Potentials on a~compact Riemann surface
Informatics and Automation, Complex analysis, mathematical physics, and applications, Tome 301 (2018), pp. 287-319.

Voir la notice de l'article provenant de la source Math-Net.Ru

Fundamental concepts of potential theory on compact Riemann surfaces are defined that generalize the corresponding concepts of logarithmic potential theory on the complex plane. The standard properties of these quantities are proved, and relationships between them are established.
@article{TRSPY_2018_301_a20,
     author = {E. M. Chirka},
     title = {Potentials on a~compact {Riemann} surface},
     journal = {Informatics and Automation},
     pages = {287--319},
     publisher = {mathdoc},
     volume = {301},
     year = {2018},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TRSPY_2018_301_a20/}
}
TY  - JOUR
AU  - E. M. Chirka
TI  - Potentials on a~compact Riemann surface
JO  - Informatics and Automation
PY  - 2018
SP  - 287
EP  - 319
VL  - 301
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/TRSPY_2018_301_a20/
LA  - ru
ID  - TRSPY_2018_301_a20
ER  - 
%0 Journal Article
%A E. M. Chirka
%T Potentials on a~compact Riemann surface
%J Informatics and Automation
%D 2018
%P 287-319
%V 301
%I mathdoc
%U http://geodesic.mathdoc.fr/item/TRSPY_2018_301_a20/
%G ru
%F TRSPY_2018_301_a20
E. M. Chirka. Potentials on a~compact Riemann surface. Informatics and Automation, Complex analysis, mathematical physics, and applications, Tome 301 (2018), pp. 287-319. http://geodesic.mathdoc.fr/item/TRSPY_2018_301_a20/

[1] A. I. Aptekarev, G. López Lagomasino, A. Martínez-Finkelshtein, “On Nikishin systems with discrete components and weak asymptotics of multiple orthogonal polynomials”, Russ. Math. Surv., 72:3 (2017), 389–449 | DOI | DOI | MR | Zbl

[2] A. I. Aptekarev, V. G. Lysov, “Systems of Markov functions generated by graphs and the asymptotics of their Hermite–Padé approximants”, Sb. Math., 201:2 (2010), 183–234 | DOI | DOI | MR | Zbl

[3] Buser P., Geometry and spectra of compact Riemann surfaces, Birkhäuser, Boston, 1992 | MR | Zbl

[4] V. I. Buslaev, S. P. Suetin, “On equilibrium problems related to the distribution of zeros of the Hermite–Padé polynomials”, Proc. Steklov Inst. Math., 290 (2015), 256–263 | DOI | DOI | MR | Zbl

[5] E. M. Chirka, Riemann Surfaces, Lekts. Kursy Nauchno-Obrazov. Tsentra, 1, Steklov Math. Inst., Moscow, 2006 | DOI | Zbl

[6] E. M. Chirka, “Harnack inequalities, Kobayashi distances and holomorphic motions”, Proc. Steklov Inst. Math., 279 (2012), 194–206 | DOI | MR | Zbl

[7] E. M. Chirka, “On the $\bar\partial$-problem with $L^2$-estimates on a Riemann surface”, Proc. Steklov Inst. Math., 290 (2015), 264–276 | DOI | DOI | MR | Zbl

[8] de Rham G., Differentiable manifolds: Forms, currents, harmonic forms, Springer, Berlin, 1984 | MR | Zbl

[9] Farkas H. V., Kra I., Riemann surfaces, Springer, New York, 1980 | MR | Zbl

[10] Frostman O., Potentiel d'équilibre et capacité des ensembles avec quelques applications à la théorie des fonctions, Meddelanden Lunds Univ. Mat. Sem., 3, Gleerup, Lund, 1935

[11] A. A. Gonchar, E. A. Rakhmanov, “On convergence of simultaneous Padé approximants for systems of functions of Markov type”, Proc. Steklov Inst. Math., 157 (1983), 31–50 | MR | Zbl | Zbl

[12] A. A. Gonchar, E. A. Rakhmanov, “Equilibrium measure and the distribution of zeros of extremal polynomials”, Math. USSR-Sb., 53:1 (1986), 119–130 | DOI | MR | Zbl

[13] A. A. Gonchar, E. A. Rakhmanov, “On the equilibrium problem for vector potentials”, Russ. Math. Surv., 40:4 (1985), 183–184 | DOI | MR | MR | Zbl

[14] A. A. Gonchar, E. A. Rakhmanov, “Equilibrium distributions and degree of rational approximation of analytic functions”, Math. USSR-Sb., 62:2 (1989), 305–348 | DOI | MR | Zbl | Zbl

[15] A. A. Gonchar, E. A. Rakhmanov, V. N. Sorokin, “Hermite–Padé approximants for systems of Markov-type functions”, Sb. Math., 188:5 (1997), 671–696 | DOI | DOI | MR | Zbl

[16] Hayman W. K., Kennedy P. B., Subharmonic functions, v. 1, Acad. Press, London, 1976 | MR | Zbl

[17] Hörmander L., The analysis of linear partial differential operators, v. 1, Distribution theory and Fourier analysis, Springer, Berlin, 1983 ; Khërmander L., Analiz lineinykh differentsialnykh operatorov s chastnymi proizvodnymi, v. 1, Teoriya raspredelenii i analiz Fure, Mir, M., 1986 | MR | Zbl | MR

[18] Hörmander L., Notions of convexity, Birkhäuser, Basel, 1994 | MR | Zbl

[19] Kang N.-G., Makarov N. G., Calculus of conformal fields on a compact Riemann surface, E-print, 2017, arXiv: 1708.07361[math-ph]

[20] A. V. Komlov, R. V. Palvelev, S. P. Suetin, E. M. Chirka, “Hermite–Padé approximants for meromorphic functions on a compact Riemann surface”, Russ. Math. Surv., 72:4 (2017), 671–706 | DOI | DOI | MR | Zbl

[21] N. S. Landkof, Foundations of Modern Potential Theory, Springer, Berlin, 1972 | MR | MR | Zbl

[22] Mumford D., Tata lectures on theta, v. I, Birkhäuser, Boston, 1983 ; v. II, 1984; Mamford D., Lektsii o teta-funktsiyakh, Mir, M., 1988 | MR | Zbl | MR

[23] E. A. Rakhmanov, “The Gonchar–Stahl $\rho^2$-theorem and associated directions in the theory of rational approximations of analytic functions”, Sb. Math., 207:9 (2016), 1236–1266 | DOI | DOI | MR | Zbl

[24] Ross J., Witt Nyström D., The Dirichlet problem for the complex homogeneous Monge–Ampère equation, E-print, 2017, arXiv: 1712.00405[math.CV]

[25] Rudin W., Functional analysis, McGraw-Hill, New York, 1973 ; Rudin U., Funktsionalnyi analiz, Mir, M., 1975 | MR | Zbl | MR

[26] Skinner B., Logarithmic potential theory on Riemann surfaces, PhD Thesis, Calif. Inst. Technol., Pasadena, 2015 | MR

[27] S. P. Suetin, “Distribution of the zeros of Padé polynomials and analytic continuation”, Russ. Math. Surv., 70:5 (2015), 901–951 | DOI | DOI | MR | Zbl

[28] Tsuji M., Potential theory in modern function theory, Maruzen Co., Tokyo, 1959 | MR | Zbl

[29] I. N. Vekua, Generalized Analytic Functions, 1st ed., Pergamon, Oxford, 1962 | MR | MR | Zbl