Tom-50

2012

Data publikacji: 13.06.2013

Licencja: Żadna

Redakcja

Sekretarz redakcji Piotr Kościelniak

Zawartość numeru

Kolodziej's subsolution theorem for unbounded pseudoconvex domains

Per Åhag, Rafał Czyż

Universitatis Iagellonicae Acta Mathematica, Tom 50, 2012, s. 7-23

https://doi.org/10.4467/20843828AM.12.001.1119

In this paper we generalize Kolodziej's subsolution theorem to bounded and unbounded pseudoconvex domains, and in that way we are able to solve complex Monge-Ampère equations on general pseudoconvex domains. We then give a negative answer to a question of Cegrell and Kolodziej by constructing a compactly supported Radon measure µ that vanishes on all pluripolar sets in Cⁿ such that µ(Cⁿ) = (2π)ⁿ, and forwhich there is no function u in L₊such that (dd^cu)=µ. We end this paper by solving a Monge-Ampère type equation. Furthermore, we proveuniqueness and stability of the solution.

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Permanence and positive bounded solutions of Kolmogorov predator-prey system

Trinh Tuan Anh, Pham Minh Thong

Universitatis Iagellonicae Acta Mathematica, Tom 50, 2012, s. 25-46

https://doi.org/10.4467/20843828AM.12.002.1120

Our main purpose is to present some criteria for the permanence and existence of a positive bounded solution of Kolmogorov predator-prey system. Under certain conditions, it is shown that the system is permanent and there exists a solution which is defined on the whole R and whose components are bounded from above and from below by positive constants.

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On computation of skew-symmetric generator for an orthogonal matrix

Iwo Biborski

Universitatis Iagellonicae Acta Mathematica, Tom 50, 2012, s. 47-51

https://doi.org/10.4467/20843828AM.12.003.1121

In this paper, we constructively prove that for any matrix A over a field of characteristic 0 and its eigenvalue λ ≠ 0 there exists a diagonal matrix D with diagonal coefficients ±1 such that DA has no eigenvalue λ. Hence and by the canonical result on Cayley transformation, for each orthogonal matrix U one can find a diagonal matrix D and a skew-symmetric matrix S such that U = D(S − I )⁻¹ (S + I ).

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On the Ohsawa–Takegoshi extension theorem

Zbigniew Błocki

Universitatis Iagellonicae Acta Mathematica, Tom 50, 2012, s. 53-61

https://doi.org/10.4467/20843828AM.12.003.1122

Motivated by a recent work by B.-Y. Chen we prove a new estimate for the ∂¯-operator, which easily implies the Ohsawa–Takegoshi extension theorem. We essentially only use the classical H¨ormander esti- mate. This method gives the same constant as the one recently obtained by Guan–Zhou–Zhu.

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Vector bundles on real algebraic curves

Łukasz Maciejewski

Universitatis Iagellonicae Acta Mathematica, Tom 50, 2012, s. 63-68

https://doi.org/10.4467/20843828AM.12.003.1123

We prove that any topological real line bundle on a compact real algebraic curve X is isomorphic to an algebraic line bundle. The result is then generalized to vector bundles of an arbitrary constant rank. As a consequence we prove that any continuous map from X into a real Grassmannian can be approximated by regular maps.

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Subsolution theorem for the complex Hessian equation

Ngoc Cuong Nguyen

Universitatis Iagellonicae Acta Mathematica, Tom 50, 2012, s. 69-88

https://doi.org/10.4467/20843828AM.12.003.1124

We prove the subsolution theorem for a complex Hessian equation in a smoothly bounded strongly m-pseudoconvex domain in Cⁿ.

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The degenerate J-flow and the Mabuchi energy on minimal surfaces of general type

Jian Song, Ben Weinkove

Universitatis Iagellonicae Acta Mathematica, Tom 50, 2012, s. 89-106

https://doi.org/10.4467/20843828AM.12.003.1125

We prove existence, uniqueness and convergence of solutions of the degenerate J- ow on Kahler surfaces. As an application, we establish the properness of the Mabuchi energy for Kahler classes in a certain subcone of the Kahler cone on minimal surfaces of general type.

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Remark on the Calabi flow with bounded curvature

Gábor Székelyhidi

Universitatis Iagellonicae Acta Mathematica, Tom 50, 2012, s. 107-115

https://doi.org/10.4467/20843828AM.12.003.1126

In this short note we prove that if the curvature tensor is uni-formly bounded along the Calabi flow and the Mabuchi energy is proper,then the flow converges to a constant scalar curvature metric.

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Blowup behavior of the Kahler–Ricci flow on Fano manifolds

Valentino Tosatti

Universitatis Iagellonicae Acta Mathematica, Tom 50, 2012, s. 117-126

https://doi.org/10.4467/20843828AM.12.003.1127

We study the blowup behavior at infinity of the normalized Kahler-Ricci flow on a Fano manifold which does not admit Kahler-Einstein metrics. We prove an estimate for the Kahler potential away from a multiplier ideal subscheme, which implies that the volume forms along the flow converge to zero locally uniformly away from the same set. Similar results are also proved for Aubin's continuity method.

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A note on relation between Azukawa isometries at one point and global biholomorphisms

Małgorzata Zajęcka

Universitatis Iagellonicae Acta Mathematica, Tom 50, 2012, s. 127-131

https://doi.org/10.4467/20843828AM.12.003.1128

We prove that under certain assumptions holomorphic functions which are Azukawa isometries at one point are in fact biholomorphisms.

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Słowa kluczowe: Complex Monge-Ampère operator, plurisubharmonic function, Dirichlet problem, Predator-prey system, permanence, bounded solution, almost periodic solution, Orthogonal matrix, Cayley transformation, characteristic polynomial, ergman kernel, ∂-equation, Real algebraic curve, algebraic vector bundle, approximation of continuous maps by regular maps, Hessian equation, subsolution theorem, capacity, Kahler surfaces, Kahler cone, J-flow, Mabuchi energy, surfaces of general type, Kahler–Ricci flow, Kahler–Einstein metric, multiplier ideal sheaf, continuity method, Mathematics Subject Classification 32F45