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Real Monge-Ampère equations and Kähler-Ricci solitons on toric log Fano varieties

Robert J. Berman, Bo Berndtsson (2013)

Annales de la faculté des sciences de Toulouse Mathématiques

We show, using a direct variational approach, that the second boundary value problem for the Monge-Ampère equation in $ℝ^{n}$ with exponential non-linearity and target a convex body $P$ is solvable iff $0$ is the barycenter of $P .$ Combined with some toric geometry this confirms, in particular, the (generalized) Yau-Tian-Donaldson conjecture for toric log Fano varieties $(X, Δ)$ saying that $(X, Δ)$ admits a (singular) Kähler-Einstein metric iff it is K-stable in the algebro-geometric sense. We thus obtain a new proof and...

Rectifiability of defect measures, fundamental groups and density of Sobolev mappings

Fang Hua Lin (1996)

Journées équations aux dérivées partielles

Régularité de la solution d'un problème variationnel

R. Tahraoui (1992)

Annales de l'I.H.P. Analyse non linéaire

Representations of Compact Lie Groups and Elliptic Operators.

Ernst Heintze, Jochen Brüning (1978/1979)

Inventiones mathematicae

Riesz transforms associated with the Hodge Laplacian in Lipschitz subdomains of Riemannian manifolds

Steve Hofmann, Marius Mitrea, Sylvie Monniaux (2011)

Annales de l’institut Fourier

We prove $L^{p}$ -bounds for the Riesz transforms associated to the Hodge-Laplacian equipped with absolute and relative boundary conditions in a Lipschitz subdomain of a (smooth) Riemannian manifold for $p$ in a certain interval depending on the Lipschitz character of the domain.