

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 


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.mpimbonn.mpg.de/hwbllmnn/archiv/blaschke1603.pdf


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 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 of finite type with negative Euler characteristic has at most eigenvalues below a certain treshhold , a new invariant introduced by us and baptized the analytic systole of , [3,4].
We also discussed relations of with other geometric invariants of [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 , where is connected (but not necessarily complete), does not change if is coamenable in [6]. 


[ 1] Christian Bär, Werner Ballmann
Guide to elliptic boundary value problems for Diractype operators Arbeitstagung Bonn 2013 of Progr. Math. : 4380 Publisher: Birkhäuser/Springer, Cham 2016[ 2] Werner Ballmann
On etafunctions for nilmanifolds J. Differential Geom. , 97: (1): 110 2014[ 3] Werner Ballmann, Henrik Matthiesen, Sugata Mondal
Small eigenvalues of closed surfaces J. Differential Geom. , 103: (1): 113 2016[ 4] Werner Ballmann, Henrik Matthiesen, Sugata Mondal
Small eigenvalues of surfaces of finite type Compos. Math. , 153: (8): 17471768 2017 DOI: 10.1112/S0010437X17007291[ 5] Werner Ballmann, Henrik Matthiesen, Sugata Mondal
On the analytic systole of Riemannian surfaces of finite type Geom. Funct. Anal. , 27: (5): 10701105 2017 DOI: 10.1007/s000390170422y[ 6] Werner Ballmann, Henrik Matthiesen, Panagiotis Polymerakis
On the bottom of spectra under coverings eprint arXiv, online first , 1701.02130: 2017[ 7] Werner Ballmann, Jochen Brüning, Gilles Carron
Index theorems on manifolds with straight ends Compos. Math. , 148: (6): 18971968 2012 DOI: 10.1112/S0010437X12000401




• 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)


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


Paul Schmutz (1989): “Zur Anzahl kleiner Eigenwerte auf Riemannschen Flächen”
Christian Bär (1990): “Das Spektrum von DiracOperatoren”,
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


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


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