Vyacheslav Zakharyuta
Characteristics of compacta in $\mathbb C^n$
The famous classical result of the geometric complex analysis (Fekete, Szegö) states that three characteristics of a compact set $K\subset\mathbb C$, which are defined from quite different reasons, do coincide. These characteristics are: transfinite diameter $d\left(K\right)$, characterising an asymptotic size of $K$ (a geometric approach); Chebyshev constant $\tau\left(K\right)$, characterising the minimal uniform deviation of monic polynomials on $K$ (an approximation theory approach); capacity $c\left(K\right)$, describing the asymptotic behavior of the Green function $g_K\left(z\right)$ at infinity (a potential theory approach).
The starting points for studying multidimensional analogs of this classical result were: Leja's transfinite diameter $d\left(K\right)$ for a compact set $K\subset\mathbb C^n$ ([2]); the notions of pluripotential Green function, Robin function and capacities in $\mathbb C^n$ ([6],[7],[9],[5],[1], etc.); the concepts of the directional Chebyshev constants $\tau\left(K,\vartheta\right)$
and principal Chebyshev constant $\tau\left(K\right)$ and the equality $\tau \left(K\right)=d\left(K\right)$ in $\mathbb C^n$ ([8]). Recent remarkable results of Rumely and his collaborators showed that multivariate complex analytic methods, developed in connection with the above classical result, have found applications in the arithmetic geometry (see, e.g. [4]). Moreover, Rumely, using methods of arithmetic intersection theory, produced a multidimensional analog of the equality $d\left(K\right)=c\left(K\right)$ in $\mathbb C^n$ ([3]).
The aim of my talk is to cast a glance to the impressive achievements concerned with the above topics in the last decades (Siciak, Sadullaev, Zeriahi, Bloom, Levenberg, Calvi, Jędrzdejowski, Kołodziej, Szczepański et al.) and represent some fresh results based on a new approach to the transfinite diameter and related notions.
REFERENCES
[1] E. Bedford, B. A. Taylor, Plurisubharmonic functions with logarithmic capacities, Ann. Inst. Fourier (Grenoble) 38 No 4 (1988), 133-171.
[2] F. Leja, Problems á resondre poses a la conference, Colloq. Math. 7 (1959), 153.
[3] R. Rumely, A Robin formula for the Fekete-Leja transfinite diameter, Math. Ann. 337 (2007), 729-738.
[4] R. Rumely, C. F. Lau, R. Varley, Existence of the sectional capacity, AMS Memoirs 690 (145), Providence, R.I., 2000.
[5] A. Sadullaev, Plurisubharmonic measures and capacities on complex manifolds, Russian Math. Surveys 36 (1981), 61-119.
[6] J. Siciak, On some extremal functions and their applications to the theory of analytic functions of several complex variables, Trans. Amer. Math. Soc. 105 (1962), 322-357.
[7] J. Siciak, Extremal plurisubharmonic functions and capacities in $\mathbb C^n$, Ann. Polon. Math. 39 (1981), 175-211.
[8] V. Zakharyuta, Transfinite diameter, Chebyshev constants, and capacity for compacta in $\mathbb{C}^{n}$, Math. USSR Sbornik 25 (1975), 350-364.
[9] V. Zakharyuta, Extremal plurisubharmonic functions, orthogonal polynomials and Bernstein-Walsh theorem for analytic functions of several complex variables, Ann. Polon. Math. 33 (1976), 137-148.