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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 $D = D ₁ \times \dots \times D_{s}$ , 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 $f : \partial Ω_{π} \to ℝ$ that can be extended to a real-valued pluriharmonic function in $Ω_{π}$ .

Pluriharmonic interpolation and hulls of C1 curves in the unit sphere.

Herbert Alexander, Joaquim Bruna (1995)

Revista Matemática Iberoamericana

Pluriharmonically dentable complex Banach spaces.

N:, Maurey, B. Ghoussoub (1989)

Journal für die reine und angewandte Mathematik

Pluriharmonicity in terms of harmonic slices.

Frank Forelli (1977)

Mathematica Scandinavica

Pluripotential theory, semigroups and boundary behavior of infinitesimal generators in strongly convex domains

Filippo Bracci, Manuel Contreras, Santiago Díaz-Madrigal (2010)

Journal of the European Mathematical Society

Plurisubharmonic extension in non-degenerate analytic polyhedra.

Åhag, Per, Czyż, Rafał, Lodin, Sam, Wikström, Frank (2007)

Zeszyty Naukowe Uniwersytetu Jagiellońskiego. Universitatis Iagellonicae Acta Mathematica

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.

Plurisubharmonic functions with logarithmic singularities

E. Bedford, B. A. Taylor (1988)

Annales de l'institut Fourier

To a plurisubharmonic function $u$ on $C^{n}$ with logarithmic growth at infinity, we may associate the Robin function $ρ_{u} (z) = \underset{λ \to \infty}{lim sup} u (λ z) - log (λ z)$ defined on $P^{n - 1}$ , the hyperplane at infinity. We study the classes $L_{+}$ , and (respectively) $L_{p}$ of plurisubharmonic functions which have the form $u = log (1 + | z |) + O (1)$ and (respectively) for which the function $ρ_{u}$ is not identically $- \infty$ . We obtain an integral formula which connects the Monge-Ampère measure on the space $C^{n}$ with the Robin function on $P^{n - 1}$ . As an application we obtain a criterion on the convergence of the Monge-Ampère...

Plurisubharmonic saddles

Siegfried Momm (1996)

Annales Polonici Mathematici

A certain linear growth of the pluricomplex Green function of a bounded convex domain of $ℂ^{N}$ at a given boundary point is related to the existence of a certain plurisubharmonic function called a “plurisubharmonic saddle”. In view of classical results on the existence of angular derivatives of conformal mappings, for the case of a single complex variable, this allows us to deduce a criterion for the existence of subharmonic saddles.

Poincaré inequality and Hajłasz-Sobolev spaces on nested fractals

Katarzyna Pietruska-Pałuba, Andrzej Stós (2013)

Studia Mathematica

Given a nondegenerate harmonic structure, we prove a Poincaré-type inequality for functions in the domain of the Dirichlet form on nested fractals. We then study the Hajłasz-Sobolev spaces on nested fractals. In particular, we describe how the "weak"-type gradient on nested fractals relates to the upper gradient defined in the context of general metric spaces.

Pointwise and locally uniform convergence of holomorphic and harmonic functions.

Šťepničková, Libuše (1999)

Commentationes Mathematicae Universitatis Carolinae

Pointwise and locally uniform convergence of holomorphic and harmonic functions

Libuše Štěpničková (1999)

Commentationes Mathematicae Universitatis Carolinae

We shall characterize the sets of locally uniform convergence of pointwise convergent sequences. Results obtained for sequences of holomorphic functions by Hartogs and Rosenthal in 1928 will be generalized for many other sheaves of functions. In particular, our Hartogs-Rosenthal type theorem holds for the sheaf of solutions to the second order elliptic PDE's as well as it has applications to the theory of harmonic spaces.

Pointwise inequalities and approximation in fractional Sobolev spaces

David Swanson (2002)

Studia Mathematica

We prove that a function belonging to a fractional Sobolev space $L^{α, p} (ℝ ⁿ)$ may be approximated in capacity and norm by smooth functions belonging to $C^{m, λ} (ℝ ⁿ)$ , 0 < m + λ < α. Our results generalize and extend those of [12], [4], [14], and [11].

Pointwise inequalities for Sobolev functions and some applications

Bogdan Bojarski, Piotr Hajłasz (1993)

Studia Mathematica

We get a class of pointwise inequalities for Sobolev functions. As a corollary we obtain a short proof of Michael-Ziemer’s theorem which states that Sobolev functions can be approximated by $C^{m}$ functions both in norm and capacity.

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