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Contact toric manifolds
, 2002
"... We complete the classification of compact connected contact toric manifolds initiated by Banyaga and Molino and by Galicki and Boyer. As an application we prove the conjectures of Toth and Zelditch on toric integrable systems on the ntorus and the 2sphere. ..."
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Cited by 71 (6 self)
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We complete the classification of compact connected contact toric manifolds initiated by Banyaga and Molino and by Galicki and Boyer. As an application we prove the conjectures of Toth and Zelditch on toric integrable systems on the ntorus and the 2sphere.
Riemannian manifolds with uniformly bounded eigenfunctions
 Duke Math. J
, 2000
"... The standard eigenfunctions φλ = e i〈λ,x 〉 on flat tori R n /L have L ∞norms bounded independently of the eigenvalue. In the case of irrational flat tori, it follows that L 2normalized eigenfunctions have uniformly bounded L ∞norms. Similar bases exist on other flat manifolds. Does this property ..."
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Cited by 31 (6 self)
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The standard eigenfunctions φλ = e i〈λ,x 〉 on flat tori R n /L have L ∞norms bounded independently of the eigenvalue. In the case of irrational flat tori, it follows that L 2normalized eigenfunctions have uniformly bounded L ∞norms. Similar bases exist on other flat manifolds. Does this property characterize flat manifolds? We give an affirmative answer for compact Riemannian manifolds with quantum completely integrable Laplacians. This paper is concerned with the relation between the dynamics of the geodesic flow G t on the unit sphere bundle S ∗ M of a compact Riemannian manifold (M, g) and the growth rate of the L ∞norms of its L 2normalized �eigenfunctions (or “modes”) {φλ}. Let Vλ: = {φ: �φλ = λφλ} denote the λeigenspace for λ ∈ Sp(�), and define
TORIC GEOMETRY OF CONVEX QUADRILATERALS
, 2009
"... We provide an explicit resolution of the Abreu equation on convex labeled quadrilaterals. This confirms a conjecture of Donaldson in this particular case and implies a complete classification of the explicit toric Kähler– Einstein and toric Sasaki–Einstein metrics constructed in [6, 23, 14]. As a ..."
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Cited by 8 (3 self)
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We provide an explicit resolution of the Abreu equation on convex labeled quadrilaterals. This confirms a conjecture of Donaldson in this particular case and implies a complete classification of the explicit toric Kähler– Einstein and toric Sasaki–Einstein metrics constructed in [6, 23, 14]. As a byproduct, we obtain a wealth of extremal toric (complex) orbisurfaces, including Kähler–Einstein ones, and show that for a toric orbisurface with 4 fixed points of the torus action, the vanishing of the Futaki invariant is a necessary and sufficient condition for the existence of Kähler metric with constant scalar curvature. Our results also provide explicit examples of relative K–unstable toric orbisurfaces that do not admit extremal metrics.
Existence and nonuniqueness of constant scalar curvature toric Sasaki metrics
 Compos. Math
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Sasakian geometry: the recent work Of Krzysztof Galicki
, 2008
"... This is a mainly expository article honoring my recently deceased friend and collaborator Krzysztof Galicki who died after a tragic hiking accident. I give a review of our recent work in Sasakian geometry. A few new results are also presented. ..."
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Cited by 4 (1 self)
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This is a mainly expository article honoring my recently deceased friend and collaborator Krzysztof Galicki who died after a tragic hiking accident. I give a review of our recent work in Sasakian geometry. A few new results are also presented.
TORIC INTEGRABLE GEODESIC FLOWS
, 2001
"... Abstract. By studying completely integrable torus actions on contact manifolds we prove a conjecture of Toth and Zelditch that toric integrable geodesic flows on tori must have flat metrics. 1. ..."
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Cited by 1 (1 self)
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Abstract. By studying completely integrable torus actions on contact manifolds we prove a conjecture of Toth and Zelditch that toric integrable geodesic flows on tori must have flat metrics. 1.