A Maximum Principle for Dirichlet-Finite Harmonic Functions on Riemannian Spaces

1970 ◽  
Vol 22 (4) ◽  
pp. 855-862
Author(s):  
Y. K. Kwon ◽  
L. Sario
Keyword(s):  
Maximum Principle ◽  
Harmonic Functions ◽  
Riemann Surfaces ◽  
Decomposition Theorem ◽  
Higher Dimensions ◽  
Integral Representations ◽  
Classification Theory ◽  
Topological Properties ◽  
Harmonic Decomposition ◽  
Bounded Harmonic Functions

Representations of harmonic functions by means of integrals taken over the harmonic boundary ΔR of a Riemann surface R enable one to study the classification theory of Riemann surfaces in terms of topological properties of ΔR (cf. [6; 4; 1; 7]). In deducing such integral representations, essential use is made of the fact that the functions in question attain their maxima and minima on ΔR.The corresponding maximum principle in higher dimensions was discussed for bounded harmonic functions in [3]. In the present paper we consider Dirichlet-finite harmonic functions. We shall show that every such function on a subregion G of a Riemannian N-space R attains its maximum and minimum on the set , where ∂G is the relative boundary of G in R and the closures are taken in Royden's compactification R*. As an application we obtain the harmonic decomposition theorem relative to a compact subset K of R* with a smooth ∂(K ∩ R).

A Maximum Principle for Bounded Harmonic Functions on Riemannian Spaces

1970 ◽  
Vol 22 (4) ◽  
pp. 847-854 ◽  
Author(s):  
Y. K. Kwon ◽  
L. Sario
Keyword(s):  
Harmonic Functions ◽  
Riemann Surfaces ◽  
Higher Dimensions ◽  
Classification Theory ◽  
Superharmonic Function ◽  
Maximum Principles ◽  
Topological Properties ◽  
Positive Continuous Function ◽  
Dirichlet Integral ◽  
Bounded Harmonic Functions

Harmonic functions with certain boundedness properties on a given open Riemann surface R attain their maxima and minima on the harmonic boundary ΔB of R. The significance of such maximum principles lies in the fact that the classification theory of Riemann surfaces related to harmonic functions reduces to a study of topological properties of Δ(cf. [11; 8; 3; 12].For the corresponding problem in higher dimensions we shall first show that the complement of ΔR with respect to the Royden boundary ΓR of a Riemannian N-space R is harmonically negligible: given any non-empty compact subset E of ΓR – ΔR there exists an Evans superharmonic function v, i.e., a positive continuous function on R* = R ∪ ΓR, superharmonic on R, with v = 0 on ΔR, v ≡ ∞ on E, and with a finite Dirichlet integral over R.

scholarly journals A boundary Schwarz lemma for pluriharmonic mappings between the unit polydiscs of any dimensions

Filomat ◽  
2020 ◽  
Vol 34 (9) ◽  
pp. 3151-3160
Author(s):  
Ziyan Huang ◽  
Di Zhao ◽  
Hongyi Li
Keyword(s):  
Harmonic Functions ◽  
Higher Dimensions ◽  
Schwarz Lemma ◽  
Boundary Schwarz Lemma ◽  
Bounded Harmonic Functions

In this paper, we present a boundary Schwarz lemma for pluriharmonic mappings between the unit polydiscs of any dimensions, which extends the classical Schwarz lemma for bounded harmonic functions to higher dimensions.

scholarly journals Plemelj–Sokhotski isomorphism for quasicircles in Riemann surfaces and the Schiffer operators

2019 ◽  
Vol 378 (3-4) ◽  
pp. 1613-1653 ◽  
Author(s):  
Eric Schippers ◽  
Wolfgang Staubach
Keyword(s):  
Harmonic Functions ◽  
Riemann Surfaces ◽  
Inner Product ◽  
Connected Components ◽  
Cauchy Type ◽  
Cauchy Type Integral ◽  
Jordan Curves ◽  
Type Integral ◽  
Algebraic Constraints ◽  
Bounded Harmonic Functions

Abstract Let R be a compact Riemann surface and $$\Gamma $$ Γ be a Jordan curve separating R into connected components $$\Sigma _1$$ Σ 1 and $$\Sigma _2$$ Σ 2 . We consider Calderón–Zygmund type operators $$T(\Sigma _1,\Sigma _k)$$ T ( Σ 1 , Σ k ) taking the space of $$L^2$$ L 2 anti-holomorphic one-forms on $$\Sigma _1$$ Σ 1 to the space of $$L^2$$ L 2 holomorphic one-forms on $$\Sigma _k$$ Σ k for $$k=1,2$$ k = 1 , 2 , which we call the Schiffer operators. We extend results of Max Schiffer and others, which were confined to analytic Jordan curves $$\Gamma $$ Γ , to general quasicircles, and prove new identities for adjoints of the Schiffer operators. Furthermore, let V be the space of anti-holomorphic one-forms which are orthogonal to $$L^2$$ L 2 anti-holomorphic one-forms on R with respect to the inner product on $$\Sigma _1$$ Σ 1 . We show that the restriction of the Schiffer operator $$T(\Sigma _1,\Sigma _2)$$ T ( Σ 1 , Σ 2 ) to V is an isomorphism onto the set of exact holomorphic one-forms on $$\Sigma _2$$ Σ 2 . Using the relation between this Schiffer operator and a Cauchy-type integral involving Green’s function, we also derive a jump decomposition (on arbitrary Riemann surfaces) for quasicircles and initial data which are boundary values of Dirichlet-bounded harmonic functions and satisfy the classical algebraic constraints. In particular we show that the jump operator is an isomorphism on the subspace determined by these constraints.

scholarly journals Classification of Locally Euclidean Spaces

1965 ◽  
Vol 25 ◽  
pp. 87-111 ◽  
Author(s):  
Leo Sario
Keyword(s):  
Riemann Surface ◽  
Riemann Surfaces ◽  
Higher Dimensions ◽  
Classification Theory ◽  
Euclidean Spaces ◽  
Main Interest ◽  
Euclidean Metrics ◽  
Riemannian Spaces

The classification of Riemann surfaces has largely reached its completion. The purpose of the present paper is to lay the foundation for a new intriguing field in the classification theory: Riemannian spaces with Euclidean metrics. The paper is self-contained, both for the Riemann surface expert and the reader whose main interest is with higher dimensions.

scholarly journals On the existence of various bounded harmonic functions with given periods, II

1975 ◽  
Vol 56 ◽  
pp. 1-5
Author(s):  
Masaru Hara
Keyword(s):  
Riemann Surface ◽  
Harmonic Function ◽  
Harmonic Functions ◽  
Dirichlet Integral ◽  
Period Function ◽  
Boundedness Property ◽  
One Dimensional ◽  
Bounded Harmonic Functions

Given a harmonic function u on a Riemann surface R, we define a period functionfor every one-dimensional cycle γ of the Riemann surface R. Γx(R) denote the totality of period functions Γu such that harmonic functions u satisfy a boundedness property X. As for X, we let B stand for boundedness, and D for the finiteness of the Dirichlet integral.

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