W. D. Neumann :
“A calculus for plumbing applied to the topology of complex surface singularities and degenerating complex curves Trans. Am. Math. Soc.
268 : 2
(1981 ),
pp. 299–344 .
MR
632532 Zbl
0546.57002 article
Abstract
BibTeX
@article {key632532m,
AUTHOR = {Neumann, Walter D.},
TITLE = {A calculus for plumbing applied to the
topology of complex surface singularities
and degenerating complex curves},
JOURNAL = {Trans. Am. Math. Soc.},
FJOURNAL = {Transactions of the American Mathematical
Society},
VOLUME = {268},
NUMBER = {2},
YEAR = {1981},
PAGES = {299--344},
DOI = {10.2307/1999331},
NOTE = {MR:632532. Zbl:0546.57002.},
ISSN = {0002-9947},
}
L. Birbrair, A. Fernandes, and W. D. Neumann :
“Bi-Lipschitz geometry of weighted homogeneous surface singularities Math. Ann.
342 : 1
(2008 ),
pp. 139–144 .
MR
2415318 Zbl
1153.14003 ArXiv
0704.2041 article
Abstract
People
BibTeX
We show that a weighted homogeneous complex surface singularity is metrically conical (i.e., bi-Lipschitz equivalent to a metric cone) only if its two lowest weights are equal. We also give an example of a pair of weighted homogeneous complex surface singularities that are topologically equivalent but not bi-Lipschitz equivalent.
@article {key2415318m,
AUTHOR = {Birbrair, Lev and Fernandes, Alexandre
and Neumann, Walter D.},
TITLE = {Bi-{L}ipschitz geometry of weighted
homogeneous surface singularities},
JOURNAL = {Math. Ann.},
FJOURNAL = {Mathematische Annalen},
VOLUME = {342},
NUMBER = {1},
YEAR = {2008},
PAGES = {139--144},
DOI = {10.1007/s00208-008-0225-4},
NOTE = {ArXiv:0704.2041. MR:2415318. Zbl:1153.14003.},
ISSN = {0025-5831},
}
L. Birbrair, A. Fernandes, and W. D. Neumann :
“Bi-Lipschitz geometry of complex surface singularities Geom. Dedicata
139
(2009 ),
pp. 259–267 .
MR
2481850 Zbl
1164.32005 ArXiv
0804.0194 article
Abstract
People
BibTeX
@article {key2481850m,
AUTHOR = {Birbrair, Lev and Fernandes, Alexandre
and Neumann, Walter D.},
TITLE = {Bi-{L}ipschitz geometry of complex surface
singularities},
JOURNAL = {Geom. Dedicata},
FJOURNAL = {Geometriae Dedicata},
VOLUME = {139},
YEAR = {2009},
PAGES = {259--267},
DOI = {10.1007/s10711-008-9333-2},
NOTE = {ArXiv:0804.0194. MR:2481850. Zbl:1164.32005.},
ISSN = {0046-5755},
}
L. Birbrair, A. Fernandes, and W. D. Neumann :
“Separating sets, metric tangent cone and applications for complex algebraic germs Selecta Math. (N.S.)
16 : 3
(2010 ),
pp. 377–391 .
MR
2734336 Zbl
1200.14010 ArXiv
0905.4312 article
Abstract
People
BibTeX
An explanation is given for the initially surprising ubiquity of separating sets in normal complex surface germs. It is shown that they are quite common in higher dimensions too. The relationship between separating sets and the geometry of the metric tangent cone of Bernig and Lytchak is described. Moreover, separating sets are used to show that the inner Lipschitz type need not be constant in a family of normal complex surface germs of constant topology.
@article {key2734336m,
AUTHOR = {Birbrair, Lev and Fernandes, Alexandre
and Neumann, Walter D.},
TITLE = {Separating sets, metric tangent cone
and applications for complex algebraic
germs},
JOURNAL = {Selecta Math. (N.S.)},
FJOURNAL = {Selecta Mathematica. New Series},
VOLUME = {16},
NUMBER = {3},
YEAR = {2010},
PAGES = {377--391},
DOI = {10.1007/s00029-010-0024-0},
NOTE = {ArXiv:0905.4312. MR:2734336. Zbl:1200.14010.},
ISSN = {1022-1824},
}
L. Birbrair, A. Fernandes, and W. D. Neumann :
“On normal embedding of complex algebraic surfaces ,”
pp. 17–22
in
Real and complex singularities
(São Carlos, Brazil, 27 July–2 August 2008 ).
Edited by M. Manoel, M. C. Romero Fuster, and C. T. C. Wall .
London Mathematical Society Lecture Note Series 380 .
Cambridge University Press ,
2010 .
Dedicated to our friends Maria (Cidinha) Ruas and Terry Gaffney in connection to their 60th birthdays.
MR
2759086 Zbl
1215.14057 ArXiv
0901.0030 incollection
Abstract
People
BibTeX
@incollection {key2759086m,
AUTHOR = {Birbrair, L. and Fernandes, A. and Neumann,
W. D.},
TITLE = {On normal embedding of complex algebraic
surfaces},
BOOKTITLE = {Real and complex singularities},
EDITOR = {Manoel, M. and Romero Fuster, M. C.
and Wall, C. T. C.},
SERIES = {London Mathematical Society Lecture
Note Series},
NUMBER = {380},
PUBLISHER = {Cambridge University Press},
YEAR = {2010},
PAGES = {17--22},
NOTE = {(S\~ao Carlos, Brazil, 27 July--2 August
2008). Dedicated to our friends Maria
(Cidinha) Ruas and Terry Gaffney in
connection to their 60th birthdays.
ArXiv:0901.0030. MR:2759086. Zbl:1215.14057.},
ISSN = {0076-0552},
ISBN = {9780521169691},
}
W. D. Neumann and A. Pichon :
Lipschitz geometry of complex surfaces: Analytic invariants and equisingularity .
Technical report ,
November 2012 .
ArXiv
1211.4897 techreport
Abstract
People
BibTeX
We prove that the outer Lipschitz geometry of a germ \( (X,0) \) of a normal complex surface singularity determines a large amount of its analytic structure. In particular, it follows that any analytic family of normal surface singularities with constant Lipschitz geometry is Zariski equisingular. We also prove a strong converse for families of normal complex hypersurface singularities in \( \mathbb{C}^3 \) : Zariski equisingularity implies Lipschitz triviality. So for such a family Lipschitz triviality, constant Lipschitz geometry and Zariski equisingularity are equivalent to each other.
@techreport {key1211.4897a,
AUTHOR = {Neumann, Walter D. and Pichon, Anne},
TITLE = {Lipschitz geometry of complex surfaces:
{A}nalytic invariants and equisingularity},
MONTH = {November},
YEAR = {2012},
PAGES = {61},
NOTE = {ArXiv:1211.4897.},
}
L. Birbrair, W. D. Neumann, and A. Pichon :
“The thick-thin decomposition and the bilipschitz classification of normal surface singularities Acta Math.
212 : 2
(2014 ),
pp. 199–256 .
MR
3207758 Zbl
1303.14016 ArXiv
1105.3327 article
Abstract
People
BibTeX
We describe a natural decomposition of a normal complex surface singularity \( (X,0) \) into its “thick” and “thin” parts. The former is essentially metrically conical, while the latter shrinks rapidly in thickness as it approaches the origin. The thin part is empty if and only if the singularity is metrically conical; the link of the singularity is then Seifert fibered. In general the thin part will not be empty, in which case it always carries essential topology. Our decomposition has some analogy with the Margulis thick-thin decomposition for a negatively curved manifold. However, the geometric behavior is very different; for example, often most of the topology of a normal surface singularity is concentrated in the thin parts.
By refining the thick-thin decomposition, we then give a complete description of the intrinsic bilipschitz geometry of \( (X,0) \) in terms of its topology and a finite list of numerical bilipschitz invariants.
@article {key3207758m,
AUTHOR = {Birbrair, Lev and Neumann, Walter D.
and Pichon, Anne},
TITLE = {The thick-thin decomposition and the
bilipschitz classification of normal
surface singularities},
JOURNAL = {Acta Math.},
FJOURNAL = {Acta Mathematica},
VOLUME = {212},
NUMBER = {2},
YEAR = {2014},
PAGES = {199--256},
DOI = {10.1007/s11511-014-0111-8},
NOTE = {ArXiv:1105.3327. MR:3207758. Zbl:1303.14016.},
ISSN = {0001-5962},
}
W. D. Neumann and A. Pichon :
“Lipschitz geometry of complex curves J. Singul.
10
(2014 ),
pp. 225–234 .
MR
3300297 Zbl
1323.14003 ArXiv
1302.1138 article
Abstract
People
BibTeX
We describe the Lipschitz geometry of complex curves. To a large part this is well known material, but we give a stronger version even of known results. In particular, we give a quick proof, without any analytic restrictions, that the outer Lipschitz geometry of a germ of a complex plane curve determines and is determined by its embedded topology. This was first proved by Pham and Teissier, but in an analytic category. We also show the embedded topology of a plane curve determines its ambient Lipschitz geometry.
@article {key3300297m,
AUTHOR = {Neumann, Walter D. and Pichon, Anne},
TITLE = {Lipschitz geometry of complex curves},
JOURNAL = {J. Singul.},
FJOURNAL = {Journal of Singularities},
VOLUME = {10},
YEAR = {2014},
PAGES = {225--234},
DOI = {10.5427/jsing.2014.10o},
NOTE = {ArXiv:1302.1138. MR:3300297. Zbl:1323.14003.},
ISSN = {1949-2006},
}
W. D. Neumann, H. M. Pedersen, and A. Pichon :
“A characterization of Lipschitz normally embedded surface singularities J. Lond. Math. Soc. (2)
101 : 2
(2020 ),
pp. 612–640 .
MR
4093968 Zbl
1441.14015 ArXiv
1806.11240 article
Abstract
People
BibTeX
Any germ of a complex analytic space is equipped with two natural metrics: the outer metric induced by the hermitian metric of the ambient space and the inner metric , which is the associated riemannian metric on the germ. These two metrics are in general nonequivalent up to bilipschitz homeomorphism. We give a necessary and sufficient condition for a normal surface singularity to be Lipschitz normally embedded (LNE), that is, to have bilipschitz equivalent outer and inner metrics. In a partner paper (Neumann, Pedersen and Pichon, J. London Math. Soc. (2020)), we apply it to prove that rational surface singularities are LNE if and only if they are minimal.
@article {key4093968m,
AUTHOR = {Neumann, Walter D. and Pedersen, Helge
M\o ller and Pichon, Anne},
TITLE = {A characterization of {L}ipschitz normally
embedded surface singularities},
JOURNAL = {J. Lond. Math. Soc. (2)},
FJOURNAL = {Journal of the London Mathematical Society.
Second Series},
VOLUME = {101},
NUMBER = {2},
YEAR = {2020},
PAGES = {612--640},
DOI = {10.1112/jlms.12279},
NOTE = {ArXiv:1806.11240. MR:4093968. Zbl:1441.14015.},
ISSN = {0024-6107},
}
W. D. Neumann, H. M. Pedersen, and A. Pichon :
“Minimal surface singularities are Lipschitz normally embedded J. Lond. Math. Soc. (2)
101 : 2
(2020 ),
pp. 641–658 .
MR
4093969 Zbl
1441.14016 ArXiv
1503.03301 article
Abstract
People
BibTeX
Any germ of a complex analytic space is equipped with two natural metrics: the outer metric induced by the hermitian metric of the ambient space and the inner metric , which is the associated riemannian metric on the germ. We show that minimal surface singularities are Lipschitz normally embedded, that is, the identity map is a bilipschitz homeomorphism between outer and inner metrics, and that they are the only rational surface singularities with this property.
@article {key4093969m,
AUTHOR = {Neumann, Walter D. and Pedersen, Helge
M\o ller and Pichon, Anne},
TITLE = {Minimal surface singularities are {L}ipschitz
normally embedded},
JOURNAL = {J. Lond. Math. Soc. (2)},
FJOURNAL = {Journal of the London Mathematical Society.
Second Series},
VOLUME = {101},
NUMBER = {2},
YEAR = {2020},
PAGES = {641--658},
DOI = {10.1112/jlms.12280},
NOTE = {ArXiv:1503.03301. MR:4093969. Zbl:1441.14016.},
ISSN = {0024-6107},
}
Introduction to Lipschitz geometry of singularities Cuernavaca, Mexico, 11–22 June 2018 ).
Edited by W. Neumann and A. Pichon .
Lecture Notes in Mathematics 2280 .
Springer (Cham, Switzerland ),
2020 .
MR
4200092 Zbl
1456.58002 book
People
BibTeX
@book {key4200092m,
TITLE = {Introduction to {L}ipschitz geometry
of singularities},
EDITOR = {Neumann, Walter and Pichon, Anne},
SERIES = {Lecture Notes in Mathematics},
NUMBER = {2280},
PUBLISHER = {Springer},
ADDRESS = {Cham, Switzerland},
YEAR = {2020},
PAGES = {xvi+346},
DOI = {10.1007/978-3-030-61807-0},
NOTE = {(Cuernavaca, Mexico, 11--22 June 2018).
MR:4200092. Zbl:1456.58002.},
ISSN = {0075-8434},
ISBN = {9783030618063},
}
