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Nonlinear potential theory and quasiregular mappings on Riemannian manifolds. (English) Zbl 0698.31010

Annales Academiæ Scientiarum Fennicæ. Series A I. Mathematica. Dissertationes 74. Helsinki: Academia Scientiarum Fennica; Helsinki: Univ. of Helsinki, Faculty of Science (Diss.) (ISBN 951-41-0612-1). 45 p. (1990).
The author extends (nonlinear) potential theory associated with degenerated elliptic equations similar to the \(p\)-Laplacian \[ -\operatorname{div}(| \nabla u|^{p-2}\nabla u)=0,\quad 1<p\le n, \] to Riemannian \(n\)-manifolds. There exist several papers on this nonlinear potential theory in \(\mathbb{R}^n\) written by combinations of Finnish mathematicians: S. Granlund, J. Heinonen, P. Lindqvist, O. Martio, and the reviewer, see e.g. [Trans. Am. Math. Soc. 277, 43–73 (1983; Zbl 0518.30024); Acta Math. 155, 153–171 (1985; Zbl 0607.35042); Ark. Mat. 26, 87–105 (1988; Zbl 0652.31006); Indiana Univ. Math. J. 38, No. 2, 253–275 (1989; Zbl 0688.31005); Ann. Inst. Fourier 39, No. 2, 293–318 (1989; Zbl 0659.35038)]. In the present paper, the author first indicates how basic properties of \(p\)-harmonic functions can be proved on manifolds using suitable chart mappings.
The main emphasis is on a notion corresponding to Green’s function in linear theory. The resulting function is still called Green’s function although it naturally looses some of its effectiveness as no linear structure is present. Green’s function is constructed in regular subsets and then defined on the whole manifold via an exhaustion. It turns out that the manifold \(M\) possesses Green’s function if and only if the \(p\)-capacity of its ideal boundary is zero.
The uniqueness of Green’s function is established in the conformally invariant case \(p=n\); cases \(p<n\) remain open. He also studies more general singular solutions and the classification theory of Riemannian manifolds.
The nonlinear potential theory in case \(p=n\) is closely connected with the theory of quasiregular mappings. As an application, the author considers the following problem: suppose that \(f\) is a nonconstant \(K\)-quasiregular mapping of \(\mathbb{R}^n\), \(n\ge 3\), into a punctured sphere \(M = S^n\setminus \{a_1, \ldots, a_q\}\) and \(M\) is equipped with a given Riemannian metric. Is there an upper bound \(q_0\), depending only on \(n\) and \(K\), such that no such mapping exists if \(q\ge q_0\)? It is shown that there exists a nontrivial class of admissible metrics on \(M\) such that the answer is affirmative. This result extends S. Rickman’s deep Picard type theorem [J. Anal. Math. 37, 100–117 (1980; Zbl 0451.30012)].

MSC:

31C12 Potential theory on Riemannian manifolds and other spaces
31-02 Research exposition (monographs, survey articles) pertaining to potential theory
31C05 Harmonic, subharmonic, superharmonic functions on other spaces
35J70 Degenerate elliptic equations
31C15 Potentials and capacities on other spaces
53C20 Global Riemannian geometry, including pinching
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