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Hi, harmonicity in 2d is preserved under mappings that satisfy the Cauchy-Riemann equations.

What about 3D? What conditions should a mapping satisfy to preserve harmonicity?

is there a general characterization a la CR for 3D?

Here is an example of non-trivial such mapping

Let $u(x,y,z)=U(X,Y,Z)$ where $$X=xy+z,~~~~ Y= \frac{\sqrt3}4 (x^2-y^2) - \frac{xy}{2}+z ,~~~~ Z= -\frac{\sqrt3}4 (x^2-y^2) - \frac{xy}{2}+z$$

(I found this example by first assuming $X=xy+z$ then guessing for Y,Z from the overdetermined system that they satisfy... hope it's right...)

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    $\begingroup$ The correct notion is the one of pluriharmonicity, see en.wikipedia.org/wiki/Pluriharmonic_function Then pluriharmonicity is preserved by pull-back via holomorphic functions of several complex variables. Of course, none of this makes any sense in $R^3$. $\endgroup$
    – Misha
    Commented May 14, 2012 at 17:26
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    $\begingroup$ Your example does not work. It is true that $\partial_XU$, $\partial_YU$, $\partial_ZU$ are orthogonal to each other. But they have different norms and that prevents the mapping to preserve harmonicity. $\endgroup$
    – Denis Serre
    Commented May 15, 2012 at 9:00

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Actually, the correct notion of maps preserving harmonicity is that of 'harmonic morphisms': A map $f:(M,g)\to (N,h)$ between Riemannian manifolds is a harmonic morphism if it pulls back $h$-harmonic functions on $N$ to $g$-harmonic functions on $M$. There are many, many nontrivial examples, and there is a large literature on the subject.

There is an extensive Atlas of Harmonic Morphisms (see http://www.maths.lth.se/matematiklu/personal/sigma/harmonic/atlas.html) that contains a useful bibliography.

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    $\begingroup$ To expound: the paper "Bent Fuglede, Harmonic morphisms between Riemannian manifolds, Ann. Inst. Fourier (Grenoble) 28 (1978)" is (I think) the first paper where the term 'harmonic morphism' is defined. There is some overlap with the paper by Ishihara in Igor's answer as these were published around the same time. Note that harmonic morphisms are not the same as harmonic maps or maps that commute with the Laplacian ("Bill Watson, Manifold maps commuting with the Laplacian, J. Differential Geometry 8 (1973)"). $\endgroup$
    – Paul Reynolds
    Commented May 14, 2012 at 22:04
  • $\begingroup$ I wrote down a system of overdetermined PDE's that such harmonic morphism must satisfy. But it seems like a large system containing 8 equations; I am at a loss as to how to reduce it... lots of structure though... $\endgroup$
    – curious
    Commented May 15, 2012 at 3:19
  • $\begingroup$ ps: i tried using maple's "casesplit" command but it blew up my computer [4gig]... $\endgroup$
    – curious
    Commented May 15, 2012 at 3:20
  • $\begingroup$ The link to the Atlas now points to another link, which is dead. $\endgroup$
    – Ben McKay
    Commented Oct 26, 2021 at 11:43
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See

MR0545705 (80k:58045) Ishihara, Tôru A mapping of Riemannian manifolds which preserves harmonic functions. J. Math. Kyoto Univ. 19 (1979), no. 2, 215–229. 58E20 (31C12)

The author shows that a mapping has the title property in dimension at least three if and only if it is either constant or a Riemannian covering up to homothety.. For $\mathbb{R}^n$ this means a composition of translations, rotations, and homothety ($x\rightarrow a x,$ for some $a>0.$)

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Your "examples" usually don't work. Up to a complex conjugation (= symmetry), a $2$D-mapping $\phi$ that preserves harmonicity is a holomorphic function, via $z=x_1+ix_2$ and $f(z)=\phi_1+i\phi_2$. Now, let us form $\psi(x_1,x_2,x_3)=(\phi(x_1,x_2),x_3)$. What happens is that $$(\partial_1^2+\partial_2^2)(u\circ\phi)=|f'|^2(\partial_1^2u+\partial_2^2u)\circ\phi.$$ From this, you see that $\Delta(u\circ\psi)$ is not proportional to $(\Delta u)\circ\psi$ if $|f'|\ne1$ (notice that if $|f'|\equiv1$, then $f$ is an affine isometry). Hence $\psi$ does not preserve harmonicity. More generally, a function $\phi$ preserves harmonicity ($u$ harmonic implies $u\circ \phi$ is harmonic) if and only if ${\rm D}\phi(x)$ is a similitude, that is the product $\rho(x)R_x$ of some isometry and of a homothety.

On the contrary, a theorem due to Liouville says that a mapping in ${\mathbb R}^3$ that preserves angles must be an affine similitude. This applies in particular to mappings that preserve harmonicity. Edit (after comments below): Liouville's Theorem actually says that a direct $C^4$ conformal map is the composition of an affine similarity and possibly of inversions $x\mapsto a\|x-x_0\|^{-2}(x-x_0)$.

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    $\begingroup$ @curious Note that $\Delta(u\circ \psi)=|f'|^2((\partial_1^2+\partial_2^2)u+\partial_3^2 u)\circ \psi$, which is not proportional to $(\Delta u)\circ \psi$ if $|f'|^2\neq 1$ and $\partial_3^2u\neq 0$. So Denis Serre is right that your "examples" usually don't work. However, I looked up the theorem due to Liouville in my copy of Marcel Berger's "Géométrie 1", and certain inversions also seem to preserve harmonicity. I guess they are $x \mapsto 1/||x||^n$, but I would have to check. Of course, these inversions are not defined everywhere in $\mathbb R^3$, so the remark is only misleading. $\endgroup$
    – Thomas Klimpel
    Commented May 15, 2012 at 0:58
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    $\begingroup$ The comment above contains frustratingly many typos (wrong parenthesis in the expression for $\Delta (u \circ \psi)$, $x\mapsto 1/||x||^n$ instead of $x\mapsto x/||x||^n$), but the message should be clear nevertheless. $\endgroup$
    – Thomas Klimpel
    Commented May 15, 2012 at 1:03
  • $\begingroup$ ah yes my bad. Thank you for straighting me out, my example was indeed nonsense. Let me check if inversion preserves harmonicity... any other examples? or is it an exhaustive list? $\endgroup$
    – curious
    Commented May 15, 2012 at 1:58
  • $\begingroup$ @Denis In the following post I considered another kind of "preserving", that is invariant under derivation operator(instead of composition).Do you have some extra ideas,(aside of existing answer) on this question?Can I ask you to give some comment on this question? mathoverflow.net/questions/162598/… $\endgroup$
    – Ali Taghavi
    Commented May 23, 2018 at 22:29
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I see the answer I gave yesterday as a guest is not posted, an email check revealed I did not properly register...

Ok, now being a registered thing I can say:

The Cauchy-Riemann equations directly relate to all entire functions being harmonic. In $\mathbb{C}$ you can even factorize the Laplacian operator into two things and you see that only one of those factors slams the stuff to zero so the outcome is harmonic.

Basically the CR-equations say:

$$ \frac{\partial}{\partial y} = i \cdot \frac{\partial}{\partial x} $$ so that $$ \frac{\partial^2}{\partial y^2} = - \frac{\partial^2}{\partial x^2} $$ Hence all analytic (entire) functions on $\mathbb{C}$ are also harmonic but it does not go the other way...

Now for $\mathbb{R^3}$ space there is indeed a complex multiplication possible but this does not give rise to harmonic functions. In this answer it will go too far to explain how 3D complex multiplication works, but as far as I know after 26 years of looking at that stuff:

You cannot make harmonic functions that way.

Here comes a small monkey out of the sleeve: I have a separate website related to all 3D complex numbers things:

http://3dcomplexnumbers.net/

The 3D Cauchy-Riemann equations are beautiful yet they do not give harmonic stuff. Hope I answered your question a tiny bit...

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