Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 37, 2012, 91-106

VARIATION OF GEODESIC LENGTH FUNCTIONS IN FAMILIES OF KÄHLER--EINSTEIN MANIFOLDS AND APPLICATIONS TO TEICHMÜLLER SPACE

Reynir Axelsson and Georg Schumacher

Háskóli Íslands, Department of Mathematics
Dunhaga 5, IS-107 Reykjavík, Ísland; reynir 'at' raunvis.hi.is

Philipps-Universität Marburg, Fachbereich Mathematik und Informatik
Lahnberge, D-35032 Marburg, Germany; schumac 'at' mathematik.uni-marburg.de

Abstract. In the study of Teichmüller spaces the second variation of the logarithm of the geodesic length function plays a central role. So far, it was accessible only in a rather indirect way. We treat the problem directly in the more general framework of the deformation theory of Kähler-Einstein manifolds. For the first variation we arrive at a surprisingly simple formula, which only depends on harmonic Kodaira-Spencer forms. We also compute the second variation in the general case and then apply the result to families of Riemann surfaces. Again we obtain a simple formula depending only on the harmonic Beltrami differentials. As a consequence a new proof for the plurisubharmonicity of the geodesic length function on Teichmüller space and its logarithm together with upper estimates follow. The results also apply to the previously not known cases of Teichmüller spaces of weighted punctured Riemann surfaces, where the methods of Kleinian groups are not available. We use our methods from [A-S], where the result was announced.

2010 Mathematics Subject Classification: Primary 53C55, 32G15, 32Q20.

Key words: Kähler-Einstein metrics, Teichmüller theory, geodesic length functions.

Reference to this article: R. Axelsson and G. Schumacher: Variation of geodesic length functions in families of Kähler-Einstein manifolds and applications to Teichmüller space. Ann. Acad. Sci. Fenn. Math. 37 (2012), 91-106.

Full document as PDF file

doi:10.5186/aasfm.2012.3703