Geometric Analysis and Mathematical Physics

A special issue of Axioms (ISSN 2075-1680). This special issue belongs to the section "Mathematical Physics".

Deadline for manuscript submissions: closed (20 December 2019) | Viewed by 11284

Special Issue Editor

Dipartimento di Matematica, Informatica ed Economia, Università degli Studi della Basilicata, 85100 Potenza, Italy
Interests: nonlinear PDEs systems of variational origin, e.g., harmonic map system; Bergman kernels; tangential Cauchy–Riemann equations; Lorentzian geometry; general relativity and gravitation theory

Special Issue Information

Dear Colleagues,

This Special Issue Geometric Analysis and Mathematical Physics seeks to gather contributions by experts in

1) The theory of singularities of space-times, e.g., boundary constructions and singular holonomy, with applications to black hole mathematical physics;

2) Shearfree null geodesic congruences associated with Petrov type D solutions to gravitational field equations and a mathematical analysis of tangential Cauchy–Riemann equations, with applications to gravity coupled with electromagnetism, neutrino, and Dirac equations;

3) Fermat principles and variational theory of light rays versus the variational theory of Cartan–Chern–Moser chains, with applications to gravitational lensing and neutrino trapping;

4) The boundary behaviour of massive scalar particles building on the theory of reproducing kernel Hilbert spaces theory within the quantization of mechanical systems whose phase space is a Hermitian manifold.

Contributions seeking to unify gravity with other forces in nature (e.g., the coupling of gravity with sigma models) are welcome, as well as results from other areas of mathematical physics not comprised in the themes listed above.

The common feature of contributions, which lies at the heart of the volume to be realised, should be the mathematical rigour of methods belonging to differential geometry, both real and complex (curvature and characteristic classes, theory of G-structures, foliation theory, and Satake-Thurston orbifolds), to existence and regularity theory for solutions to partial differential equations — especially subelliptic as well suited to the study of the theory of Hörmander systems of vector fields, of Cauchy–Riemann geometry, and of Webster’s pseudohermitian geometry - harmonic maps and morphisms theory, and hence methods of complex analysis (of functions of one or several complex variables), of functional analysis, and harmonic analysis.

Prof. Dr. Sorin Dragomir
Guest Editor

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Keywords

  • Tangential Cauchy–Riemann equations
  • Lorentzian metric
  • Killing vector field
  • CR structure
  • Tanaka–Webster connection
  • Graham–Lee connection
  • Fefferman metric
  • Monge-Ampère equation
  • Reproducing kernel Hilbert space
  • Weighted kernel
  • Dirac equation
  • Harmonic map
  • Cartan chain
  • Orbifold

Published Papers (4 papers)

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Research

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11 pages, 278 KiB  
Article
Harmonic Coordinates for the Nonlinear Finsler Laplacian and Some Regularity Results for Berwald Metrics
by Erasmo Caponio and Antonio Masiello
Axioms 2019, 8(3), 83; https://0-doi-org.brum.beds.ac.uk/10.3390/axioms8030083 - 23 Jul 2019
Cited by 3 | Viewed by 2660
Abstract
We prove existence of harmonic coordinates for the nonlinear Laplacian of a Finsler manifold and apply them in a proof of the Myers–Steenrod theorem for Finsler manifolds. Different from the Riemannian case, these coordinates are not suitable for studying optimal regularity of the [...] Read more.
We prove existence of harmonic coordinates for the nonlinear Laplacian of a Finsler manifold and apply them in a proof of the Myers–Steenrod theorem for Finsler manifolds. Different from the Riemannian case, these coordinates are not suitable for studying optimal regularity of the fundamental tensor, nevertheless, we obtain some partial results in this direction when the Finsler metric is Berwald. Full article
(This article belongs to the Special Issue Geometric Analysis and Mathematical Physics)

Review

Jump to: Research

33 pages, 429 KiB  
Review
The Generalized Hypergeometric Structure of the Ward Identities of CFT’s in Momentum Space in d > 2
by Claudio Corianò and Matteo Maria Maglio
Axioms 2020, 9(2), 54; https://0-doi-org.brum.beds.ac.uk/10.3390/axioms9020054 - 14 May 2020
Cited by 5 | Viewed by 2441
Abstract
We review the emergence of hypergeometric structures (of F4 Appell functions) from the conformal Ward identities (CWIs) in conformal field theories (CFTs) in dimensions d > 2. We illustrate the case of scalar 3- and 4-point functions. 3-point functions are associated to [...] Read more.
We review the emergence of hypergeometric structures (of F4 Appell functions) from the conformal Ward identities (CWIs) in conformal field theories (CFTs) in dimensions d > 2. We illustrate the case of scalar 3- and 4-point functions. 3-point functions are associated to hypergeometric systems with four independent solutions. For symmetric correlators, they can be expressed in terms of a single 3K integral—functions of quadratic ratios of momenta—which is a parametric integral of three modified Bessel K functions. In the case of scalar 4-point functions, by requiring the correlator to be conformal invariant in coordinate space as well as in some dual variables (i.e., dual conformal invariant), its explicit expression is also given by a 3K integral, or as a linear combination of Appell functions which are now quartic ratios of momenta. Similar expressions have been obtained in the past in the computation of an infinite class of planar ladder (Feynman) diagrams in perturbation theory, which, however, do not share the same (dual conformal/conformal) symmetry of our solutions. We then discuss some hypergeometric functions of 3 variables, which define 8 particular solutions of the CWIs and correspond to Lauricella functions. They can also be combined in terms of 4K integral and appear in an asymptotic description of the scalar 4-point function, in special kinematical limits. Full article
(This article belongs to the Special Issue Geometric Analysis and Mathematical Physics)
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42 pages, 569 KiB  
Review
Weighted Bergman Kernels and Mathematical Physics
by Elisabetta Barletta, Sorin Dragomir and Francesco Esposito
Axioms 2020, 9(2), 48; https://0-doi-org.brum.beds.ac.uk/10.3390/axioms9020048 - 29 Apr 2020
Viewed by 2790
Abstract
We review several results in the theory of weighted Bergman kernels. Weighted Bergman kernels generalize ordinary Bergman kernels of domains Ω C n but also appear locally in the attempt to quantize classical states of mechanical systems whose classical phase space is [...] Read more.
We review several results in the theory of weighted Bergman kernels. Weighted Bergman kernels generalize ordinary Bergman kernels of domains Ω C n but also appear locally in the attempt to quantize classical states of mechanical systems whose classical phase space is a complex manifold, and turn out to be an efficient computational tool that is useful for the calculation of transition probability amplitudes from a classical state (identified to a coherent state) to another. We review the weighted version (for weights of the form γ = | φ | m on strictly pseudoconvex domains Ω = { φ < 0 } C n ) of Fefferman’s asymptotic expansion of the Bergman kernel and discuss its possible extensions (to more general classes of weights) and implications, e.g., such as related to the construction and use of Fefferman’s metric (a Lorentzian metric on Ω × S 1 ). Several open problems are indicated throughout the survey. Full article
(This article belongs to the Special Issue Geometric Analysis and Mathematical Physics)
64 pages, 661 KiB  
Review
A Comprehensive Survey on Parallel Submanifolds in Riemannian and Pseudo-Riemannian Manifolds
by Bang-Yen Chen
Axioms 2019, 8(4), 120; https://0-doi-org.brum.beds.ac.uk/10.3390/axioms8040120 - 30 Oct 2019
Cited by 1 | Viewed by 2878
Abstract
A submanifold of a Riemannian manifold is called a parallel submanifold if its second fundamental form is parallel with respect to the van der Waerden–Bortolotti connection. From submanifold point of view, parallel submanifolds are the simplest Riemannian submanifolds next to totally geodesic ones. [...] Read more.
A submanifold of a Riemannian manifold is called a parallel submanifold if its second fundamental form is parallel with respect to the van der Waerden–Bortolotti connection. From submanifold point of view, parallel submanifolds are the simplest Riemannian submanifolds next to totally geodesic ones. Parallel submanifolds form an important class of Riemannian submanifolds since extrinsic invariants of a parallel submanifold do not vary from point to point. In this paper, we provide a comprehensive survey on this important class of submanifolds. Full article
(This article belongs to the Special Issue Geometric Analysis and Mathematical Physics)
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