Geodesic
Examples of malformed subsets of a Riemann surface
Glasgow mathematical journal, Tome 24 (1983) no. 2, pp. 101-106

Voir la notice de l'article provenant de la source Cambridge University Press

Let R be a hyperbolic Riemann surface and W an open subset of R with ∂W piecewise analytic. Denote by the space of Dirichlet finite Tonelli functions on R and by π the harmonic projection of . Consider the relative HD–class on W, HD(W;∂W) = {u∈ │ u │ W∈HD(W) and u │ R\W = 0}. The extremization operation μis the linear mapping of HD(W;∂W) into HD(R) defined by μ. Since π preserves values of functions at the Royden harmonic boundary, the maximum principle implies that μis an order preserving injection and that Mμ is an isometry with respect to the supremum norms. 
Glasner, Moses. Examples of malformed subsets of a Riemann surface. Glasgow mathematical journal, Tome 24 (1983) no. 2, pp. 101-106. doi: 10.1017/S0017089500005152
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