New York Journal of Mathematics
Volume 22 (2016) 1111-1133


Simone Calamai, David Petrecca, and Kai Zheng

On the geodesic problem for the Dirichlet metric and the Ebin metric on the space of Sasakian metrics

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Published: September 23, 2016
Keywords: Space of Kähler metrics, geodesic equation, Cauchy problem, space of Sasakian metrics
Subject: 53C55 (primary); 32Q15, 58D27 (secondary)

We study the geodesic equation for the Dirichlet (gradient) metric in the space of Kähler potentials. We first solve the initial value problem for the geodesic equation of the combination metric, including the gradient metric. We then discuss a comparison theorem between it and the Calabi metric. As geometric motivation of the combination metric, we find that the Ebin metric restricted to the space of type II deformations of a Sasakian structure is the sum of the Calabi metric and the gradient metric.


S. C. is supported by the PRIN Project "Varietà reali e complesse: geometria, topologia e analisi armonica'', by SIR 2014 AnHyC "Analytic aspects in complex and hypercomplex geometry" (code RBSI14DYEB), and by GNSAGA of INdAM.
D. P. is supported by the Research Training Group 1463 "Analysis, Geometry and String Theory'' of the DFG, as well as the GNSAGA of INdAM.
The work of K. Z. has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 703949, and was also partially supported by the Engineering and Physical Sciences Research Council (EPSRC) on a Programme Grant entitled "Singularities of Geometric Partial Differential Equations" reference number EP/K00865X/1.

Author information

Simone Calamai:
Dip. di Matematica e Informatica "U. Dini'' - Università di Firenze, Viale Morgagni 67A - Firenze - Italy

David Petrecca:
Institut für Differentialgeometrie, Leibniz Universität Hannover, Welfengarten 1, Hannover, Germany

Kai Zheng:
Mathematics Institute, University of Warwick, Coventry, CV4 7AL, UK