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30
A Brunn-Minkowski type inequality for Fano manifolds and the BandoMabuchi uniqueness theorem
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Viscosity solutions to degenerate complex Monge-Ampère equations
, 2010
"... Degenerate complex Monge-Ampère equations on compact Kähler manifolds have been recently intensively studied using tools from pluripotential theory. We develop an alternative approach based on the concept of viscosity solutions and compare systematically viscosity concepts with pluripotential theor ..."
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Cited by 19 (7 self)
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Degenerate complex Monge-Ampère equations on compact Kähler manifolds have been recently intensively studied using tools from pluripotential theory. We develop an alternative approach based on the concept of viscosity solutions and compare systematically viscosity concepts with pluripotential theoretic ones. This approach works only for a rather restricted type of degenerate complex Monge-Ampère equations. Nevertheless, we prove that the local potentials of the singular Kähler-Einstein metrics constructed previously by the authors are continuous plurisubharmonic functions. They were previously known to be locally bounded. Another application is a lower order construction with a C 0-estimate of the solution to the Calabi conjecture which does not use Yau’s celebrated theorem.
Moser-Trudinger type inequalities for complex Monge-Ampère operators and Aubins hypothèse fondamentale
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Holomorphic Morse inequalities and asymptotic cohomology groups: a tribute to Bernhard Riemann
- IN THE PROCEEDINGS OF THE RIEMANN INTERNATIONAL SCHOOL OF MATHEMATICS, ADVANCES IN NUMBER THEORY AND GEOMETRY, HELD AT VERBANIA
, 2009
"... The goal of this note is to present the potential relationships between certain Monge-Ampère integrals appearing in holomorphic Morse inequalities, and asymptotic cohomology estimates for tensor powers of line bundles, as recently introduced by algebraic geometers. The expected most general statemen ..."
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Cited by 7 (4 self)
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The goal of this note is to present the potential relationships between certain Monge-Ampère integrals appearing in holomorphic Morse inequalities, and asymptotic cohomology estimates for tensor powers of line bundles, as recently introduced by algebraic geometers. The expected most general statements are still conjectural and oweadebt to Riemann’s pioneering work, which ledto the concept of Hilbert polynomials and to the Hirzebruch-Riemann-Roch formula during the XX-th century.
On Pointwise Gradient estimates for the complex MongeAmpère equation
- in Advances in Geometric Analysis, 87–96, Adv. Lect. Math. (ALM) 21, International
, 2012
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