi$. Since solutions of the Einstein-massless scalar field equations, or more generally, of the Einstein field equations with source the wave map $\phi$ solves those generalized Bach’s equations, we include the latter in our analysis providing a systematic study for them, relying on the recent concept of $\phi$-curvatures. We take the opportunity to discuss the related topic of warped product solutions.

This paper is devoted to the study of conformal and projective structures, and especially their connections, in the language of 2-frames, or $G$-structures of second order. While their normal Cartan connections are well-known, we use the dressing field method to obtain their local, mostly gauge invariant, connections. Lastly, we recall how to obtain the tractor bundle from conformal structures, and then show that we can define a projective tractor bundle using the same kind of construction.

It is shown that there is no quasi-sphere packing of the lattice grid ℤ^d+1 or a co-compact hyperbolic lattice of or the 3-regular tree × ℤ, in ℝ^d, for all d. A similar result is proved for some other graphs too. Rather than using a direct geometrical approach, the main tools we are using are from nonlinear potential theory.