### Erasmo Caponio

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Dettagli Personali

Data di nascita: 15 Febbraio 1971

Luogo di Nascita: Gioia del Colle (BA)

Nazionalità: Italiana

Posizione Accademica:

[07/2015--oggi] Professore Associato di Analisi Matematica presso il Politecnico di Bari (SSD MAT/05)

[10/2002-06/2015] Ricercatore Universitario presso il Politecnico di Bari (SSD MAT/05).

Istruzione e Formazione:

[03/2000-10/2002] Dottorato di Ricerca in Matematica presso l'Università di Firenze. Titolo conseguito il 30/09/2003.

Laurea in Matematica presso l'Università di Bari, conseguita il 18/03/1999 con voto 110/110 e lode.

Interessi di Ricerca: Metodi variazionali e topologici nello studio di equazioni differenziali su varietà e applicazioni alle proprietà globali di varietà Riemanniane, Finsleriane e Lorentziane.

Affiliazioni Accademiche: UMI (Unione Matematica Italiana) GNAMPA (Gruppo Nazionale per l'Analisi Matematica la Probabilità e le loro Applicazioni).

## Pubblicazioni

*Il seguente elenco è solo una parte della Produzione scientifica del docente.*

Per maggiori informazioni consultare il Catalogo Istituzionale dei prodotti della Ricerca (IRIS) .

Caponio E and Stancarone G.

**Standard static Finsler spacetimes**.*INTERNATIONAL JOURNAL OF GEOMETRIC METHODS IN MODERN PHYSICS*13, 2016. DOI BibTeX@article{ 11589_25715, author = "Caponio E and Stancarone G", title = "Standard static Finsler spacetimes", year = 2016, journal = "INTERNATIONAL JOURNAL OF GEOMETRIC METHODS IN MODERN PHYSICS", volume = 13, doi = "http://dx.doi.org/10.1142/S0219887816500407" }

Caponio E, Germinario A V and Sanchez M.

**Convex regions of stationary spacetimes and Randers spaces. Applications to lensing and asymptotic flatness**.*THE JOURNAL OF GEOMETRIC ANALYSIS*26:791–836, 2016. URL, DOI BibTeX@article{ 11589_6387, author = "Caponio E and Germinario A V and Sanchez M", title = "Convex regions of stationary spacetimes and Randers spaces. Applications to lensing and asymptotic flatness", year = 2016, journal = "THE JOURNAL OF GEOMETRIC ANALYSIS", volume = 26, abstract = "By using stationary-to-Randers correspondence (SRC, see Caponio et al. in Rev Mat Iberoamericana 27:919–952, 2011), a characterization of light and time- convexity of the boundary of a region of a standard stationary (n + 1)-spacetime is obtained, in terms of the convexity of the boundary of a domain in a Finsler n or (n + 1)-space of Randers type. The latter convexity is analyzed in depth and, as a consequence, the causal simplicity and the existence of causal geodesics confined in the region and connecting a point to a stationary line are characterized. Applications to asymptotically flat spacetimes include the light-convexity of hypersurfaces S n−1 (r ) × R, where S n−1 (r ) is a sphere of large radius in a spacelike section of an end, as well as the characterization of their time-convexity with natural physical interpretations. The lens effect of both light rays and freely falling massive particles with a finite lifetime, (i.e., the multiplicity of such connecting curves) is characterized in terms of the focalization of the geodesics in the underlying Randers manifolds.", keywords = "Stationary spacetime, Finsler manifold, Randers metric, convex boundary, timelike and lightlike geodesics, gravitational lensing, asymptotic flatness", url = "http://dx.doi.org/10.1007/s12220-015-9572-z", doi = "10.1007/s12220-015-9572-z", pages = "791--836" }

Caponio E, Javaloyes M A and Sanchez M.

**Wind Finslerian structures: from Zermelo's navigation to the causality of spacetimes**. 2015. URL BibTeX@misc{ 11589_25535, author = "Caponio E and Javaloyes M A and Sanchez M", title = "Wind Finslerian structures: from Zermelo's navigation to the causality of spacetimes", year = 2015, url = "http://arxiv.org/abs/1407.5494" }

Caponio E, Devillanova G, Maddalena F, Masiello A and Solimini S.

**Problems in Calculus of Variations and Nonlinear Analysis**. In*Proceedings of the “1st Workshop on the State of the Art and Challenges of Research Efforts at POLIBA, 03–05 dicembre 2014, Bari, Italy*. 2014, 227–331. BibTeX@conference{ 11589_20518, author = "Caponio E and Devillanova G and Maddalena F and Masiello A and Solimini S", title = "Problems in Calculus of Variations and Nonlinear Analysis", year = 2014, publisher = "Cangemi Editore", booktitle = "Proceedings of the ``1st Workshop on the State of the Art and Challenges of Research Efforts at POLIBA, 03--05 dicembre 2014, Bari, Italy", pages = "227--331" }

Caponio E and Stancarone G.

**Causality properties of static Finsler spacetimes**. In*Proceedings of the ”1st Workshop on the State of the Art and Challenges of Research Efforts at POLIBA" track C2 – Research Contributions*. 2014, 21–25. BibTeX@conference{ 11589_18376, author = "Caponio E and Stancarone G", title = "Causality properties of static Finsler spacetimes", year = 2014, publisher = "Cangemi Editore", address = "Roma", booktitle = {Proceedings of the ''1st Workshop on the State of the Art and Challenges of Research Efforts at POLIBA" track C2 -- Research Contributions}, pages = "21--25" }

Caponio E, Javaloyes M A and Masiello A.

**Addendum to "Morse theory of causal geodesics in a stationary spacetime via Morse theory of geodesics of a Finsler metric", Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010) 857–876**.*ANNALES DE L INSTITUT HENRI POINCARÉ. ANALYSE NON LINÉAIRE*30:961–968, 2013. DOI BibTeX@article{ 11589_52324, author = "Caponio E and Javaloyes M A and Masiello A", title = {Addendum to "Morse theory of causal geodesics in a stationary spacetime via Morse theory of geodesics of a Finsler metric", Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010) 857--876}, year = 2013, journal = "ANNALES DE L INSTITUT HENRI POINCARÉ. ANALYSE NON LINÉAIRE", volume = 30, abstract = "We give the details of the proof of the existence of an isomorphism between some homological groups related to the critical groups of the energy functional of a Finsler manifold in the two end-points boundary conditions.", keywords = "Morse theory, critical groups, Finsler metric", doi = "10.1016/j.anihpc.2013.03.005", pages = "961--968" }

Caponio E.

. Volume 26, pages 163–177, Springer Science+Business Media, LLC 2011, 2013. DOI BibTeX**Infinitesimal and local convexity of a hypersurface in a semi-Riemannian manifold**@inbook{ 11589_11804, author = "Caponio E", title = "Infinitesimal and local convexity of a hypersurface in a semi-Riemannian manifold", year = 2013, publisher = "Springer Science+Business Media, LLC 2011", address = "New york", journal = "SPRINGER PROCEEDINGS IN MATHEMATICS & STATISTICS", volume = 26, booktitle = "Recent Trends in Lorentzian Geometry", abstract = "Given a Riemannian manifold (M, g) and an embedded hypersurface H in M, a result by R. L. Bishop states that infinitesimal convexity on a neighborhood of a point in H implies local convexity. Such result was extended very recently to Finsler manifolds by the author et al. in [2]. We show in this note that the techniques in [2], unlike the ones in Bishop’s paper, can be used to prove the same result when (M, g) is semi-Riemannian. We make some remarks for the case when only time-like, null, or space-like geodesics are involved. The notion of geometric convexity is also reviewed, and some applications to geodesic connectedness of an open subset of a Lorentzian manifold are given.", keywords = "convexity; geodesics; semi-Riemannian manifolds", doi = "10.1007/978-1-4614-4897-6_6", pages = "163--177" }

Caponio E and Javaloyes M A.

**A remark on the Morse Theorem about infinitely many geodesics between two points**. In*International Meeting on Differential Geometry, Cordoba 2010*. 2013, 39–48. URL BibTeX@conference{ 11589_22441, author = "Caponio E and Javaloyes M A", title = "A remark on the Morse Theorem about infinitely many geodesics between two points", year = 2013, publisher = "Ediciones Don Folio", address = "Cordoba", booktitle = "International Meeting on Differential Geometry, Cordoba 2010", url = "http://arxiv.org/abs/1105.3923v2", pages = "39--48" }

Caponio E, Javaloyes MA and Masiello A.

**On the energy functional on Finsler manifolds and applications to stationary spacetimes**.*MATHEMATISCHE ANNALEN*351:365–392, 2011. DOI BibTeX@article{ 11589_52081, author = "Caponio E and Javaloyes MA and Masiello A", title = "On the energy functional on Finsler manifolds and applications to stationary spacetimes", year = 2011, journal = "MATHEMATISCHE ANNALEN", volume = 351, abstract = "In this paper, we first study some global properties of the energy functional on a non-reversible Finsler manifold. In particular we present a fully detailed proof of the Palais--Smale condition under the completeness of the Finsler metric. Moreover we define a Finsler metric of Randers type, which we call Fermat metric, associated to a conformally standard stationary spacetime. We shall study the influence of the Fermat metric on the causal properties of the spacetime, mainly the global hyperbolicity. Moreover we study the relations between the energy functional of the Fermat metric and the Fermat principle for the light rays in the spacetime. This allows one to obtain existence and multiplicity results for light rays, using the Finsler theory. Finally the case of timelike geodesics with fixed energy is considered. The research that led to the present paper was partially supported by a grant of the group GNAMPA of INdAM", keywords = "Finsler metric, critical point theory, causality, geodesics", doi = "10.1007/s00208-010-0602-7", pages = "365--392" }

Bartolo R, Caponio E, Germinario AV and Sanchez M.

**Convex domains of Finsler and Riemannian manifolds**.*CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS*40:335–356, 2011. DOI BibTeX@article{ 11589_52220, author = "Bartolo R and Caponio E and Germinario AV and Sanchez M", title = "Convex domains of Finsler and Riemannian manifolds", year = 2011, journal = "CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS", volume = 40, abstract = {"A detailed study of the notions of convexity for a hypersurface in a Finsler manifold is carried out. In particular, the infinitesimal and local notions of convexity are shown to be equivalent. Our approach differs from Bishop's one in his classical result (Bishop, Indiana Univ Math J 24:169-172, 1974) for the Riemannian case. Ours not only can be extended to the Finsler setting but it also reduces the typical requirements of differentiability for the metric and it yields consequences on the multiplicity of connecting geodesics in the convex domain defined by the hypersurface."}, keywords = "Finsler metric, convex boundary, geodesic", doi = "10.1007/s00526-010-0343-1", pages = "335--356" }

Caponio E, Javaloyes MA and Sanchez M.

**On the interplay between Lorentzian Causality and Finsler metrics of Randers type**.*REVISTA MATEMATICA IBEROAMERICANA*27:919–952, 2011. DOI BibTeX@article{ 11589_2007, author = "Caponio E and Javaloyes MA and Sanchez M", title = "On the interplay between Lorentzian Causality and Finsler metrics of Randers type", year = 2011, journal = "REVISTA MATEMATICA IBEROAMERICANA", volume = 27, abstract = "We obtain some results in both Lorentz and Finsler geometries, by using a correspondence between the conformal structure (Causality) of standard stationary spacetimes on M = R x S and Randers metrics on S. In particular: (1) For stationary spacetimes: we give a simple characterization of when R x S is causally continuous or globally hyperbolic (including in the latter case, when S is a Cauchy hypersurface), in terms of an associated Randers metric. Consequences for the computability of Cauchy developments are also derived. (2) For Finsler geometry: Causality suggests that the role of completeness in many results of Riemannian Geometry (geodesic connectedness by minimizing geodesics, Bonnet-Myers, Synge theorems) is played by the compactness of symmetrized closed balls in Finslerian Geometry. Moreover, under this condition we show that for any Randers metric R there exists another Randers metric R with the same pregeodesics and geodesically complete. Even more, results on the differentiability of Cauchy horizons in spacetimes yield consequences for the differentiability of the Randers distance to a subset, and vice versa.", keywords = "Causality in Lorentzian manifold; Cauchy horizons; Finsler metric", doi = "10.4171/RMI/658", pages = "919--952" }

Bartolo R, Candela A M and Caponio E.

**An Avez-Seifert type theorem for normal geodesics on a stationary spacetime**. In*Advances in Lorentzian Geometry, AMS/IP Stud. Adv. Math. 49, Amer. Math. Soc., Providence, RI*. 2011, 1–9. BibTeX@conference{ 11589_52712, author = "Bartolo R and Candela A M and Caponio E", title = "An Avez-Seifert type theorem for normal geodesics on a stationary spacetime", year = 2011, booktitle = "Advances in Lorentzian Geometry, AMS/IP Stud. Adv. Math. 49, Amer. Math. Soc., Providence, RI", pages = "1--9" }

Caponio E.

**The index of a geodesic in a Randers space and some remarks about the lack of regularity of the energy functional of a Finsler metric**.*ACTA MATHEMATICA ACADEMIAE PAEDAGOGICAE NYÍREGYHÁZIENSIS*26:265–274, 2010. BibTeX@article{ 11589_5293, author = "Caponio E", title = "The index of a geodesic in a Randers space and some remarks about the lack of regularity of the energy functional of a Finsler metric", year = 2010, journal = "ACTA MATHEMATICA ACADEMIAE PAEDAGOGICAE NYÍREGYHÁZIENSIS", volume = 26, abstract = "In a series of recent papers by the author and collaborators, the relations existing between the metric properties of Randers spaces and the conformal geometry of stationary Lorentzian manifolds were discovered and investigated. These relations were called Stationary-to-Randers Correspondence (SRC). In this paper we focus on one aspect of SRC, the equality between the index of a geodesic in a Randers space and that of its lightlike lift in the associated conformal stationary spacetime. Moreover we make some remarks about regularity of the energy functional of a Finsler metric on the infinite dimensional manifold of $H^1$ curves connecting two points, in connection with infinite dimensional techniques in Morse Theory.", keywords = "index theorem; Morse theory; Finsler metric", pages = "265--274" }

Caponio E, Javaloyes M and Piccione P.

**Maslov index in semi-Riemannian submersions**.*ANNALS OF GLOBAL ANALYSIS AND GEOMETRY*38:57–75, 2010. DOI BibTeX@article{ 11589_8837, author = "Caponio E and Javaloyes M and Piccione P", title = "Maslov index in semi-Riemannian submersions", year = 2010, journal = "ANNALS OF GLOBAL ANALYSIS AND GEOMETRY", volume = 38, abstract = "We study focal points and Maslov index of a horizontal geodesic gamma in the total space of a semi-Riemannian submersion by determining an explicit relation with the corresponding objects along the projected geodesic in the base space. We use this result to calculate the focal Maslov index of a (spacelike) geodesic in a stationary spacetime which is orthogonal to a timelike Killing vector field.", keywords = "submersion; Maslov index; semi-Riemannian manifold", doi = "10.1007/s10455-010-9200-x", pages = "57--75" }

Caponio E, Javaloyes MA and Masiello A.

**Finsler geodesics in the presence of a convex function and their applications**.*JOURNAL OF PHYSICS. A, MATHEMATICAL AND THEORETICAL*43, 2010. DOI BibTeX@article{ 11589_6389, author = "Caponio E and Javaloyes MA and Masiello A", title = "Finsler geodesics in the presence of a convex function and their applications", year = 2010, journal = "JOURNAL OF PHYSICS. A, MATHEMATICAL AND THEORETICAL", volume = 43, abstract = "In this paper, we obtain a result about the existence of only a finite number of geodesics between two fixed non-conjugate points in a Finsler manifold endowed with a convex function. We apply it to Randers and Zermelo metrics. As a by-product, we also get a result about the finiteness of the number of lightlike and timelike geodesics connecting an event to a line in a standard stationary spacetime.", keywords = "convexity; Finsler metric; geodesic", doi = "10.1088/1751-8113/43/13/135207" }

Bartolo R, Candela AM and Caponio E.

**Normal Geodesics Connecting two Non-necessarily Spacelike Submanifolds in a Stationary Spacetime**.*ADVANCED NONLINEAR STUDIES*10:851–866, 2010. BibTeX@article{ 11589_292, author = "Bartolo R and Candela AM and Caponio E", title = "Normal Geodesics Connecting two Non-necessarily Spacelike Submanifolds in a Stationary Spacetime", year = 2010, journal = "ADVANCED NONLINEAR STUDIES", volume = 10, abstract = "In this paper we obtain an existence theorem for normal geodesics joining two given submanifolds in a globally hyperbolic stationary spacetime M. The proof is based on both variational and geometric arguments involving the causal structure of M, the completeness of suitable Finsler metrics associated to it and some basic properties of a submersion. By this interaction, unlike previous results on the topic, also non-spacelike submanifolds can be handled.", keywords = "spacelike submanifold; normal geodesic; submersion", pages = "851--866" }

Caponio E, Javaloyes MA and Masiello A.

**Morse theory of causal geodesics in a stationary spacetime via Morse theory of geodesics of a Finsler metric**.*ANNALES DE L INSTITUT HENRI POINCARÉ. ANALYSE NON LINÉAIRE*27:857–876, 2010. DOI BibTeX@article{ 11589_2008, author = "Caponio E and Javaloyes MA and Masiello A", title = "Morse theory of causal geodesics in a stationary spacetime via Morse theory of geodesics of a Finsler metric", year = 2010, journal = "ANNALES DE L INSTITUT HENRI POINCARÉ. ANALYSE NON LINÉAIRE", volume = 27, abstract = "We show that the index of a lightlike geodesic in a conformally standard stationary spacetime is equal to the index of its spatial projection as a geodesic of a Finsler metric. Moreover we obtain the Morse relations of lightlike geodesics connecting a point to a curve by using Morse theory on the Finsler manifold. To this end, we prove a splitting lemma for the energy functional of a Finsler metric. Finally, we show that the reduction to Morse theory of a Finsler manifold can be done also for timelike geodesics.", keywords = "Morse theory; Finsler metric; Index theorem", doi = "10.1016/j.anihpc.2010.01.001", pages = "857--876" }

Caponio E.

**Trajectories of charged particles in the Reissner-Nordström spacetime**. In*Recent Developments in Gravitational Physics. Institute of Physics Conference Series*176. 2006, 341–347. BibTeX@conference{ 11589_22573, author = "Caponio E", title = "Trajectories of charged particles in the Reissner-Nordström spacetime", year = 2006, publisher = "Taylor & Francis", address = "New York", journal = "INSTITUTE OF PHYSICS CONFERENCE SERIES", volume = 176, booktitle = "Recent Developments in Gravitational Physics. Institute of Physics Conference Series", abstract = "We study existence and multiplicity of timelike global solutions to the Lorentz force equation with stationary gravitational and electromagnetic background fields. The theory is applied to obtain some results about the number of trajectories for a charged test particle, connecting two points in the outer region of the Reissner- Nordstr¨om spacetime.", keywords = "Reissner-Nordstrom spacetime; charge-to-mass ratio; Lorentz force law", pages = "341--347" }

Caponio E and Minguzzi E.

**Solutions to the Lorentz force equation with fixed charge-to-mass ratio in globally hyperbolic space-times**.*JOURNAL OF GEOMETRY AND PHYSICS*49:176–186, 2004. DOI BibTeX@article{ 11589_6391, author = "Caponio E and Minguzzi E", title = "Solutions to the Lorentz force equation with fixed charge-to-mass ratio in globally hyperbolic space-times", year = 2004, journal = "JOURNAL OF GEOMETRY AND PHYSICS", volume = 49, abstract = "We extend the classical Avez-Seifert theorem to trajectories of charged test particles with fixed charge-to-mass ratio. In particular, given two events x(0) and x(1), with x(1) in the chronological future of x(0), we find an interval I =] - R, R[ such that for any q/m epsilon I there is a timelike connecting solution of the Lorentz force equation. Moreover, under the assumption that there is no null geodesic connecting x(0) and x(1), we prove that to any value of \q/m\ there correspond at least two connecting timelike solutions which coincide only if they are geodesics. (C) 2003 Elsevier Science B.V. All rights reserved.", keywords = "Kaluza-Klein; charge-to-mass ratio; Lorentz force law", doi = "10.1016/S0393-0440(03)00073-1", pages = "176--186" }

Caponio E.

**Proprietà globali dell'equazione relativistica di Lorentz**.*BOLLETTINO DELL'UNIONE MATEMATICA ITALIANA. A*7:455–458, 2004. BibTeX@article{ 11589_10631, author = "Caponio E", title = "Proprietà globali dell'equazione relativistica di Lorentz", year = 2004, journal = "BOLLETTINO DELL'UNIONE MATEMATICA ITALIANA. A", volume = 7, keywords = "Legge della forza di Lorentz; traiettorie di tipo tempo; Relatività Generale", pages = "455--458" }

## Attività Didattiche

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## Attività di Ricerca

Metodi variazionali e topologici nello studio di equazioni differenziali su varietà e applicazioni alle proprietà globali di varietà Riemanniane, Finsleriane e Lorentziane (incluse le proprietà causali di queste ultime).