Problem of the maximum of the product of the conformal radii of nonoverlapping domains

Kuz'mina, G. V.

doi:10.1007/bf01885516

Cited by 6 publications

(6 citation statements)

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“…Evaluating S * 0,n (ξ i ; α i ) has a rich and fruitful history in geometric function theory and it goes under the name of the extremal decomposition of Riemann sphere, see [32][33][34][35][36][37][44][45][46][47]. Actually, S * 0,4 (ξ i ) is claimed to have a closed-form expression in term of Jacobi elliptic functions [45,46], and the partial results are known for n ≥ 5, although these expressions seem to be implicit at best. There may be value of investigating them further to derive a closed-form expression for S * 0,n (ξ i ; α i ), however this problem appears to be intractable/impractical to the author.…”

Section: Jhep05(2023)186mentioning

confidence: 99%

Bootstrapping closed string field theory

Fırat

2023

J. High Energ. Phys.

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The determination of the string vertices of closed string field theory is shown to be a conformal field theory problem solvable by combining insights from Liouville theory, hyperbolic geometry, and conformal bootstrap. We first demonstrate how Strebel differentials arise from hyperbolic string vertices by performing a WKB approximation to the associated Fuchsian equation, which we subsequently use it to derive a Polyakov-like conjecture for Strebel differentials. This result implies that the string vertices are generated by the interactions of n zero momentum tachyons, or equivalently, a certain limit of suitably regularized on-shell Liouville action. We argue that the latter can be related to the interaction of three zero momentum tachyons on a generalized cubic vertex through classical conformal blocks. We test this claim for the quartic vertex and discuss its generalization to higher-string interactions.

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Section: Jhep05(2023)186mentioning

confidence: 99%

Bootstrapping closed string field theory

Fırat

2023

J. High Energ. Phys.

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“…Presently, solutions for Problems 1 and 2 are known only in the cases n = 2, 3, 4. For n = 2 the solutions of these problems are given, respectively, by [8], see also [7]). Parallel with Problem 2, one has considered an analogous problem on the extremal partition of the upper halfplane (or unit circle).…”

Section: ~ Letmentioning

confidence: 99%

Properties of associated quadratic differentials in certain extremal problems

Kuznetsov¹

1991

J Math Sci

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“…In the case of n = 4 the problem (1) was reduced to the smallest capacity problem in the certain continuum family and was solved in [3]. Currently for the case of n ⩾ 5 complete solution of this problem is unknown.…”

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Application of upper estimates for products of inner radii to distortion theorems for univalent functions

Denega,

Zabolotnyi

2023

Mat. Stud.

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In 1934 Lavrentiev solved the problem of maximum ofproduct of conformal radii of two non-overlapping simply connected domains. In the case of three or more points, many authors considered estimates of a more general Mobius invariant of the form$$T_{n}:={\prod\limits_{k=1}^nr(B_{k},a_{k})}{\bigg(\prod\limits_{1\leqslant k<p\leqslant n}|a_{k}-a_{p}|\bigg)^{-\frac{2}{n-1}}},$$where $r(B,a)$ denotes the inner radius of the domain $B$ with respect to the point $a$ (for an infinitely distant point under the corresponding factor we understand the unit).In 1951 Goluzin for $n=3$ obtained an accurate evaluation for $T_{3}$.In 1980 Kuzmina showedthat the problem of the evaluation of $T_{4}$ isreduced to the smallest capacity problem in the certain continuumfamily and obtained the exact inequality for $T_{4}$.No other ultimate results in this problem for $n \geqslant 5$ are known at present.In 2021 \cite{Bakhtin2021,BahDen22} effective upper estimates are obtained for $T_{n}$, $n \geqslant 2$.Among the possible applications of the obtained results in other tasks of the function theory are the so-called distortion theorems.In the paper we consider an application of upper estimates for products of inner radii to distortion theorems for univalent functionsin disk $U$, which map it onto a star-shaped domains relative to the origin.

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Problem of the maximum of the product of the conformal radii of nonoverlapping domains

Cited by 6 publications

References 6 publications

Bootstrapping closed string field theory

Bootstrapping closed string field theory

Properties of associated quadratic differentials in certain extremal problems

Application of upper estimates for products of inner radii to distortion theorems for univalent functions

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