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Dear all,

I am looking for reference books about real-valued harmonic functions on complete Riemannian surfaces, do you have any reference in your mind about this? I found some books about harmonic functions on the plane (but mainly discussing on conformal maps) or about general harmonic maps (which is too general to possess nice properties). They do not fit my need...

Precisely, I would like to understand the zero set (for example, is it a one dimensional manifold?), the existence and linearity of such functions.

Thanks for your help in advance!

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The zero set will will have prong singularities, like the zeroes of $Re(z^n)$ or $Im(z^n)$, at least. Meeks used to carry around a book by Rick Schoen that had lots of information about harmonic maps on Riemann surfaces. I tried to find it but its been too long ago to remember. Lots of theorems about minimal surfaces are proved via arguments with harmonic mappings as the coordinate functions are harmonic in the conformal structure underlying the pull back Riemannina metric. –  Charlie Frohman Aug 10 '11 at 13:46
    
Thank you Charlie for the examples. (I will try to find the book of R. Schoen.) –  Chih-Wei Chen Aug 12 '11 at 14:36
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1 Answer

Two remarks:

  1. the notion of harmonic function on a surface is conformally invariant, so if your question is local the it is about harmonic function on $C$.

  2. on a surface, a harmonic function is the same as the real part of a holomorphic function, see wikipedia. So you can reformulate your question on zeros as a question on the inverse image of the real line by a holomorphic function defined on (a subset of) $C$.

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Thank you Jean-Marc, I had found these two remarks but I cannot figure out more "explicit" properties from them in general. However I think they will be helpful when certain particular cases are discussed. Thanks again. –  Chih-Wei Chen Aug 12 '11 at 14:34
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