Volume 52

2015

Publication date: 09.12.2015

Licence: None

Editorial team

Editor-in-Chief Sławomir Kołodziej

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On the Abhyankar-Moh inequality

Roland D. Barrolleta, Evelia R Garcia Barroso, Arkadiusz Ploski

Universitatis Iagellonicae Acta Mathematica, Volume 52, 2015, pp. 7-14

https://doi.org/10.4467/20843828AM.15.001.3727

In their fundamental paper on the embeddings of the line in the plane, Abhyankar and Moh proved an important inequality which can be stated in terms of the semigroup associated with the branch at infinity of a plane algebraic curve. In this note we study the semigroups of integers satisfying the Abhyankar-Moh inequality and describe such semigroups with the maximum conductor.

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Counterpart of the Aubin-Yau functional for real Riemannian manifolds

Dongwei Gu

Universitatis Iagellonicae Acta Mathematica, Volume 52, 2015, pp. 15-21

https://doi.org/10.4467/20843828AM.15.002.3728

In this note we construct a functional which is an analogue of the Aubin–Yau functional on any connected compact real Riemannian manifold. The construction is similar to the one in the K¨ahler case and they coincide with each other when the manifold is of complex dimension 1.

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A remark on Fejér and Mittag-Leffler theorems

Marek Jarnicki, Józef Siciak

Universitatis Iagellonicae Acta Mathematica, Volume 52, 2015, pp. 23-28

https://doi.org/10.4467/20843828AM.15.003.3729

We discuss some generalizations of the classical Fejer and Mittag-Leffler theorems to the case of several complex variables with applications to the Shilov and Bergman boundaries.

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Degeneration of Kähler-Ricci solitons on Fano manifolds

D.H. Phong, Jian Song, Jacob Sturm

Universitatis Iagellonicae Acta Mathematica, Volume 52, 2015, pp. 29-43

https://doi.org/10.4467/20843828AM.15.004.3730

We consider the space KR(n, F) of Kähler–Ricci solitons on n-dimensional Fano manifolds with Futaki invariant bounded by F. We prove a partial C0 estimate for KR(n, F) as a generalization of the recent work of Donaldson-Sun for Fano Kähler–Einstein manifolds. In particular, any sequence in KR(n, F) has a convergent subsequence in the Gromov- Hausdorff topology to a Kähler–Ricci soliton on a Fano variety with log terminal singularities.

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Merton portfolio problem with one indivisible asset

Jakub Trybuła

Universitatis Iagellonicae Acta Mathematica, Volume 52, 2015, pp. 45-56

https://doi.org/10.4467/20843828AM.15.005.3909

In this paper we consider a modification of the classical Merton portfolio optimization problem. Namely, an investor can trade in financial asset and consume his capital. He is additionally endowed with a one unit of an indivisible asset which he can sell at any time. We give a numerical example of calculating the optimal time to sale the indivisible asset, the optimal consumption rate and the value function.

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Słowa kluczowe: Branches at infinity, semigroups, sequences of divisors, Abhyankar-Moh inequality, Infinite-dimensional manifold, volume element, Aubin-Yau functional, Fejer theorem, Mittag-Leffler theorem, balanced domain, star-like domain, Shilov boundary, Bergman boundary., Bergman kernel, partial C 0 estimate, conformal transformations, heeger–Colding theory, Perelman pseudo-locality, Kodaira embeddings, nvestor problem, optimal stopping, HJB equation, indivisible asset

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