Conformal Deformation Research Articles

Hadamard’s Variational Formula for Simple Eigenvalues

Here, we study Hadamard’s variational formula for simple eigenvalues under dynamical and conformal deformations. Particularly, harmonic convexity of the first eigenvalue of the Laplacian under the mixed boundary condition is established for a two-dimensional domain, which implies several new inequalities.

Self-Adhesive Electronic Skin with Bio-Inspired 3D Architecture for Mechanical Stimuli Monitoring and Human-Machine Interactions.

Recent development of wearable devices is revolutionizing the way of artificial electronic skins (E-skin), physiological health monitoring and human-machine interactions (HMI). However, challenge remains to fit flexible electronic devices to the human skin with conformal deformation and identifiable electrical feedback according to the mechanical stimuli. Herein, an adhesive E-skin is developed that can firmly attach on the human skin for mechanical stimuli perception. The laser-induced adhesive layer serves as the essential component to ensure the conformal attachment of E-skin on curved surface, which ensures the accurate conversion from mechanical deformation to precise electrical readouts. Especially, the 3D architecture facilitates the non-overlapping outputs that bi-directional joint bending and distinguishes strain/pressure. The optimized E-skin with bio-inspired micro-cilia exhibited significantly improved sensing performances with sensitivity of 0.652 kPa-1 in 0-4kPa and gauge factor of 8.13 for strain (0-15%) with robustness. Furthermore, the adhesive E-skin can distinguish inward/outward joint bending in non-overlapping behaviors, allowing the establishment of ternary system to expand communication capacity for logic outputs such as effective Morse code and intelligent control. It expects that the adhesive E-skin can serve as a functional bridge between human and electrical terminals for applications from daily mechanical monitoring to efficient HMI.

On the path of spin-[formula omitted] Berry phase

This paper investigates the relation between the Berry phase of a spinor accumulated along a path corresponding to an adiabatically varying magnetic field and the conformal deformation along the path in the parameter space. The first part demonstrates that the accumulated phase can be controlled by continuous deformation characterized by a single conformal parameter. The source of the corresponding field strength responsible for the deformed phase path is shown to be a conformally deformed monopole located at the origin in the parameter space with the fundamental strength g=±1/2. In the second part, we model a rotating magnetic field coupled to a charged spinor with a time-varying amplitude subjected to the conformal deformation, realizing a single-variable deformation of the evolution process of the phase path.

Rigidity of Schouten tensor under conformal deformation

Rigidity of Schouten tensor under conformal deformation

Potentials on the conformally compactified Minkowski spacetime and their application to quark deconfinement

In this paper, we study a class of conformal metric deformations in the quasi-radial coordinate parametrizing the three-sphere in the conformally compactified Minkowski spacetime [Formula: see text]. Prior to reduction of the associated Laplace–Beltrami operators to a Schrödinger form, a corresponding class of exactly solvable potentials (each one containing a scalar and a gradient term) is found. In particular, the scalar piece of these potentials can be exactly or quasi-exactly solvable, and among them we find the finite range confining trigonometric potentials of Pöschl–Teller, Scarf and Rosen–Morse. As an application of the results developed in the paper, the large compactification radius limit of the interaction described by some of these potentials is studied, and this regime is shown to be relevant to a quantum mechanical quark deconfinement mechanism.

Gravitational action for a massive Majorana fermion in 2d quantum gravity

We compute the gravitational action of a free massive Majorana fermion coupled to two-dimensional gravity on compact Riemann surfaces of arbitrary genus. The structure is similar to the case of the massive scalar. The small-mass expansion of the gravitational yields the Liouville action at zeroth order, and we can identify the Mabuchi action at first order. While the massive Majorana action is a conformal deformation of the massless Majorana CFT, we find an action different from the one given by the David-Distler-Kawai (DDK) ansatz.

Kirigami-Triggered Spoof Plasmonic Interconnects for Radiofrequency Elastronics.

The flexible and conformal interconnects for electronic systems as a potential signal transmission device have great prospects in body-worn or wearable applications. High-efficiency wave propagation and conformal structure deformation around human body at radio communication are still confronted with huge challenges due to the lack of methods to control the wave propagation and achieve the deformable structure simultaneously. Here, inspired by the kirigami technology, a new paradigm to construct spoof plasmonic interconnects (SPIs) that support radiofrequency (RF) surface plasmonic transmission is proposed, together with high elasticity, strong robustness, and multifunction performance. Leveraging the strong field-confinement characteristic of spoof surface plasmons polaritons, the Type-I SPI opens its high-efficiency transmission band after stretching from a simply connected metallic surface. Meanwhile, the broadband transmission of the kirigami-based SPI exhibits strong robustness and excellent stability undergoing complex deformations, i.e., bending, twisting, and stretching. In addition, the prepared Type-II SPI consisting of 2 different subunit cells can achieve band-stop transmission characteristics, with its center frequency dynamically tunable by stretching the buckled structure. Experimental measurements verify the on-off switching performance in kirigami interconnects triggered by stretching. Overcoming the mechanical limitation of rigid structure with kirigami technology, the designer SPIs exhibit high stretchability through out-of-plane structure deformation. Such kirigami-based interconnects can improve the elastic functionality of wearable RF electronics and offer high compatibility to large body motion in future body network systems.

Egg-Yolk principle for uniformizing Gromov hyperbolic domains

Recently, there are two kinds of conformal deformations of Gromov hyperbolic domains, introduced respectively by Bonk-Heinonen-Koskela and the first author, such that the deformed spaces are uniform and Loewner spaces. Given a Gromov hyperbolic domain Ω in Rn, we prove that uniform and Loewner subdomains of Ω are also uniform and Loewner in the uniformizations of the domain Ω, respectively. As a tool, we establish certain characterizations of uniform metric spaces. Finally, we provide a new proof for quasiconformal subinvariance of uniform domains.

BGG Sequences with Weak Regularity and Applications

We investigate some Bernstein–Gelfand–Gelfand complexes consisting of Sobolev spaces on bounded Lipschitz domains in Rn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {R}}^{n}$$\\end{document}. In particular, we compute the cohomology of the conformal deformation complex and the conformal Hessian complex in the Sobolev setting. The machinery does not require algebraic injectivity/surjectivity conditions between the input spaces, and allows multiple input complexes. As applications, we establish a conformal Korn inequality in two space dimensions with the Cauchy–Riemann operator and an additional third-order operator with a background in Möbius geometry. We show that the linear Cosserat elasticity model is a Hodge–Laplacian problem of a twisted de Rham complex. From this cohomological perspective, we propose potential generalizations of continuum models with microstructures.

A kirigami-enabled electrochromic wearable variable-emittance device for energy-efficient adaptive personal thermoregulation.

For centuries, people have put effort to improve the thermal performance of clothing to adapt to varying temperatures. However, most clothing we wear today only offers a single-mode insulation. The adoption of active thermal management devices, such as resistive heaters, Peltier coolers, and water recirculation, is limited by their excessive energy consumption and form factor for long-term, continuous, and personalized thermal comfort. In this paper, we developed a wearable variable-emittance (WeaVE) device, enabling the tunable radiative heat transfer coefficient to fill the missing gap between thermoregulation energy efficiency and controllability. WeaVE is an electrically driven, kirigami-enabled electrochromic thin-film device that can effectively tune the midinfrared thermal radiation heat loss of the human body. The kirigami design provides stretchability and conformal deformation under various modes and exhibits excellent mechanical stability after 1,000 cycles. The electronic control enables programmable personalized thermoregulation. With less than 5.58 mJ/cm2 energy input per switching, WeaVE provides 4.9°C expansion of the thermal comfort zone, which is equivalent to a continuous power input of 33.9 W/m2. This nonvolatile characteristic substantially decreases the required energy while maintaining the on-demand controllability, thereby providing vast opportunities for the next generation of smart personal thermal managing fabrics and wearable technologies.

Cohomology and deformation quantization of Poisson conformal algebras

In this paper, we first recall the notion of (noncommutative) Poisson conformal algebras and give some constructions of them. Then we introduce the notion of conformal formal deformations of commutative associative conformal algebras and show that Poisson conformal algebras are the corresponding semi-classical limits. At last, we develop the cohomology theory of noncommutative Poisson conformal algebras and use this cohomology to study their deformations.

Quantization and variational problem of the Gubser-Rocha Einstein-Maxwell-Dilaton model, conformal and non-conformal deformations, and its proper thermodynamics

We show that the strongly coupled field theory holographically dual to the Gubser-Rocha anti-de-Sitter Einstein-Maxwell-Dilaton theory describes not a single non-trivial AdS2 IR fixed point, but a one-parameter family. It is dual to a local quantum critical phase instead of a quantum critical point. This result follows from a detailed analysis of the possible quantizations of the gravitational theory that is consistent with the thermodynamics of the analytical Gubser-Rocha black hole solution. The analytic Gubser-Rocha black hole is only a 2-parameter subset of all possible solutions, and we construct other members numerically. These new numerical solutions correspond to turning on an additional scalar charge. Moreover, each solution has multiple holographic interpretations depending on the quantization chosen. In one particular quantization involving a multitrace deformation the scalar charge is a marginal operator. In other quantizations where the marginal multitrace operator is turned off, the analytic Gubser-Rocha black hole does not describe a finite temperature conformal fluid.

Sensitivity‐Enhanced Skin‐Attachable Strain Sensors Using Multi‐Material Printed Bioinspired Heterogeneous Moduli Structures for Proprioceptive Training

AbstractProprioception evaluated by the repetition performance of limbs’ movement is an important indicator of sports rehabilitation. Since motion monitoring using wearable sensors is hampered by challenges in sensitivity, conformability, and stability, proprioception assessments still rely heavily on bulky apparatus. This work reports a bioinspired strain sensor with sensitivity‐enhanced structures based on strain redistribution to achieve proprioceptive training of joints, showing high sensitivity and stability even under large deformation. The sensing structure consists of units with different moduli and is independent of the sensing materials and sensors’ shape, and thus can be universally applied to various devices. A new method of multi‐material 3D printing is developed to construct the complex substrates with heterogeneous moduli architecture and automatically fabricate the entire sensor. To fulfill effective proprioception exercise, the conformal deformation of printed sensors attached to joints during the whole stretch period is verified in computed tomography imaging, which indicates the output of sensors can precisely reflect the movement of joints. A complete recovery cycle of 4 weeks is demonstrated, during which an injured patient training with our skin‐attachable strain sensors restored his ability in limb control. This proprioceptive training paradigm with intelligent sensors paves the way to home‐based rehabilitation.

Perturbative techniques in conformal geometry

This paper reviews some results, mostly perturbative in nature, concerning the prescription of scalar curvature via conformal deformation of the metric on a compact manifold. This problem consists in solving a nonlinear elliptic equation with critical power, leading to non-compactness phenomena. We will see how some finite-dimensional reduction of the problem in certain situations, jointly with the variational structure, can help in finding existence and multiplicity results.

A novel conformal deformation based sparse subspace clustering

A novel conformal deformation based sparse subspace clustering

The 2+1-Dimensional Special Relativity

In the new mathematical description of special relativity in terms of the relativistic velocity space, many physical aspects acquire new geometric meanings. Performing conformal deformations upon the 2-dimensional relativistic velocity space for the (2+1)-dimensional special relativity, we find that these conformal deformations correspond to the generalized Lorentz transformations, which are akin to the ordinary Lorentz transformation, but are morphed by a global rescaling of the polar angle and correspondingly characterized by a topological integral index. The generalized Lorentz transformations keep the two fundamental principles of special relativity intact, suggesting that the indexed generalization may be related to the Bondi–Metzner–Sachs (BMS) group of the asymptotic symmetries of the spacetime metric. Furthermore, we investigate the Doppler effect of light, the Planck photon rocket, and the Thomas precession, affirming that they all remain in the same forms of the standard special relativity under the generalized Lorentz transformation. Additionally, we obtain the general formula of the Thomas precession, which gives a clear geometric meaning from the perspective of the gauge field theory in the relativistic velocity space.

Conformal Deformation, Monopoles, and Their Application to Control the Measurement, Berry Phase’s Path, and Resonant Frequency of a Spin

The effects of the conformal deformation of physical parameters on the measurement, Berry phase, and resonance of a spin are investigated. We first construct a form-invariant yet measurement-tunable representation of a spinor from the deformed parameter surface. Then, it is shown that the evolution path of the Berry phase is tunable by the conformal index. Third, the source of the path-tunable Berry phase is shown to be the conformal monopole revealed by the recent investigation of the transformation design of quantum states. The dual fields produced by the dual conformal monopole are presented. It is demonstrated that the strengths of the pole and its dual partner are determined by the reciprocally quantized conformal index. Finally, we discuss the influence of the deformation on the resonance of the spin, and show that the resonant frequency follows the quantization rule ω¯R=ωRconformalfactorn/n with ωR as the Rabi frequency without deformation.

Kähler Finsler metrics and conformal deformations

The conformai properties of complex Finsler metrics are studied. We first give a characterization of a compact complex Finsler manifold to be globally conformai Kahler. By considering the total holomorphic curvature and total Ricci curvature in the volume preserved conformal classes, we then study the variational properties of Kahler Finsler metrics. By studying the spectral properties of two average metrics, the stabilities of critical Kähler Finsler metrics are verified. Finally, a Yamabe type problem for mean holomorphic Ricci curvature is considered, and a partial existence result is obtained.

Pion form factor from an AdS deformed background

We consider a bottom-up AdS/QCD model with a conformal exponential deformation ekIz2 on a Lorentz invariant AdS background, where kI stands for the scale kπ that fixes confinement in the pion case and kγ for the kinematical energy scale associated with the virtual photon. In this model we assume the conformal dimension associated with the operator that creates pions at the boundary as Δ=3, as in the original bottom-up AdS/QCD proposals. Regarding the geometric slope related to photon field kγ, we analyze two cases: constant and depending on the transferred momentum q. In these two cases we computed the electromagnetic pion form factor as well as the pion radius. We compare our results with experimental data as well as other theoretical (holographic and non-holographic) models. In particular, for the momentum dependent scale, we find good agreement with the available experimental data as well as non-holographic models.

Prescribed Webster Scalar Curvatures on Compact Pseudo-Hermitian Manifolds

In this paper, we investigate the problem of prescribing Webster scalar curvatures on compact pseudo-Hermitian manifolds. In terms of the method of upper and lower solutions and the perturbation theory of self-adjoint operators, we can describe some sets of Webster scalar curvature functions which can be realized through pointwise CR conformal deformations and CR conformally equivalent deformations respectively from a given pseudo-Hermitian structure.