Conformal Deformation Research Articles

A Polyakov Formula for Sectors

We consider finite area convex Euclidean circular sectors. We prove a variational Polyakov formula which shows how the zeta-regularized determinant of the Laplacian varies with respect to the opening angle. Varying the angle corresponds to a conformal deformation in the direction of a conformal factor with a logarithmic singularity at the origin. We compute explicitly all the contributions to this formula coming from the different parts of the sector. In the process, we obtain an explicit expression for the heat kernel on an infinite area sector using Carslaw–Sommerfeld’s heat kernel. We also compute the zeta-regularized determinant of rectangular domains of unit area and prove that it is uniquely maximized by the square.

Classical and Quantum Superintegrability of Stäckel Systems

In this paper we discuss maximal superintegrability of both classical and quantum St\"ackel systems. We prove a sufficient condition for a flat or constant curvature St\"ackel system to be maximally superintegrable. Further, we prove a sufficient condition for a St\"ackel transform to preserve maximal superintegrability and we apply this condition to our class of St\"ackel systems, which yields new maximally superintegrable systems as conformal deformations of the original systems. Further, we demonstrate how to perform the procedure of minimal quantization to considered systems in order to produce quantum superintegrable and quantum separable systems.

A Schwarz-type lemma for noncompact manifolds with boundary and geometric applications

We prove a Schwarz-type lemma for noncompact manifolds with possibly noncompact boundary. The result is a consequence of a suitable form of the weak maximum principle of independent interest. The paper is enriched with applications to conformal deformations of noncompact manifolds with boundary, among them a generalization of a classical result by Escobar.

On the conformal scalar curvature equations on Finsler manifolds

In this paper, we study conformal deformations and [Formula: see text]-conformal deformations of Ricci-directional and second type scalar curvatures on Finsler manifolds. Then we introduce the best equation to study the Yamabe problem on Finsler manifolds. Finally, we restrict conformal deformations of metrics to [Formula: see text]-conformal deformations and derive the Yamabe functional and the Yamabe flow in Finsler geometry.

Biharmonic maps from biconformal transformations with respect to isoparametric functions

By exploiting biconformal transformations of the metric we construct biharmonic functions and mappings from Riemannian manifolds. Isoparametric functions, characterized by the property that their level sets are parallel and of constant mean curvature, play an important role in the construction of examples. We extend our method to include triconformal deformations of the metrics with respect to the Hopf map from R4 to R3, which, in addition to a biconformal deformation of the domain, incorporates a conformal deformation of the codomain.

Twisted Reality Condition for Dirac Operators

Motivated by examples obtained from conformal deformations of spectral triples and a spectral triple construction on quantum cones we propose a new twisted reality condition for the Dirac operator.

Pitfalls of the Fourier Transform Method in Affine Models, and Remedies

We study sources of potentially serious errors of popular numerical realizations of the Fourier method in affine models, and explain that, in many cases, a calibration procedure based on such a realization will be able to find a correct parameter set only in a rather small region of the parameter space, with a blind spot: an interval of strikes depending on the model and time to maturity, where accurate calculations are extremely time consuming. We explain how to construct more accurate and faster pricing and calibration procedures. Efficient realizations require analytic continuation of the characteristic function not only into a strip around the real axis (the problem of finding the maximal strip of analyticity has been extensively studied in the literature under the name moment explosion) but into much wider regions of the complex plane, and an appropriate conformal deformation of the contour of integration. In the paper, we study analytic continuation of the characteristic functions in wide classes of affine models, models with jumps including. An important ingredient of our method is the study of the associated system of generalized Riccati equations. As a byproduct, we show that the straightforward application of the Runge-Kutta method may lead to sizable errors, and suggest certain remedies. The methodology of the paper can be applied to other models (e.g. quadratic term structure models, Wishart dynamics, 3/2-model), where European options and other options can be priced using the Fourier transform techniques.

An inclusive Conformal Geometric Algebra GPU animation interpolation and deformation algorithm

In the last years, Geometric Algebra with its Euclidean, Homogeneous and Conformal models attracts the research interest in many areas of Computer Science and Engineering and particularly in Comput...

EFFICIENT PRICING OF BARRIER OPTIONS AND CREDIT DEFAULT SWAPS IN LÉVY MODELS WITH STOCHASTIC INTEREST RATE

AbstractRecently, advantages of conformal deformations of the contours of integration in pricing formulas for European options have been demonstrated in the context of wide classes of Lévy models, the Heston model, and other affine models. Similar deformations were used in one‐factor Lévy models to price options with barrier and lookback features and credit default swaps (CDSs). In the present paper, we generalize this approach to models, where the dynamics of the assets is modeled as , where X is a Lévy process, and the interest rate is stochastic. Assuming that X and r are independent, and , the infinitesimal generator of the pricing semigroup in the model for the short rate, satisfies weak regularity conditions, which hold for popular models of the short rate, we develop a variation of the pricing procedure for Lévy models which is almost as fast as in the case of the constant interest rate. Numerical examples show that about 0.15 second suffices to calculate prices of 8 options of same maturity in a two‐factor model with the error tolerance and less; in a three‐factor model, accuracy of order 0.001–0.005 is achieved in about 0.2 second. Similar results are obtained for quanto CDS, where an additional stochastic factor is the exchange rate. We suggest a class of Lévy models with the stochastic interest rate driven by 1–3 factors, which allows for fast calculations. This class can satisfy the current regulatory requirements for banks mandating sufficiently sophisticated credit risk models.

COTTON TENSOR AND CONFORMAL DEFORMATIONS OF THREE-DIMENSIONAL RICCI FLOW

In this paper, we study the deformation of the three-dimensional conformal structures by the Ricci flow. We drive the evolution equation of the Cotton--York tensor and the $L^{1}$-norm of it under the Ricci flow. In particular, we investigate the behavior of the $L^{1}$-norm of the Cotton--York tensor under the Ricci flow on three-dimensional simply-connected Riemannian homogeneous spaces which admit compact quotients. For a non-homogeneous case, we also investigate the behavior of the $L^{1}$-norm for the product metric of the Rosenau solution for the Ricci flow on $S^{2}$ and the standard metric of $S^{1}$.

Brown–York Mass and Compactly Supported Conformal Deformations of Scalar Curvature

In this article, we find a connection between Brown–York mass and the first Dirichlet eigenvalue of a Schrodinger type operator. In particular, we prove a local positive mass type theorem for metrics conformal to the background one with suitable presumptions. As applications, we investigate compactly supported conformal deformations which either increase or decrease scalar curvature. We find local conformal rigidity phenomena occur in both cases for small domains and as for manifolds with nonpositive scalar curvature it is even more rigid in particular. On the other hand, such deformations exist for closed or a type of non-compact manifolds with positive scalar curvature. These results together give an answer to a question that arises naturally in (Corvino in Commun Math Phys 214:137–189, 2000; Lohkamp in Math Ann 313:385–407, 1999).

Yamabe type equations with a sign-changing nonlinearity, and the prescribed curvature problem

In this paper, we investigate the prescribed scalar curvature problem on a non-compact Riemannian manifold (M,〈,〉), namely the existence of a conformal deformation of the metric 〈,〉 realizing a given function s˜(x) as its scalar curvature. In particular, the work focuses on the case when s˜(x) changes sign. Our main achievement are two new existence results requiring minimal assumptions on the underlying manifold, and ensuring a control on the stretching factor of the conformal deformation in such a way that the conformally deformed metric be bi-Lipschitz equivalent to the original one. The topological–geometrical requirements we need are all encoded in the spectral properties of the standard and conformal Laplacians of M. Our techniques can be extended to investigate the existence of entire positive solutions of quasilinear equations of the typeΔpu+a(x)up−1−b(x)uσ=0 where Δp is the p-Laplacian, σ>p−1>0, a,b∈Lloc∞(M) and b changes sign, and in the process of collecting the material for the proof of our theorems, we have the opportunity to give some new insight on the subcriticality theory for the Schrödinger type operatorQV′:φ⟼−Δpφ−a(x)|φ|p−2φ. In particular, we prove sharp Hardy-type inequalities in some geometrically relevant cases, notably for minimal submanifolds of the hyperbolic space.

A simple proof of an isoperimetric inequality for Euclidean and hyperbolic cone-surfaces

We prove that the isoperimetric inequalities in the Euclidean and hyperbolic plane hold for all Euclidean, respectively hyperbolic, cone-metrics on a disk with singularities of negative curvature. This is a discrete analog of the theorems of Weil and Bol that deal with Riemannian metrics of curvature bounded from above by 0, respectively by −1. A stronger discrete version was proved by A.D. Alexandrov, with a subsequent extension by approximation to metrics of bounded integral curvature.Our proof uses “discrete conformal deformations” of the metric that eliminate the singularities and increase the area. Therefore it resembles Weil's argument that uses the uniformization theorem and the harmonic minorant of a subharmonic function.

On scalar curvature rigidity of vacuum static spaces

In this paper we extend the local scalar curvature rigidity result in Brendle and Marques (J Differ Geom 88:379–394, 2011) to a small domain on general vacuum static spaces, which confirms the interesting dichotomy of local surjectivity and local rigidity about the scalar curvature in general in the light of the paper (Corvino, Commun Math Phys 214:137–189, 2000). We obtain the local scalar curvature rigidity of bounded domains in hyperbolic spaces. We also obtain the global scalar curvature rigidity for conformal deformations of metrics in the domains, where the lapse functions are positive, on vacuum static spaces with positive scalar curvature, and show such domains are maximal, which generalizes the work in Hang and Wang (Commun Anal Geom 14:91–106, 2006).

Close-to-conformal deformations of volumes

Conformal deformations are infinitesimal scale-rotations, which can be parameterized by quaternions. The condition that such a quaternion field gives rise to a conformal deformation is nonlinear and in any case only admits Möbius transformations as solutions. We propose a particular decoupling of scaling and rotation which allows us to find near to conformal deformations as minimizers of a quadratic, convex Dirichlet energy. Applied to tetrahedral meshes we find deformations with low quasiconformal distortion as the principal eigenvector of a (quaternionic) Laplace matrix. The resulting algorithms can be implemented with highly optimized standard linear algebra libraries and yield deformations comparable in quality to far more expensive approaches.

Eigenvalues of the sub-Laplacian and deformations of contact structures on a compact CR manifold

Given a compact strictly pseudoconvex CR manifold M, we study the differentiability of the eigenvalues of the sub-Laplacian Δb,θ associated with a compatible contact form (i.e. a pseudo-Hermitian structure) θ on M, under conformal deformations of θ. As a first application, we show that the property of having only simple eigenvalues is generic with respect to θ, i.e. the set of structures θ such that all the eigenvalues of Δb,θ are simple, is residual (and hence dense) in the set of all compatible positively oriented contact forms on M. In the last part of the paper, we introduce a natural notion of critical pseudo-Hermitian structure of the functional θ↦λk(θ), where λk(θ) is the k-th eigenvalue of the sub-Laplacian Δb,θ, and obtain necessary and sufficient conditions for a pseudo-Hermitian structure to be critical.

Noncommutative geometry and conformal geometry. III. Vafa–Witten inequality and Poincaré duality

This paper is the third part of a series of papers whose aim is to use the framework of twisted spectral triples to study conformal geometry from a noncommutative geometric viewpoint. In this paper we reformulate the inequality of Vafa–Witten [42] in the setting of twisted spectral triples. This involves a notion of Poincaré duality for twisted spectral triples. Our main results have various consequences. In particular, we obtain a version in conformal geometry of the original inequality of Vafa–Witten, in the sense of an explicit control of the Vafa–Witten bound under conformal changes of metrics. This result has several noncommutative manifestations for conformal deformations of ordinary spectral triples, spectral triples associated with conformal weights on noncommutative tori, and spectral triples associated with duals of torsion-free discrete cocompact subgroups satisfying the Baum–Connes conjecture.

Efficient Pricing Barrier Options and CDS in LLvy Models with Stochastic Interest Rate

Recently, advantages of conformal deformations of the contours of integration in pricing formulas for European options have been demonstrated in the context of wide classes of L'evy models, the Heston model and other affine models. Similar deformations were used in one-factor L'evy models to price options with barrier and lookback features and CDSs. In the present paper, we generalize this approach to models of structural default, where the dynamics of assets follows an exponential L'evy process $X_t$, and the interest rate $r_t$ is stochastic. Assuming that $X_t$ and $r_t$ are independent, and the infinitesimal generator of the pricing semigroup in the model for the short rate, is (block)-diagonalizable, we develop a variation of the pricing procedure for L'evy models which is almost as fast as in the case of the constant interest rate. Numerical examples show that about 0.15 sec suffice to calculate prices of 8 options of same maturity in a two-factor model with the error tolerance $5\cdot 10^{-5}$ sec. and less; in a three-factor model, accuracy of order 0.001-0.005 is achieved in about 0.2 sec. Similar results are obtained for quanto CDS, where an additional stochastic factor is the exchange rate. We suggest a class of L'evy models with the stochastic interest rate driven by 1-2 (possibly, 3) factors, which allows for fast calculations. This class can satisfy the current requirements by regulators for banks to have sufficiently sophisticated credit risk models.

Discrete Conformal Deformation: Algorithm and Experiments

In this paper, we introduce the definition of discrete conformality for triangulated surfaces with flat cone metrics and describe an algorithm for solving the problem of prescribing curvature, which is to deform the metric discrete conformally so that the curvature of the resulting metric coincides with the prescribed curvature. We explicitly construct a discrete conformal map between the input triangulated surface and the deformed triangulated surface. Our algorithm can handle a surface with any topology, with or without boundary, and can find a deformed metric for any prescribed curvature satisfying the Gauss--Bonnet formula. In addition, we present the numerical examples to show the convergence of our discrete conformality and to demonstrate the efficiency and the robustness of our algorithm.

Thermally-induced transition of lamellae orientation in block-copolymer films on 'neutral' nanoparticle-coated substrates.

Block-copolymer orientation in thin films is controlled by the complex balance between interfacial free energies, including the inter-block segregation strength, the surface tensions of the blocks, and the relative substrate interactions. While block-copolymer lamellae orient horizontally when there is any preferential affinity of one block for the substrate, we recently described how nanoparticle-roughened substrates can be used to modify substrate interactions. We demonstrate how such 'neutral' substrates can be combined with control of annealing temperature to generate vertical lamellae orientations throughout a sample, at all thicknesses. We observe an orientational transition from vertical to horizontal lamellae upon heating, as confirmed using a combination of atomic force microscopy (AFM), neutron reflectometry (NR) and rotational small-angle neutron scattering (RSANS). Using molecular dynamics (MD) simulations, we identify substrate-localized distortions to the lamellar morphology as the physical basis of the novel behavior. In particular, under strong segregation conditions, bending of horizontal lamellae induce a large energetic cost. At higher temperatures, the energetic cost of conformal deformations of lamellae over the rough substrate is reduced, returning lamellae to the typical horizontal orientation. Thus, we find that both surface interactions and temperature play a crucial role in dictating block-copolymer lamellae orientation. Our combined experimental and simulation findings suggest that controlling substrate roughness should provide a useful and robust platform for controlling block-copolymer orientation in applications of these materials.