Geodesic
A note on surfaces with radially symmetric nonpositive Gaussian curvature
Mathematica Bohemica, Tome 130 (2005) no. 2, pp. 167-176

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MR Zbl
It is easily seen that the graphs of harmonic conjugate functions (the real and imaginary parts of a holomorphic function) have the same nonpositive Gaussian curvature. The converse to this statement is not as simple. Given two graphs with the same nonpositive Gaussian curvature, when can we conclude that the functions generating their graphs are harmonic? In this paper, we show that given a graph with radially symmetric nonpositive Gaussian curvature in a certain form, there are (up to) four families of harmonic functions whose graphs have this curvature. Moreover, the graphs obtained from these functions are not isometric in general.
It is easily seen that the graphs of harmonic conjugate functions (the real and imaginary parts of a holomorphic function) have the same nonpositive Gaussian curvature. The converse to this statement is not as simple. Given two graphs with the same nonpositive Gaussian curvature, when can we conclude that the functions generating their graphs are harmonic? In this paper, we show that given a graph with radially symmetric nonpositive Gaussian curvature in a certain form, there are (up to) four families of harmonic functions whose graphs have this curvature. Moreover, the graphs obtained from these functions are not isometric in general.
DOI : 10.21136/MB.2005.134135
Classification : 35C05, 53A05
Keywords: Gaussian curvature; holomorphic function 
Shomberg, Joseph. A note on surfaces with radially symmetric nonpositive Gaussian curvature. Mathematica Bohemica, Tome 130 (2005) no. 2, pp. 167-176. doi: 10.21136/MB.2005.134135
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