Profile
Profile

Prof. Dr. Werner Ballmann

E-Mail: hwbllmnn(at)mpim-bonn.mpg.de
Telefon: +49 228 402240
Fax: +49 228 73 62268
Homepage: http://www.mpim-bonn.mpg.de/node/103
Raum: N2.012
Standort: Max Planck Institute for Mathematics
Institute: Max Planck Institute for Mathematics
Mathematical Institute
Forschungsbereiche: Research Area A
Research Area G (until 10/2012)
Former Research Area F
Geburtsdatum: 11.Apr 1951

Academic Career

1979

Dr. rer. nat., University of Bonn

1979 - 1985

Assistant Professor, University of Bonn

1984 - 1986

Associate Professor, University of Maryland, College Park, MD, USA

1986 - 1987

Professor (C3), University of Bonn

1987 - 1989

Ordinarius, University of Zürich, Switzerland

1989 - 2016

Professor (C4), University of Bonn

Since 2007

Scientific Member and Director, Max Planck Institute for Mathematics, Bonn

Research Profile

The present foci of my work are spectral theory on one hand and the generalized Blaschke conjecture on the other. In spectral theory, my recent work deals with small eigenvalues of the Laplacian on complete Riemannian surfaces and an associated new invariant, the analytic systole. I also study the bottom of the spectrum of the Laplacian under Riemannian coverings, extending and generalizing previous work of Brooks. All this is joint work and project with three young mathematicians, Sugata Mondal, Henrik Matthiesen, and Panagiotis Polymerakis.

In the near future, I intend to concentrate more on the generalized Blasche conjecture. Say that a closed Riemannian manifold is a Blaschke manifold if its injectivity radius and diameter coincide. The generalized Blaschke conjecture then asserts that a Blaschke manifold is a compact rank one symmetric space (cross). Blaschke manifolds do have the cohomology of a cross and fall into corresponding types. Whereas the spherical type was solved by Berger, Kazdan, Weinstein, and Yang, the conjecture is open for the other types. This is a joint project with Karsten Grove. So far our main result is the uncovering of a serious mistake in one of the classical papers on the subject, pertaining to the diffeomorphism class of Blaschke manifolds of the complex projective type.
See also http://people.mpim-bonn.mpg.de/hwbllmnn/archiv/blaschke1603.pdf

Contribution to Research Areas


Research Area A
The article [1] contains an introduction to the boundary value theory of Dirac type operators as Christian Baer and I developed it earlier, but contains also some related new work. In the paper [2], I discuss an <br>eta-invariant occuring in index theorems for Dirac operators on finite volume manifolds of pinched negative sectional curvature.

In the last years, I have been working on the spectrum of the Laplace operator on functions on Riemannian manifolds. My work concerns two problems, 1) small eigenvalues of surfaces and 2) the behaviour of the bottom of the spectrum of the Laplacian under coverings. My collaborators on the first problem were Henrik Matthiesen and Sugata Mondal. Generalizing work of Otal and Rosas, we showed that a complete Riemannian surface S of finite type with negative Euler characteristic <br>chi(S) has at most -<br>chi(S) eigenvalues below a certain treshhold <br>Lambda(S), a new invariant introduced by us and baptized the analytic systole of S, [3,4].

We also discussed relations of <br>Lambda(S) with other geometric invariants of S [5]. My collaborators on the second problem were Henrik Matthiesen and Panagiotis Polymerakis. Extending work of Brooks and others, we showed that the bottom of the spectrum under a Riemannian covering M_1<br>to M_0, where M_0 is connected (but not necessarily complete), does not change if <br>pi_1(M_1) is coamenable in <br>pi_1(M_0) [6].

Selected Publications

[1] Christian Bär, Werner Ballmann
Guide to elliptic boundary value problems for Dirac-type operators
Arbeitstagung Bonn 2013
of Progr. Math. : 43--80
Publisher: Birkhäuser/Springer, Cham
2016
[2] Werner Ballmann
On eta-functions for nilmanifolds
J. Differential Geom. , 97: (1): 1--10
2014
[3] Werner Ballmann, Henrik Matthiesen, Sugata Mondal
Small eigenvalues of closed surfaces
J. Differential Geom. , 103: (1): 1--13
2016
[7] Werner Ballmann, Jochen Brüning, Gilles Carron
Index theorems on manifolds with straight ends
Compos. Math. , 148: (6): 1897--1968
2012

Publication List

MathSciNet Publication List (external link)

Editorships

• International Journal of Mathematics (1989 - 1995)
• Geometriae Dedicata (1989 - 1999)
• Inventiones mathematicae (1996 - 2007)
• Mathematische Zeitschrift (1997 - 2003)
• Journal für die reine und angewandte Mathematik (Advisory Board, since 1990)
• Mathematical Proceedings, Cambridge Philosophical Society (since 2006)

Habilitations

Christian Bär (1993), now Professor, University of Potsdam

Bernhard Leeb (1997), now Professor, LMU Munich

Vicente Cortés (1999), now Professor, University of Hamburg

Dorothee Schueth (2000), now Professor, HU Berlin

Chand Devchand (2003), now Privatdozent, University of Potsdam

Alina Vdovina (2005), now Lecturer, Newcastle University, England, UK

Gregor Weingart (2005), now Researcher, National Autonomous University of Mexico, Mexico

Alexander Lytchak (2007), now Professor, University of Cologne

Thomas Vogel (2014), now Professor, LMU München

Selected PhD students

Paul Schmutz (1989): “Zur Anzahl kleiner Eigenwerte auf Riemannschen Flächen”

Christian Bär (1990): “Das Spektrum von Dirac-Operatoren”,
now Professor, University of Potsdam

Dorothee Schüth (1993): “Stetige isospektrale Deformationen”,
now Professor, HU Berlin

Vicente Cortés (1994): “Alekseevskiis quaternionische Kählermannigfaltigkeiten”,
now Professor, University of Hamburg

Gregor Weingart (1998): “Moduli Spaces of Minimal Isometric Immersions”,
now Researcher, National Autonomous University of Mexico, Mexico

Alexander Lytchak (2001): “Allgemeine Theorie der Submetrien und verwandte mathematische Probleme”,
now Professor, University of Cologne

Anna Wienhard (2004): “Bounded Cohomology and Geometry”,
now Professor, University of Heidelberg

Supervised Theses

  • Master theses: 10
  • Diplom theses: 45
  • PhD theses: 14, currently 3
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