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Perron-Frobenius operators and the Klein-Gordon equation

Francisco Canto-Martín, Håkan Hedenmalm, Alfonso Montes-Rodríguez (2014)

Journal of the European Mathematical Society

For a smooth curve Γ and a set Λ in the plane ℝ 2 , let A C ( Γ ; Λ ) be the space of finite Borel measures in the plane supported on Γ , absolutely continuous with respect to the arc length and whose Fourier transform vanishes on Λ . Following [12], we say that ( Γ , Λ ) is a Heisenberg uniqueness pair if A C ( Γ ; Λ ) = { 0 } . In the context of a hyperbola Γ , the study of Heisenberg uniqueness pairs is the same as looking for uniqueness sets Λ of a collection of solutions to the Klein-Gordon equation. In this work, we mainly address the...

Plateau-Stein manifolds

Misha Gromov (2014)

Open Mathematics

We study/construct (proper and non-proper) Morse functions f on complete Riemannian manifolds X such that the hypersurfaces f(x) = t for all −∞ < t < +∞ have positive mean curvatures at all non-critical points x ∈ X of f. We show, for instance, that if X admits no such (not necessarily proper) function, then it contains a (possibly, singular) complete (possibly, compact) minimal hypersurface of finite volume.

Plurifine potential theory

Jan Wiegerinck (2012)

Annales Polonici Mathematici

We give an overview of the recent developments in plurifine pluripotential theory, i.e. the theory of plurifinely plurisubharmonic functions.

Pluriharmonic extension in proper image domains

Rafał Czyż (2009)

Annales Polonici Mathematici

Let D j be a bounded hyperconvex domain in ℂ n j and set D = D ₁ × ⋯ × D s , j=1,...,s, s ≥ 3. Also let Ω π be the image of D under the proper holomorphic map π. We characterize those continuous functions f : ∂ Ω π → ℝ that can be extended to a real-valued pluriharmonic function in Ω π .

Plurisubharmonic functions on compact sets

Rafał Czyż, Lisa Hed, Håkan Persson (2012)

Annales Polonici Mathematici

Poletsky has introduced a notion of plurisubharmonicity for functions defined on compact sets in ℂⁿ. We show that these functions can be completely characterized in terms of monotone convergence of plurisubharmonic functions defined on neighborhoods of the compact.

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