2
$\begingroup$

Conformal mappings from $U$ to $V$, both subsets of $\mathbb{C}$, locally preserve angles. But, in general, such mappings neither preserve areas nor preserve perimeters.

Q. Are there examples of analytic conformal mappings that preserve areas but not perimeters? Or vice versa?


          enter image description here
          The conformal mapping $w=z^2$ in rectangular coordinates: (John Mathews.)


$\endgroup$
  • 1
    $\begingroup$ regarding deformations that preserve areas and perimeters, my question Work on “Churning Polygons” and especially the accepted answer may also shed some light on the problem $\endgroup$ – Manfred Weis Oct 7 '18 at 11:54
15
$\begingroup$

No.

Either condition implies $|\,f'(z)| = 1$ for all $z$ in the domain of $\,f$, which in turn implies that $\,f'(z)$ is locally constant (for instance using the open mapping theorem) and thus that $\,f$ is a Euclidean isometry on each connected component of its domain.

| cite | improve this answer | |
$\endgroup$
9
$\begingroup$

See this handout where Brian Conrad talks about the mapmaker's paradox. A mapmaker would like to draw a map that is area preserving and conformal, however a $C^{\infty}$ isomorphism between Riemannian manifolds with corners is conformal and volume preserving if and only if it is an isometry (theorem 2.4)! The case of perimeter preserving conformal maps was discussed in an older question MO172764. Therefore cartographers make do with maps that are conformal but not area preserving, or nonconformal and area preserving, like examples given in the handout.

| cite | improve this answer | |
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

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