Existence and uniqueness theorems for weak solutions of a complex Monge-Ampère equation are established, extending the Bedford-Taylor pluripotential theory. As a consequence, using the Tian-Yau-Zelditch theorem, it is shown that geodesics in the space of Kähler potentials can be approximated by geodesics in the spaces of Bergman metrics. Motivation from Donaldson’s program on constant scalar curvature metrics and Yau’s strategy of approximating Kähler metrics by Bergman metrics is also discussed.
@incollection{JEDP_2005____A10_0,
author = {Phong, D.H. and Sturm, Jacob},
title = {Monge-Amp\`ere {Equations,} {Geodesics} and {Geometric} {Invariant} {Theory}},
booktitle = {},
series = {Journ\'ees \'equations aux d\'eriv\'ees partielles},
eid = {10},
pages = {1--15},
publisher = {Groupement de recherche 2434 du CNRS},
year = {2005},
doi = {10.5802/jedp.22},
mrnumber = {2352778},
language = {en},
url = {http://www.numdam.org/articles/10.5802/jedp.22/}
}
TY - JOUR AU - Phong, D.H. AU - Sturm, Jacob TI - Monge-Ampère Equations, Geodesics and Geometric Invariant Theory JO - Journées équations aux dérivées partielles PY - 2005 SP - 1 EP - 15 PB - Groupement de recherche 2434 du CNRS UR - http://www.numdam.org/articles/10.5802/jedp.22/ DO - 10.5802/jedp.22 LA - en ID - JEDP_2005____A10_0 ER -
%0 Journal Article %A Phong, D.H. %A Sturm, Jacob %T Monge-Ampère Equations, Geodesics and Geometric Invariant Theory %J Journées équations aux dérivées partielles %D 2005 %P 1-15 %I Groupement de recherche 2434 du CNRS %U http://www.numdam.org/articles/10.5802/jedp.22/ %R 10.5802/jedp.22 %G en %F JEDP_2005____A10_0
Phong, D.H.; Sturm, Jacob. Monge-Ampère Equations, Geodesics and Geometric Invariant Theory. Journées équations aux dérivées partielles (2005), article no. 10, 15 p. doi : 10.5802/jedp.22. http://www.numdam.org/articles/10.5802/jedp.22/
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