The following theorem is well-known

**Theorem** (Liouville). Let be a simply connected domain in . Then all real solutions of

in where a constant, are of the form

where is a locally univalent meromorphic function in .

In geometry, our PDE

says that under the case , it holds

where denotes the standard metric on with constant curvature . Thus we have

**Corollary**. All solutions of the PDE in with and

are of the form

.

The proof of this corollary is quite simple.

*Proof*. By the integrability assumption cannot have an essential singularity at infinity, for otherwise, would conver (possible except one point) infinitely many times near infinity, which is impossible.

Therefore

for some . By composing with an inversion, we may assume the former case holds. Then maps onto . Since cannot cover (notice that for all ), does not have poles in . This means is a covering map and therefore it assumes the form

for some and . A substitution into our PDE gives the desired conclusion.

It is worth noticing that our Corollary was also proved by W. Chen and C. Li by the method of moving planes [here] and Y.Y. Li and M. Zhu by the method of moving spheres [here].

Besides, one can see that the intergrability condition is also necessary for asymptotic radial symmetry. All non-radial solutions, which arise from transcendental functions, satisfy

.

Let’s go back to the theorem, it does not hold for domains which are not simply connected. For instance, the function

is a solution of our PDE in the punctured disc with an isolated singularity at the origin. Such a result for this domain was derived by K.Chou and Y. Wan [here].

A natural extension, but has its own interest, is the so-called Toda system SU(n). The simplest case, known as SU(2), is the following

in .

The solutions of SU(2) are already known, precisely, they can be written as follows

and

where are real and are complex numbers. We refer the reader to a paper due to del Pino, M. Kowalczyk and J. Wei for details [here].

The following pictures demonstrate the shape of these solutions for a particular choice of eight parameters (i.e. , , ).

For

For

.

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