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March  2004, 3(1): 41-51. doi: 10.3934/cpaa.2004.3.41

Liouville's formula under the viewpoint of minimal surfaces

Francisco Brito 1, , Maria Luiza Leite 2, and  Vicente de Souza Neto 3,

1. 

Departamento de Matematica, Universidade Federal de Pernambuco, Recife, 50740-540, Brazil
2. 

Departamento de Estatıstica, Universidade Federal de Pernambuco, Recife, 50740-540, Brazil
3. 

Universidade Catolica de Pernambuco, Rua do Principe, 526, Recife, 50050-900, Brazil

Received  January 2003 Revised  May 2003 Published  January 2004

We study the equation $\Delta u=e^{-2u}$ in dimension two and review the Liouville's formula for a solution $u$ in terms of the Weierstrass representation of a minimal surface in $\mathbb R^3.$ We list minimal surfaces corresponding to classical solutions and point out a gap of $100$ years from Bonnet's family of minimal surfaces to solutions discovered by physicists in the sixties. We also prove Chen-Li's symmetry theorem in the context of minimal surfaces theory.
Keywords: minimal surfaces, Weierstrass representation., Liouville formula.
Mathematics Subject Classification: 35C99, 53A1.
Citation: Francisco Brito, Maria Luiza Leite, Vicente de Souza Neto. Liouville's formula under the viewpoint of minimal surfaces. Communications on Pure and Applied Analysis, 2004, 3 (1) : 41-51. doi: 10.3934/cpaa.2004.3.41
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