New development of the research of Lefschetz fibrations by Teichmuller theory

2013 Fiscal Year Final Research Report

• Project/Area Number: 23654024
• Research Category: Grant-in-Aid for Challenging Exploratory Research
• Allocation Type: Multi-year Fund
• Research Field: Geometry
• Research Institution: Tokyo Institute of Technology
• Principal Investigator: SHIGA Hiroshige 東京工業大学, 理工学研究科, 教授 (10154189)
• Co-Investigator(Kenkyū-buntansha): MIYACHI Hideki 大阪大学, 大学院・理学研究科, 准教授 (40385480) ; ENDO Hisaaki 東京工業大学, 大学院・理工学研究科, 教授 (20323777)
• Project Period (FY): 2011 – 2013
• Keywords: レフシェッツファイバー空間 / タイヒミュラー空間

Research Abstract

Shiga obtained effective bounds of holomorphic families of Riemann surfaces, and also showed the rigidity and the finiteness of Teichmuller curves. In a joint work with Miyachi, Shiga studied relations of the holonomy and the slope inequality of Lefschetz fibrations.
Miyachi clarified the boundary behavior of the Teichmuller distance.
In a joint work with T. Mark and J. Van Horn-Morries, Endo found monodromy permutations for rational blow-downs and developed some method to study Lefschetz fibrations by using finite graphs.

Research Products

• [Journal Article] A differential formula for Extremal length, In the tradition of Ahlfors-Bers, VI (2013)
  • Author(s): H. Miyachi
  • Journal Title: Contemp. Math., AMS Volume: 590 Pages: 137-152
  • DOI: 10.1090/conm/590
  • Peer Reviewed
• [Journal Article] On the number of holomorphic families of Riemann surfaces (2012)
  • Author(s): H. Shiga
  • Journal Title: Contemp. Math., AMS Volume: 575 Pages: 331-342
  • DOI: 10.1090/conm/575
  • Peer Reviewed
• [Journal Article] Extending holomorphic motions and monodromy (2012)
  • Author(s): M. Beck, Y. Jiang, S. Mitra and H. Shiga
  • Journal Title: Ann. Acad. Sci. Fenn. Math Volume: 37 Pages: 53-67
  • DOI: 10.5186/aasfm.2012.3713
  • Peer Reviewed
• [Journal Article] Holonomies and the slope inequality of Lefschetz fibrations (2011)
  • Author(s): H. Miyachi and H. Shiga
  • Journal Title: Proc. Amer. Math. Soc Volume: 139 Pages: 1299-1307
  • DOI: 10.1090/S0002-9939-2010-10563-4
  • Peer Reviewed
• [Journal Article] Jeremy Van Horn-Morris, Monodromy substitutions and rational blowdowns (2011)
  • Author(s): H. Endo, Thomas E. Mark
  • Journal Title: Journal of Topology Volume: 4 Pages: 227-253
  • DOI: 10.1112/jtopol/jtq041
  • Peer Reviewed
• [Journal Article] Chart description for hyperelliptic Lefschetz fibrations and their stabilization
  • Author(s): H. Endo and S. Kamada
  • Journal Title: Topology and its Applications Volume: (掲載決定)
  • URL: http://arxiv.org/pdf/1306.2707.pdf
  • Peer Reviewed
• [Presentation] Chart description for hyperelliptic Lefschetz fibrations and their stabilization (2014)
  • Author(s): H. Endo
  • Organizer: Seminaire de topologie et de geometrie algebrique
  • Place of Presentation: Universite de Nantes, France
  • Year and Date: 2014-03-13
  • Invited
• [Presentation] Complex analytic properties of deformation spaces of Kleinian groups (2014)
  • Author(s): H. Shiga
  • Organizer: New Trends in Teichmüller Theory and Mapping Class Groups
  • Place of Presentation: Mathematisches Forschungsinstitut Oberwolfach, Germany
  • Year and Date: 2014-02-10
  • Invited
• [Presentation] The geometry of extremal length, I, II, III, Special Program on Teichmüller Theory (2013)
  • Author(s): H. Miyachi
  • Organizer: The Erwin Schrodinger International Institute for Mathematical Physics
  • Place of Presentation: Wien, Austria
  • Year and Date: 20130319-20
  • Invited
• [Presentation] Isometry of Teichmüller space (2013)
  • Author(s): H. Miyachi
  • Organizer: Advanced School and discussion meeting on Knot theory and its application
  • Place of Presentation: IISER Mohali, India
  • Year and Date: 2013-12-12
  • Invited
• [Presentation] Bers-Thurston Classification of the mapping class group (2013)
  • Author(s): H. Miyachi
  • Organizer: Advanced School and discussion meeting on Knot theory and its application
  • Place of Presentation: IISER Mohali, India
  • Year and Date: 2013-12-11
  • Invited
• [Presentation] Introduction to Teichmüller space (2013)
  • Author(s): H. Miyachi
  • Organizer: Advanced School and discussion meeting on Knot theory and its application
  • Place of Presentation: IISER Mohali, India
  • Year and Date: 2013-12-10
  • Invited
• [Presentation] On some analytic properties of deformation spaces of Kleinian groups (2013)
  • Author(s): H. Shiga
  • Organizer: Conformal Geometry and Riemann Surfaces
  • Place of Presentation: Queens College, City University of New York, USA
  • Year and Date: 2013-10-27
  • Invited
• [Presentation] Chart description for hyperelliptic Lefschetz fibrations and their stabilization (2013)
  • Author(s): H. Endo
  • Organizer: International Conference on Topology and Geometry 2013, Joint with the 6th Japan-Mexico Topology Symposium
  • Place of Presentation: Shimane University, Matsue(島根県)
  • Year and Date: 2013-09-02
  • Invited
• [Presentation] Chart description for hyperelliptic Lefschetz fibrations and their stabilization (2013)
  • Author(s): H. Endo
  • Organizer: Workshop and Conference on the Topology and Invariants of Smooth 4-Manifolds
  • Place of Presentation: University of Minnesota, Twin Cities, Minneapolis, USA
  • Year and Date: 2013-08-08
  • Invited
• [Presentation] Geometry on Teichmüller space via extremal length (2013)
  • Author(s): H. Miyachi
  • Organizer: The XXIIth Rolf Nevanlinna Colloquium
  • Place of Presentation: University of Helsinki, Finnland
  • Year and Date: 2013-08-06
  • Invited
• [Presentation] Lefschetz fibrations and the signature cocycle (2013)
  • Author(s): H. Endo
  • Organizer: Conference on interactions between low dimensional topology and mapping class groups
  • Place of Presentation: Max Plank Institute for Mathematics, Bonn, Germany
  • Year and Date: 2013-07-05
  • Invited
• [Presentation] On some analytic properties of deformation spaces of Kleinian groups (2013)
  • Author(s): H. Shiga
  • Organizer: Analysis and Geometry of Riemann Surfaces and Related Topics
  • Place of Presentation: 東京工業大学(東京都)
  • Year and Date: 2013-06-22
  • Invited
• [Presentation] Geometry of Teichmüller distance (2013)
  • Author(s): H. Miyachi
  • Organizer: Analysis and Geometry of Riemann Surfaces and Related Topics
  • Place of Presentation: 東京工業大学(東京都)
  • Year and Date: 2013-06-21
  • Invited
• [Presentation] Geometry on Extremal length in Teichmüller theory (2013)
  • Author(s): H. Miyachi
  • Organizer: AMS Sectional meeting : Special Session on Analysis
  • Place of Presentation: Iowa State University, USA
  • Year and Date: 2013-04-28
  • Invited
• [Presentation] Holomorphic families of Riemann surfaces and their degenerations (2012)
  • Author(s): H. Shiga
  • Organizer: Dynamics and Geometry Seminar
  • Place of Presentation: Harvard University, USA
  • Year and Date: 2012-09-26
  • Invited
• [Presentation] Some geometric properties of configuration spaces and moduli spaces, Centre for Quantum Geometry of Moduli Spaces (QGM), Science and Technology (2012)
  • Author(s): H. Shiga
  • Organizer: QGM Seminar
  • Place of Presentation: Aarhus University, Denmark
  • Year and Date: 2012-08-17
  • Invited
• [Presentation] Extremal length geometry of Teichmüller space (2012)
  • Author(s): H. Miyachi
  • Organizer: Progress in Low-dimensional topology : Teichmüller theory and 3-manifold groups, Centre for Quantum Geometry of Moduli Spaces (QGM), Science and Technology
  • Place of Presentation: Aarhus University, Denmark
  • Year and Date: 2012-08-11
  • Invited
• [Presentation] On holomorphic families of Kleinian groups (2012)
  • Author(s): H. Shiga
  • Organizer: Group actions and applications in Geometry, Topology and Analysis
  • Place of Presentation: Kunming University of Science and Technology, China
  • Year and Date: 2012-07-24
  • Invited
• [Presentation] On holomorphic families of Kleinian groups (2012)
  • Author(s): H. Shiga
  • Organizer: Colloquium of Complex Analysis
  • Place of Presentation: Fudan University, China
  • Year and Date: 2012-05-22
  • Invited
• [Presentation] On holomorphic families of Kleinian groups (2012)
  • Author(s): H. Shiga
  • Organizer: Complex Analysis seminar
  • Place of Presentation: Shenzen University, China
  • Year and Date: 2012-05-18
  • Invited
• [Presentation] On geometric properties of configuration spaces and moduli spaces (2012)
  • Author(s): H. Shiga
  • Organizer: Colloquium of Mathematical Department of Gouzhou University
  • Place of Presentation: 中山大学数学教室(中国)
  • Year and Date: 2012-05-17
  • Invited
• [Presentation] Geometry of limit sets of Kleinian surface groups (2011)
  • Author(s): H. Miyachi
  • Organizer: BOREL SEMINAR 2011, Topics in Real and Complex Hyperbolic Geometry
  • Place of Presentation: Maison des Congrès, Les Diablerets, Switzerland
  • Year and Date: 2011-08-24
  • Invited
• [Presentation] On the boundary behavior of Cauchy integrals on Riemann surface (2011)
  • Author(s): H. Shiga
  • Organizer: International Conference on Analysis
  • Place of Presentation: Academy of Mathematics and System Science, Chinese Academy of Sciences, China
  • Year and Date: 2011-08-16
  • Invited
• [Presentation] Kleinian groups and geometric function theory (2011)
  • Author(s): H. Shiga
  • Organizer: Complex Analysis Seminar
  • Place of Presentation: Morinig Center of Mathematics, Beijing, China
  • Year and Date: 2011-08-12
  • Invited
• [Presentation] Teichmüller spaces and configuration spaces (2011)
  • Author(s): H. Shiga
  • Organizer: Complex Analysis Seminar
  • Place of Presentation: Morinig Center of Mathematics, Beijing, China
  • Year and Date: 2011-08-11
  • Invited
• [Presentation] Limits of geodesic rays and non-visible points of Teichmüller space (2011)
  • Author(s): H. Miyachi
  • Organizer: Aspects of hyperbolicity in geometry, topology, and dynamics
  • Place of Presentation: University of Warwick, UK
  • Year and Date: 2011-07-26
  • Invited
• [Presentation] On generalizations of Plemelj's theorem on Cauchy integrals (2011)
  • Author(s): H. Shiga
  • Organizer: XVIth Conference on Analytic Functions and Related Topics
  • Place of Presentation: Institute of Mathematics and Computer Science of the State School of Higher Education in Chelm, Poland
  • Year and Date: 2011-06-28
  • Invited
• [Presentation] Lipschitz algebra on Teichmüller space (2011)
  • Author(s): 宮地 秀樹
  • Organizer: Geometric and analytic approaches to representations of a group and representation spaces
  • Place of Presentation: 京都大学数理解析研究所(京都府)
  • Year and Date: 2011-06-21
  • Invited