Conformal Diffeomorphisms Research Articles

Non-existence of Yamabe Minimizers on Singular Spheres

We prove that a minimizer of the Yamabe functional does not exist for a sphere \(\mathbb S^n\) of dimension \(n\ge 3\), endowed with a standard edge-cone spherical metric of cone angle greater than or equal to \(4\pi \), along a great circle of codimension two. When the cone angle along the singularity is smaller than \(2\pi \), the corresponding metric is known to be a Yamabe metric, and we show that all Yamabe metrics in its conformal class are obtained from it by constant multiples and conformal diffeomorphisms preserving the singular set.

The Branch Set of Minimal Disks in Metric Spaces

AbstractWe study the structure of the branch set of solutions to Plateau’s problem in metric spaces satisfying a quadratic isoperimetric inequality. In our 1st result, we give examples of spaces with isoperimetric constant arbitrarily close to the Euclidean isoperimetric constant $(4\pi )^{-1}$ for which solutions have large branch set. This complements recent results of Lytchak–Wenger and Stadler stating, respectively, that any space with Euclidean isoperimetric constant is a CAT($0$) space and solutions to Plateau’s problem in a CAT($0$) space have only isolated branch points. We also show that any planar cell-like set can appear as the branch set of a solution to Plateau’s problem. These results answer two questions posed by Lytchak and Wenger. Moreover, we investigate several related questions about energy-minimizing parametrizations of metric disks: when such a map is quasisymmetric, when its branch set is empty, and when it is unique up to a conformal diffeomorphism.

Umbilical submanifolds of [formula omitted

In this article we complete the classification of the umbilical submanifolds of a Riemannian product of two space forms, addressing the case of a conformally flat product Hk×Sn−k+1, which has not been covered in previous works on the subject. We show that there exists precisely a p-parameter family of congruence classes of umbilical submanifolds of Hk×Sn−k+1 with substantial codimension p, which we prove to be at most min{k+1,n−k+2}. We study more carefully the cases of codimensions one and two and exhibit, respectively, a one-parameter family and a two-parameter family (together with three extra one-parameter families) that contain precisely one representative of each congruence class of such submanifolds. In particular, this yields another proof of the classification of all (congruence classes of) umbilical submanifolds of Sn×R, and provides a similar classification for the case of Hn×R. We determine all possible topological types, actually, diffeomorphism types, of a complete umbilical submanifold of Hk×Sn−k+1. We also show that umbilical submanifolds of the product model of Hk×Sn−k+1 can be regarded as rotational submanifolds in a suitable sense, and explicitly describe their profile curves when k=n. As a consequence of our investigations, we prove that every conformal diffeomorphism of Hk×Sn−k+1 onto itself is an isometry.

Shape Dynamics of the TT¯ Deformation

I will show how the flow triggered by deforming two-dimensional conformal field theories on a torus by the TT¯ operator is identical to the evolution generated by the (radial) quantum Shape Hamiltonian in 2 + 1 dimensions. I will discuss how the gauge invariances of the Shape Dynamics, i.e., volume-preserving conformal invariance and diffeomorphism invariance along slices of constant radius are realized as Ward identities of the deformed quantum field theory. I will also comment about the relationship between the reduction to shape space on the gravity side and the solvability of the irrelevant operator deformation of the conformal field theory

On a generalization of the Liouville conformal theorem to general rigid geometric structures

For any \(n\ge 3\), the classical Liouville theorem asserts that any local conformal diffeomorphism defined on a connected open subset of \({\mathbb {S}}^n\) can be uniquely extended to a M\(\ddot{\mathrm{o}}\)bius transformation. In this article, we formulate and prove a generalization of this theorem to general rigid geometric structures.

Ground state Dirac bubbles and Killing spinors

We prove a classification result for ground state solutions of the critical Dirac equation on \(\mathbb {R}^n\), \(n\geqslant 2\). By exploiting its conformal covariance, the equation can be posed on the round sphere \(\mathbb {S}^n\) and the non-zero solutions at the ground level are given by Killing spinors, up to conformal diffeomorphisms. Moreover, such ground state solutions of the critical Dirac equation are also related to the Yamabe equation for the sphere, for which we crucially exploit some known classification results.

CONFORMAL IMAGE REGISTRATION BASED ON CONSTRAINED OPTIMIZATION

AbstractImage registration is the process of finding an alignment between two or more images so that their appearances match. It has been widely studied and applied to several fields, including medical imaging and biology, where it is related to morphometrics. In this paper, we present a construction of conformal diffeomorphisms which is based on constrained optimization. We consider a set of different penalty terms that aim to enforce conformality, based on discretizations of the Cauchy–Riemann equations and geometric principles, and demonstrate them experimentally on a variety of images.

On conformal submersions with geodesic or minimal fibers

We prove that every conformal submersion from a round sphere onto an Einstein manifold with fibers being geodesics is—up to an isometry—the Hopf fibration composed with a conformal diffeomorphism of the complex projective space of appropriate dimension. We also show that there are no conformal submersions with minimal fibers between manifolds satisfying certain curvature assumptions.

Deformed Heisenberg charges in three-dimensional gravity

We consider the bulk plus boundary phase space for three-dimensional gravity with negative cosmological constant for a particular choice of conformal boundary conditions: the conformal class of the induced metric at the boundary is kept fixed and the mean extrinsic curvature is constrained to be one. Such specific conformal boundary conditions define so-called Bryant surfaces, which can be classified completely in terms of holomorphic maps from Riemann surfaces into the spinor bundle. To study the observables and gauge symmetries of the resulting bulk plus boundary system, we will introduce an extended phase space, where these holomorphic maps are now part of the gravitational bulk plus boundary phase space. The physical phase space is obtained by introducing two sets of Kac-Moody currents, which are constrained to vanish. The constraints are second-class and the corresponding Dirac bracket yields an infinite-dimensional deformation of the Heisenberg algebra for the spinor-valued surface charges. Finally, we compute the Poisson algebra among the generators of conformal diffeomorphisms and demonstrate that there is no central charge. Although the central charge vanishes and the boundary CFT is likely non-unitary, we will argue that a version of the Cardy formula still applies in this context, such that the entropy of the BTZ black hole can be derived from the degeneracy of the eigenstates of quasi-local energy.

THE SCHWARZIAN DERIVATIVE AND CONFORMAL TRANSFORMATION ON FINSLER MANIFOLDS

Thurston, in 1986, discovered that the Schwarzian derivative has mysterious properties similar to the curvature on a manifold. After his work, there are several approaches to develop this notion on Riemannian manifolds. Here, a tensor field is identified in the study of global conformal diffeomorphisms on Finsler manifolds as a natural generalization of the Schwarzian derivative. Then, a natural definition of a Mobius mapping on Finsler manifolds is given and its properties are studied. In particular, it is shown that Mobius mappings are mappings that preserve circles and vice versa. Therefore, if a forward geodesically complete Finsler manifold admits a Mobius mapping, then the indicatrix is conformally diffeomorphic to the Euclidean sphere $ S^{n-1}$ in $ \mathbb{R}^n $. In addition, if a forward geodesically complete absolutely homogeneous Finsler manifold of scalar flag curvature admits a non-trivial change of Mobius mapping, then it is a Riemannian manifold of constant sectional curvature.

On Conformal and Concircular Diffeomorphisms of Eisenhart’s Generalized Riemannian Spaces

We consider conformal and concircular mappings of Eisenhart’s generalized Riemannian spaces. We prove conformal and concircular invariance of some tensors in Eisenhart’s generalized Riemannian spaces. We give new generalizations of symmetric spaces via Eisenhart’s generalized Riemannian spaces. Finally, we describe some properties of covariant derivatives of tensors analogous to Yano’s tensor of concircular curvature in Eisenhart symmetric spaces of various kinds.

A contribution to the geometry in the large of conformal diffeomorphisms

Our contribution to the geometry of conformal diffeomorphisms is a series of Liouville-type theorems for conformal diffeomorphisms of complete Riemannian manifolds. In addition, we give applications of these results to the geometries in the large of conformally flat Riemannian manifolds and conharmonic transformations of complete Riemannian manifolds, as well as to the theory of general relativity.

Solitons and Nonsmooth Diffeomorphisms in Conformal Nets

We show that any solitonic representation of a conformal (diffeomorphism covariant) net on S^1 has positive energy and construct an uncountable family of mutually inequivalent solitonic representations of any conformal net, using nonsmooth diffeomorphisms. On the loop group nets, we show that these representations induce representations of the subgroup of loops compactly supported in S^1 \ {-1} which do not extend to the whole loop group. In the case of the U(1)-current net, we extend the diffeomorphism covariance to the Sobolev diffeomorphisms D^s(S^1), s > 2, and show that the positive-energy vacuum representations of Diff_+(S^1) with integer central charges extend to D^s(S^1). The solitonic representations constructed above for the U(1)-current net and for Virasoro nets with integral central charge are continuously covariant with respect to the stabilizer subgroup of Diff_+(S^1) of -1 of the circle.

Conformal deformations of a specific class of lorentzian manifolds with non-irreducible holonomy representation

Concerning holonomy theory or in the context of the existence of parallel spinors, Lorentzian manifolds with indecomposable, but non-irreducible holonomy representation have considerable significance. In this paper, we have comprehensively concentrated on conformal deformations of a particular class of four dimensional Lorentzian manifolds with indecomposable, non-irreducible holonomy representation which admit a recurrent light-like vector field. This type of Lorentzian manifolds are denoted by pr-waves and their holonomy algebra is contained in the parabolic algebra $ \big(\mathbb{R}\oplus \mbox{so(2)}\big)\ltimes \mathbb{R}^2 $. Moreover, it is mainly illustrated that for an arbitrary conformal diffeomorphism by inducing some specific structural conditions a pr-wave manifold behaves totally analogous to Einstein manifolds. Particularly, it is demonstrated that in some special circumstances the structure of a pr-wave manifold is precisely the same as a manifold equipped with a warped product metric.

Parametric rigidity of real families of conformal diffeomorphisms tangent to x→−x

We prove that one-parameter families of real germs of conformal diffeomorphisms tangent to the involution x ↦−x are rigid in the parameter. We establish a connection between the dynamics in the Poincaré and Siegel domains. Although repeatedly employed in the literature, the dynamics in the Siegel domain does not explain the intrinsic real properties of these germs. Rather, these properties are fully elucidated in the Poincaré domain, where the fixed points are linearizable. However, a detailed study of the dynamics in the Siegel domain is of crucial importance. We relate both points of view on the intersection of the Siegel normalization domains.

Local gravity theories in conformal superspace

The conformal method for the initial value formulation of GR is one of the most powerful and widely usedtools in both analytical studies and numerical simulations of solutions. As is well-known, it exploits a hidden spatial conformal symmetry that can be extracted from the equations. But why conformal symmetry? Motivated by these questions and by well-known obstacles to quantum gravity, I look for the most general geometrodynamical symmetries compatible with a reduced physical configuration space for metric gravity. I argue that they are indeed spatial conformal diffeomorphisms. The next question is: what sort of theories most naturally incorporate this symmetry? Demanding locality for an action for metric gravity compatible with these principles determines the allowed operators, both for the purely gravitational part as well as matter couplings. The symmetries guarantee that there are two gravitational propagating physical degrees of freedom, but no explicit refoliation-invariance. The simplest such system has a geometric interpretation as a geodesic model in infinite-dimensional conformal superspace. One example of solution to the equations of motion corresponds to a static Bianchi IX spatial ansatz. The unique coupling to electromagnetism forces the electromagnetic equations to be hyperbolic, enabling us to “build” a standard space–time causal structure. There are, however, deviations from the standard Maxwell equations when space–time anisotropies become too large. Regarding quantization, with the geometric interpretation and the lack of refoliation invariance, the path integral treatment of the symmetries becomes much less involved than the similar approaches to GR. The symmetries form an (infinite-dimensional) Lie algebra, and no BFV treatment is necessary. Moreover, one can use gauge-invariant variational principles for selecting the boundary conditions of the path integral. We find that the propagator around the homogeneous solution has up to 6th order spatial derivatives, giving it plausible regularization properties as in Horava–Lifschitz.

Noncommutative geometry and conformal geometry. I. Local index formula and conformal invariants

This paper is part of a series of articles on noncommutative geometry and conformal geometry. In this paper, we reformulate the local index formula in conformal geometry in such a way to take into account the action of conformal diffeomorphisms. We also construct and compute a whole new family of geometric conformal invariants associated with conformal diffeomorphisms. This includes conformal invariants associated with equivariant characteristic classes. The approach of this paper involves using various tools from noncommutative geometry, such as twisted spectral triples and cyclic theory. An important step is to establish the conformal invariance of the Connes–Chern character of the conformal Dirac spectral triple of Connes–Moscovici. Ultimately, however, the main results of the paper are stated in a purely differential-geometric fashion.

Comparing shapes of genus-zero surfaces

We present a new method to compare the shapes of genus-zero surfaces by introducing a measure of mutual stretching, the symmetric distortion energy. Given a pair of genus-zero surfaces, we establish the existence of a conformal diffeomorphism that minimizes this energy. We then prove that the energies of the minimizing diffeomorphisms produce a metric on the space of genus-zero Riemannian surfaces. This metric and the corresponding optimal diffeomorphisms are shown to have properties that are highly desirable for applications.

A conformal boundary for space-times based on light-like geodesics: The 3-dimensional case

A new causal boundary, which we will term the l-boundary, inspired by the geometry of the space of light rays and invariant by conformal diffeomorphisms for space-times of any dimension m≥3, proposed by one of the authors [R. J. Low, The Space of Null Geodesics (and a New Causal Boundary), Lecture Notes in Physics 692 (Springer, 2006), pp. 35–50] is analyzed in detail for space-times of dimension 3. Under some natural assumptions, it is shown that the completed space-time becomes a smooth manifold with boundary and its relation with Geroch-Kronheimer-Penrose causal boundary is discussed. A number of examples illustrating the properties of this new causal boundary as well as a discussion on the obtained results will be provided.

On the set of metrics without local limiting Carleman weights

In the paper [1] it is shown that the set of Riemannian metrics which do not admit global limiting Carleman weights is open and dense, by studying the conformally invariant Weyl and Cotton tensors. In the paper [7] it is shown that the set of Riemannian metrics which do not admit local limiting Carleman weights at any point is residual, showing that it contains the set of metrics for which there are no local conformal diffeomorphisms between any distinct open subsets. This paper is a continuation of [1], in order to prove that the set of Riemannian metrics which do not admit local limiting Carleman weights at any point is open and dense.