W. D. Neumann :
“Graph 3-manifolds, splice diagrams, singularities 787–817
in
Singularity theory
(Marseille, France, 24 January–25 February 2005 ).
Edited by D. Chéniot, N. Dutertre, C. Murolo, D. Trotman, and A. Pichon .
World Scientific (Hackensack, NJ ),
2007 .
Conference dedicated to Jean-Paul Brasselet on his 60th birthday.
MR
2342940 Zbl
1155.32019 incollection
Abstract
People
BibTeX
We describe how a coarse classification of graph manifolds can give clearer insight into their structure, and we relate this particularly to the manifolds that can occur as the links of points in normal complex surfaces. We relate this discussion to a special class of singularities; those of “splice type”, which turn out to play a central role among singularities of complex surfaces.
@incollection {key2342940m,
AUTHOR = {Neumann, Walter D.},
TITLE = {Graph 3-manifolds, splice diagrams,
singularities},
BOOKTITLE = {Singularity theory},
EDITOR = {Ch\'eniot, Denis and Dutertre, Nicolas
and Murolo, Claudio and Trotman, David
and Pichon, Anne},
PUBLISHER = {World Scientific},
ADDRESS = {Hackensack, NJ},
YEAR = {2007},
PAGES = {787--817},
DOI = {10.1142/9789812707499_0034},
NOTE = {(Marseille, France, 24 January--25 February
2005). Conference dedicated to Jean-Paul
Brasselet on his 60th birthday. MR:2342940.
Zbl:1155.32019.},
ISBN = {9789812704108},
}
W. D. Neumann and A. Pichon :
“Complex analytic realization of links 231–238
in
Intelligence of low dimensional topology 2006
(Hiroshima, Japan, 22–26 July 2006 ).
Edited by J. S. Carter, S. Kamada, L. H. Kauffman, A. Kawauchi, and T. Kohno .
Series on Knots and Everything 40 .
World Scientific (Hackensack, NJ ),
2007 .
MR
2371730 Zbl
1146.32013 ArXiv
math/0610348 incollection
Abstract
People
BibTeX
@incollection {key2371730m,
AUTHOR = {Neumann, Walter D. and Pichon, Anne},
TITLE = {Complex analytic realization of links},
BOOKTITLE = {Intelligence of low dimensional topology
2006},
EDITOR = {Carter, J. Scott and Kamada, Seiichi
and Kauffman, Louis H. and Kawauchi,
Akio and Kohno, Toshitake},
SERIES = {Series on Knots and Everything},
NUMBER = {40},
PUBLISHER = {World Scientific},
ADDRESS = {Hackensack, NJ},
YEAR = {2007},
PAGES = {231--238},
DOI = {10.1142/9789812770967_0029},
NOTE = {(Hiroshima, Japan, 22--26 July 2006).
ArXiv:math/0610348. MR:2371730. Zbl:1146.32013.},
ISSN = {0219-9769},
ISBN = {9789812705938},
}
A. Némethi, W. D. Neumann, and A. Pichon :
“Principal analytic link theory in homology sphere links 377–387
in
Topology of algebraic varieties and singularities: Invited papers of the conference in honor of Anatoly Libgober’s 60th birthday
(Jaca, Spain, 22–26 June 2009 ).
Edited by J. I. Cogolludo-Agustín and E. Hironaka .
Contemporary Mathematics 538 .
American Mathematical Society ,
2011 .
MR
2777831 Zbl
1272.32029 ArXiv
0909.1348 incollection
Abstract
People
BibTeX
For the link \( M \) of a normal complex surface singularity \( (X,0) \) we ask when a knot \( K \subset M \) exists for which the answer to whether \( K \) is the link of the zero set of some analytic germ \( (X,0)\to (\mathbb{C},0) \) affects the analytic structure on \( (X,0) \) . We show that if \( M \) is an integral homology sphere then such a knot exists if and only if \( M \) is not one of the Brieskorn homology spheres \( M(2,3,5) \) , \( M(2,3,7) \) , \( M(2,3,11) \) .
@incollection {key2777831m,
AUTHOR = {N\'emethi, A. and Neumann, Walter D.
and Pichon, A.},
TITLE = {Principal analytic link theory in homology
sphere links},
BOOKTITLE = {Topology of algebraic varieties and
singularities: {I}nvited papers of the
conference in honor of {A}natoly {L}ibgober's
60th birthday},
EDITOR = {Cogolludo-Agust\'{\i}n, Jos\'e Ignacio
and Hironaka, Eriko},
SERIES = {Contemporary Mathematics},
NUMBER = {538},
PUBLISHER = {American Mathematical Society},
YEAR = {2011},
PAGES = {377--387},
DOI = {10.1090/conm/538/10614},
NOTE = {(Jaca, Spain, 22--26 June 2009). ArXiv:0909.1348.
MR:2777831. Zbl:1272.32029.},
ISSN = {0271-4132},
ISBN = {9780821848906},
}
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.
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TITLE = {The thick-thin decomposition and the
bilipschitz classification of normal
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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 and A. Pichon :
“Lipschitz geometry does not determine embedded topological type 183–195
in
Singularities in geometry, topology, foliations and dynamics: A celebration of the 60th birthday of José Seade
(Mérida, Mexico, 8–19 December 2014 ).
Edited by J. L. Cisneros-Molina, D. Tráng Lê, M. Oka, and J. Snoussi .
Trends in Mathematics .
Birkhäuser (Cham, Switzerland ),
2017 .
Dedicated to José Seade for a great occasion. Happy birthday, Pepe.
MR
3706220 Zbl
1426.14003 ArXiv
1506.03841 incollection
Abstract
People
BibTeX
We investigate the relationships between the Lipschitz outer geometry and the embedded topological type of a hypersurface germ in \( (\mathbb{C}^n,0) \) . It is well known that the Lipschitz outer geometry of a complex plane curve germ determines and is determined by its embedded topological type. We prove that this does not remain true in higher dimensions. Namely, we give two normal hypersurface germs \( (X_1,0) \) and \( (X_2,0) \) in \( (\mathbb{C}^3,0) \) having the same outer Lipschitz geometry and different embedded topological types. Our pair consist of two superisolated singularities whose tangent cones form an Alexander–Zariski pair having only cusp-singularities. Our result is based on a description of the Lipschitz outer geometry of a superisolated singularity. We also prove that the Lipschitz inner geometry of a superisolated singularity is completely determined by its (non-embedded) topological type, or equivalently by the combinatorial type of its tangent cone.
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TITLE = {Lipschitz geometry does not determine
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BOOKTITLE = {Singularities in geometry, topology,
foliations and dynamics: {A} celebration
of the 60th birthday of {J}os\'e {S}eade},
EDITOR = {Cisneros-Molina, Jos\'e Luis and Tr\'ang
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SERIES = {Trends in Mathematics},
PUBLISHER = {Birkh\"auser},
ADDRESS = {Cham, Switzerland},
YEAR = {2017},
PAGES = {183--195},
DOI = {10.1007/978-3-319-39339-1_11},
NOTE = {(M\'erida, Mexico, 8--19 December 2014).
Dedicated to Jos\'e Seade for a great
occasion. Happy birthday, Pepe!. ArXiv:1506.03841.
MR:3706220. Zbl:1426.14003.},
ISSN = {2297-0215},
ISBN = {9783319393384},
}
A. Fernandes and J. E. Sampaio :
“Tangent cones of Lipschitz normally embedded sets are Lipschitz normally embedded Int. Math. Res. Not.
2019 : 15
(August 2019 ),
pp. 4880–4897 .
Appendix by Anne Pichon and Walter D. Neumann.
MR
3988673 Zbl
1457.32014 ArXiv
1705.00038 article
Abstract
People
BibTeX
We prove that tangent cones of Lipschitz normally embedded sets are Lipschitz normally embedded. We also extend to real subanalytic sets the notion of reduced tangent cone and we show that subanalytic Lipschitz normally embedded sets have reduced tangent cones. In particular, we get that Lipschitz normally embedded complex analytic sets have reduced tangent cones.
@article {key3988673m,
AUTHOR = {Fernandes, Alexandre and Sampaio, J.
Edson},
TITLE = {Tangent cones of {L}ipschitz normally
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embedded},
JOURNAL = {Int. Math. Res. Not.},
FJOURNAL = {International Mathematics Research Notices},
VOLUME = {2019},
NUMBER = {15},
MONTH = {August},
YEAR = {2019},
PAGES = {4880--4897},
DOI = {10.1093/imrn/rnx290},
NOTE = {Appendix by Anne Pichon and Walter D.
Neumann. ArXiv:1705.00038. MR:3988673.
Zbl:1457.32014.},
ISSN = {1073-7928},
}
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},
}
W. D. Neumann :
“3-manifolds and links of singularities 73–86
in
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
4200095 Zbl
1457.32071 incollection
Abstract
People
BibTeX
This chapter gives a brief overview of general 3-manifold topology, and its implications for links of isolated complex surface singularities. It does not discuss Lipschitz geometry, but it provides many examples of isolated complex surface singularities on which one can work to find their Lipschitz geometry (e.g., thick-thin decompositions and inner and/or outer bilipschitz classifications).
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AUTHOR = {Neumann, Walter D.},
TITLE = {3-manifolds and links of singularities},
BOOKTITLE = {Introduction to {L}ipschitz geometry
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EDITOR = {Neumann, Walter and Pichon, Anne},
SERIES = {Lecture Notes in Mathematics},
NUMBER = {2280},
PUBLISHER = {Springer},
ADDRESS = {Cham, Switzerland},
YEAR = {2020},
PAGES = {73--86},
DOI = {10.1007/978-3-030-61807-0_3},
NOTE = {(Cuernavaca, Mexico, 11--22 June 2018).
MR:4200095. Zbl:1457.32071.},
ISSN = {0075-8434},
ISBN = {9783030618063},
}
