Beltrami Differentials Research Articles

Mapping relativistic to ultra/non-relativistic conformal symmetries in 2D and finite sqrt{Toverline{T}} deformations

The conformal symmetry algebra in 2D (Diff(S1)⊕Diff(S1)) is shown to be related to its ultra/non-relativistic version (BMS3≈GCA2) through a nonlinear map of the generators, without any sort of limiting process. For a generic classical CFT2, the BMS3 generators then emerge as composites built out from the chiral (holomorphic) components of the stress-energy tensor, T and overline{T} , closing in the Poisson brackets at equal time slices. Nevertheless, supertranslation generators do not span Noetherian symmetries. BMS3 becomes a bona fide symmetry once the CFT2 is marginally deformed by the addition of a sqrt{Toverline{T}} term to the Hamiltonian. The generic deformed theory is manifestly invariant under diffeomorphisms and local scalings, but it is no longer a CFT2 because its energy and momentum densities fulfill the BMS3 algebra. The deformation can also be described through the original CFT2 on a curved metric whose Beltrami differentials are determined by the variation of the deformed Hamiltonian with respect to T and overline{T} . BMS3 symmetries then arise from deformed conformal Killing equations, corresponding to diffeomorphisms that preserve the deformed metric and stress-energy tensor up to local scalings. As an example, we briefly address the deformation of N free bosons, which coincides with ultra-relativistic limits only for N = 1. Furthermore, Cardy formula and the S-modular transformation of the torus become mapped to their corresponding BMS3 (or flat) versions.

Uniform bounds on harmonic Beltrami differentials and Weil–Petersson curvatures

Abstract In this article we show that for every finite area hyperbolic surface X of type {(g,n)} and any harmonic Beltrami differential μ on X, then the magnitude of μ at any point of small injectivity radius is uniform bounded from above by the ratio of the Weil–Petersson norm of μ over the square root of the systole of X up to a uniform positive constant multiplication. We apply the uniform bound above to show that the Weil–Petersson Ricci curvature, restricted at any hyperbolic surface of short systole in the moduli space, is uniformly bounded from below by the negative reciprocal of the systole up to a uniform positive constant multiplication. As an application, we show that the average total Weil–Petersson scalar curvature over the moduli space is uniformly comparable to {-g} as the genus g goes to infinity.

Chern-Simons Origin of Superstring Integrability.

We derive the AdS_{5}×S^{5} Green-Schwarz superstring from four-dimensional Beltrami-Chern-Simons theory reduced on a manifold with singular boundary conditions. In this construction, the Lax connection and spectral parameter of the integrable superstring have a simple geometric origin in four dimensions as gauge connection and reduction coordinate. κ symmetry arises as a certain class of singular gauge transformations, while the world-sheet metric comes from complex-structure-changing Beltrami differentials. Our approach offers the possibility of investigating integrable holography using traditional field theory methods.

On infinitesimally weakly non-decreasable Beltrami differentials

Z. Zhou et al. proved that in a Teichmüller equivalence class, there exists an extremal quasiconformal mapping with a weakly non-decreasable dilatation. In this paper, we prove that in an infinitesimal equivalence class, there exists a weakly non-decreasable extremal Beltrami differential.

Eigenvalue equation for genus two modular graphs

We obtain a second order differential equation on moduli space satisfied by certain modular graph functions at genus two, each of which has two links. This eigenvalue equation is obtained by analyzing the variations of these graphs under the variation of the Beltrami differentials. This equation involves seven distinct graphs, three of which appear in the integrand of the D8 mathrm{mathcal{R}} 4 term in the low momentum expansion of the four graviton amplitude at genus two in type II string theory.

Uniformizations of stable $$(\gamma ,n)$$ ( γ , n ) -gonal Riemann surfaces

A $$(\gamma ,n)$$ -gonal pair is a pair (S, f), where S is a closed Riemann surface and $$f:S \rightarrow R$$ is a degree n holomorphic map onto a closed Riemann surface R of genus $$\gamma $$ . If the signature of (S, f) is of hyperbolic type, then it admits a uniformizing pair $$(\varGamma ,G)$$ , where G is a Fuchsian group acting on the unit disc $${{\mathbb {D}}}$$ containing $$\varGamma $$ as an index n subgroup, such that f is induced by the inclusion $$\varGamma \le G$$ . The uniformizing pair is uniquely determined by (S, f), up to conjugation by holomorphic automorphisms of $${{\mathbb {D}}}$$ , and it permits to provide a natural complex orbifold structure on the Hurwitz space parametrizing (twisted) isomorphic classes of pairs topologically equivalent to (S, f). In order to produce certain compactifications of these Hurwitz spaces, one needs to consider the so called stable $$(\gamma ,n)$$ -gonal pairs, which are natural geometrical deformations of $$(\gamma ,n)$$ -gonal pairs. Due to the above, it seems interesting to search for uniformizations of stable $$(\gamma ,n)$$ -gonal pairs, in terms of certain class of Kleinian groups. In this paper we review such uniformizations by using noded Fuchsian groups, obtained from the noded Beltrami differentials of Fuchsian groups that were previously studied by Alexander Vasil’ev and the author, and which provide uniformizations of stable Riemann orbifolds. These uniformizations permit to obtain a compactification of the Hurwitz spaces together a complex orbifold structure, these being quotients of the augmented Teichmuller space of G by a suitable finite index subgroup of its modular group.

Non-decreasable extremal Beltrami differentials of non-landslide type

In this paper, we deform a uniquely-extremal Beltrami differential into different non-decreasable Beltrami differentials, and then construct non-unique extremal Beltrami differentials such that they are both non-landslide and non-decreasable.

A note on Hamilton sequences of extremal Beltrami differentials

Let X be a hyperbolic Riemann surface and let μ be an extremal Beltrami differential on X with ‖μ‖∞∈(0,1). It is proved that, if {ϕn} is a Hamilton sequence of μ, then {ϕn} must be a Hamilton sequence of any extremal Beltrami differential ν contained in [μ]. This result proved a conjecture of the first author of this paper in 1996. This result is also a generalization of two known results.

On Hyperbolic Metric and Invariant Beltrami Differentials for Rational Maps

Given a rational map R, we consider the complement of the postcritical set \(S_R\). In this paper we discuss the existence of invariant Beltrami differentials supported on an R invariant subset X of \(S_R\). Under some geometrical restrictions on X, we show the absence of invariant Beltrami differentials with support intersecting X. In particular, we show that if X has finite hyperbolic area, then X cannot support invariant Beltrami differentials except in the case where R is a Lattes map.

Quasiconformal maps of bordered Riemann surfaces with L 2 Beltrami differentials

Let Σ be a Riemann surface of genus g bordered by n curves homeomorphic to the circle S1. Consider quasiconformal maps f: Σ→Σ1 such that the restriction to each boundary curve is a Weil-Petersson class quasisymmetry. We show that any such f is homotopic to a quasiconformal map whose Beltrami differential is L 2 with respect to the hyperbolic metric on Σ. The homotopy H(t, •): Σ → Σ1 is independent of t on the boundary curves; that is, H(t, p) = f(p) for all p ∈ ∂Σ.

Equiboundedness of the Weil-Petersson metric

Given a topological type for surfaces of negative Euler characteristic, uniform bounds are developed for derivatives of solutions of the 2 2 -dimensional constant negative curvature equation and the Weil-Petersson metric for the Teichmüller and moduli spaces. The dependence of the bounds on the geometry of the underlying Riemann surface is studied. The comparisons between the C 0 C^0 , C 2 , α C^{2,\alpha } and L 2 L^2 norms for harmonic Beltrami differentials are analyzed. Uniform bounds are given for the covariant derivatives of the Weil-Petersson curvature tensor in terms of the systoles of the underlying Riemann surfaces and the projections of the differentiation directions onto pinching directions. The main analysis combines Schauder and potential theory estimates with the analytic implicit function theorem.

On trivial and locally trivial Beltrami differentials

On trivial and locally trivial Beltrami differentials

Convergence of the Weil–Petersson metric on the Teichmüller space of bordered Riemann surfaces

Consider a Riemann surface of genus [Formula: see text] bordered by [Formula: see text] curves homeomorphic to the unit circle, and assume that [Formula: see text]. For such bordered Riemann surfaces, the authors have previously defined a Teichmüller space which is a Hilbert manifold and which is holomorphically included in the standard Teichmüller space. We show that any tangent vector can be represented as the derivative of a holomorphic curve whose representative Beltrami differentials are simultaneously in [Formula: see text] and [Formula: see text], and furthermore that the space of [Formula: see text] differentials in [Formula: see text] decomposes as a direct sum of infinitesimally trivial differentials and [Formula: see text] harmonic [Formula: see text] differentials. Thus the tangent space of this Teichmüller space is given by [Formula: see text] harmonic Beltrami differentials. We conclude that this Teichmüller space has a finite Weil–Petersson Hermitian metric. Finally, we show that the aforementioned Teichmüller space is locally modeled on a space of [Formula: see text] harmonic Beltrami differentials.

Infinitesimally extremal Beltrami differentials of non-landslide type

In this paper, it is shown that there are infinitely many extremal Beltrami differentials of non-landside type and non-constant modulus in an infinitesimal equivalence class unless the class contains a unique extremal.

Extremal Beltrami Differentials of Non-Landslide Type

AbstractIn this paper it is shown that there are infinitely many extremal Beltrami differentials of non-landside type and non-constant modulus in a Teichmüller equivalence class if the class contains a landslide extremal. The result answers, affirmatively in a stronger sense, a problem posed by Li.

Holomorphic deformations of the Ricci-flat $$\partial \overline \partial $$ -manifolds

We present a construction of globally convergent power series of integrable Beltrami differentials on the Ricci-flat $$\partial \overline \partial $$ -manifolds and also a construction of global canonical family of holomorphic (n, 0)-forms on the deformation spaces of the Ricci-flat $$\partial \overline \partial $$ -manifolds.

Higher Order Deformations of Complex Structures

Deformations of complex structures by finite Beltrami differentials are considered on general Riemann surfaces. Exact formulas to any fixed order are derived for the corresponding deformations of the period matrix, Green's functions, and correlation functions in conformal field theories with vanishing total central charge. The stress tensor is shown to give a simple representation of these deformations valid to all orders. Such deformation formulas naturally enter into the evaluation of superstring amplitudes at two-loop order with Ramond punctures, and at higher loop order, in the supergravity formulation of the RNS superstring.

Diffeomorphism Spline

Conventional splines offer powerful means for modeling surfaces and volumes in three-dimensional Euclidean space. A one-dimensional quaternion spline has been applied for animation purpose, where the splines are defined to model a one-dimensional submanifold in the three-dimensional Lie group. Given two surfaces, all of the diffeomorphisms between them form an infinite dimensional manifold, the so-called diffeomorphism space. In this work, we propose a novel scheme to model finite dimensional submanifolds in the diffeomorphism space by generalizing conventional splines. According to quasiconformal geometry theorem, each diffeomorphism determines a Beltrami differential on the source surface. Inversely, the diffeomorphism is determined by its Beltrami differential with normalization conditions. Therefore, the diffeomorphism space has one-to-one correspondence to the space of a special differential form. The convex combination of Beltrami differentials is still a Beltrami differential. Therefore, the conventional spline scheme can be generalized to the Beltrami differential space and, consequently, to the diffeomorphism space. Our experiments demonstrate the efficiency and efficacy of diffeomorphism splines. The diffeomorphism spline has many potential applications, such as surface registration, tracking and animation.

Quasi-isometry and deformations of Calabi–Yau manifolds

We prove several formulas related to Hodge theory and the Kodaira–Spencer–Kuranishi deformation theory of Kahler manifolds. As applications, we present a construction of globally convergent power series of integrable Beltrami differentials on Calabi–Yau manifolds and also a construction of global canonical family of holomorphic $$(n,0)$$ -forms on the deformation spaces of Calabi–Yau manifolds. Similar constructions are also applied to the deformation spaces of compact Kahler manifolds.

Computing quasiconformal maps using an auxiliary metric and discrete curvature flow

Surface mapping plays an important role in geometric processing, which induces both area and angular distortions. If the angular distortion is bounded, the mapping is called a quasiconformal mapping (QC-Mapping). Many surface mappings in our physical world are quasiconformal. The angular distortion of a QC mapping can be represented by the Beltrami differentials. According to QC Teichmuller theory, there is a one-to-one correspondence between the set of Beltrami differentials and the set of QC surface mappings under normalization conditions. Therefore, every QC surface mapping can be fully determined by the Beltrami differential and reconstructed by solving the so-called Beltrami equation. In this work, we propose an effective method to solve the Beltrami equation on general Riemann surfaces. The solution is a QC mapping associated with the prescribed Beltrami differential. The main strategy is to define an auxiliary metric (AM) on the domain surface, such that the original QC mapping becomes conformal under the auxiliary metric. The desired QC-mapping can then be obtained by using the conventional conformal mapping method. In this paper, we first formulate a discrete analogue of QC mappings on triangular meshes. Then, we propose an algorithm to compute discrete QC mappings using the discrete Yamabe flow method. To the best of our knowledge, it is the first work to compute the discrete QC mappings for general Riemann surfaces, especially with different topologies. Numerically, the discrete QC mapping converges to the continuous solution as the mesh grid size approaches to 0. We tested our algorithm on surfaces scanned from real life with different topologies. Experimental results demonstrate the generality and accuracy of our auxiliary metric method.