MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I have three related questions:

(1) How does one describe the possible Riemannian metrics that are compatible with a conformal structure on a two dimensional surface?

(2) Can all conformable structures be realized through embeddings of the surface in Euclidean 3-space?

(3) How does one understand solutions to the equation

-exp(f) = Laplacian(f), i.e., $$- \mathrm{exp} (f) = \Delta f$$

where $f$ is a real-valued function of two variables in an open domain?

share|cite|improve this question
You should probably split this question up. In particular, the first question is too elementary and should be asked either locally or on The second question is a good one. I don't understand what you want in the third. – Deane Yang Apr 1 '11 at 17:03
I tried to make the posting more readable without changing the questions substantively. Roll-back or correct as you see fit. – Joseph O'Rourke Apr 1 '11 at 20:23
The answer to question (2) is an old thread:… – Ryan Budney Apr 1 '11 at 22:14
up vote 5 down vote accepted

I don't have much to say about (1) or (2), but (3) is, of course, a classical equation and the usual interpretation is this:

I'm assuming that you are talking about a domain in the $z$-plane, also known as the $xy$-plane, and that, by $\Delta f$ you mean the classical $f_{xx} + f_{yy}$, not the negative of this or this times some metric scalar (as some sources interpret $\Delta$). In this case, the general solution of $\Delta f = -\exp(f)$ can locally be written in the form

$$ f = \log\left({8\ h'(z)\overline{h'(z)}}\over{(1+h(z)\overline{h(z)})^2}\right) $$

where $h$ is a (locally defined) holomorphic function on the domain. In the case that the domain is simply connected, one can take $h$ to be globally defined, but you may have to allow it to be meromorphic.

Geometrically, what is going on is that the conformal metric $ds^2 = \exp(f/2)(dx^2+dy^2)$ has constant curvature $K = 1$ and hence must be constructed by pulling back the standard metric on the $2$-sphere by a conformal map (which one can take to be orientation preserving (in the weak sense), so that it is actually a holomorphic map to the $2$-sphere, thought of as $\mathbb{C}\mathbb{P}^1$ and hence $ds^2$ is the pullback via $h$ of the $K=1$ conformal metric

$$ {{4}\over{(1+w\overline{w})^2}}\ dw\circ d\overline{w}\ . $$

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.