A note on surfaces with radially symmetric nonpositive Gaussian curvature
Mathematica Bohemica (2005)
- Volume: 130, Issue: 2, page 167-176
- ISSN: 0862-7959
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topShomberg, Joseph. "A note on surfaces with radially symmetric nonpositive Gaussian curvature." Mathematica Bohemica 130.2 (2005): 167-176. <http://eudml.org/doc/249583>.
@article{Shomberg2005,
abstract = {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.},
author = {Shomberg, Joseph},
journal = {Mathematica Bohemica},
keywords = {Gaussian curvature; holomorphic function; holomorphic function},
language = {eng},
number = {2},
pages = {167-176},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {A note on surfaces with radially symmetric nonpositive Gaussian curvature},
url = {http://eudml.org/doc/249583},
volume = {130},
year = {2005},
}
TY - JOUR
AU - Shomberg, Joseph
TI - A note on surfaces with radially symmetric nonpositive Gaussian curvature
JO - Mathematica Bohemica
PY - 2005
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 130
IS - 2
SP - 167
EP - 176
AB - 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.
LA - eng
KW - Gaussian curvature; holomorphic function; holomorphic function
UR - http://eudml.org/doc/249583
ER -
References
top- Functions of One Complex Variable I, Second edition. Springer, New York, 1973. (1973) MR0447532
- Elements of Differential Geometry, Prentice-Hall, New Jersey, 1977. (1977) MR0442832
- Elementary Differential Geometry, Springer, London, 2001. (2001) Zbl0959.53001MR1800436
- Mathworld, Wolfram Research, Inc. CRC Press LLC, http://mathworld.wolfram.com, 1999. (1999)
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