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The article is 'Hadamard and Calabi Yau conjectures on negatively curved an minimal surfaces' Nadirashvili. In the proof of proposition 4.3 it asserts that the function y is holomorphic. I'm not sure about it (actually i think it is false). What do you think about it? Thank you for your help.

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I agree, this part is wrong as stated, although the ideas behind it are sound. Pascal Collin and Harold Rosenberg did a great service to the community by publishing the note

P. Collin and H. Rosenberg, Notes sur la démonstration de N. Nadirashvili des conjectures de Hadamard et Calabi-Yau, Bull. Sci. Math. 123 (1999), no. 7, 563–575.

in which they clarify and correct various details. In particular, the appendix of this note contains a cleaned-up proof of Proposition 3.4, which provides the missing details concerning the zeroes of the various functions involved.

References: Collin-Rosenberg, Nadirashvili (also preprint), and first part of this discussion.

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