The Mabuchi completion of the space of Kähler potentials

Darvas, Tamás

doi:10.1353/ajm.2017.0032

Cited by 82 publications

(131 citation statements)

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“…is indeed the unique bounded/weak geodesic joining u k t 1 and u k t 2 . The next result connects weak geodesics to the rooftop envelopes of Section 3.3: 46]). Suppose u 0 , u 1 ∈ PSH(X, ω) and t → u t is the weak geodesic connecting u 0 , u 1 .…”

Section: The Weak Geodesic Segments Of Psh(x ω)mentioning

confidence: 74%

Geometric pluripotential theory on Kähler manifolds

Darvas¹

2019

Advances in Complex Geometry

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106

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Finite energy pluripotential theory accommodates the variational theory of equations of complex Monge-Ampère type arising in Kähler geometry. Recently it has been discovered that many of the potential spaces involved have a rich metric geometry, effectively turning the variational problems in question into problems of infinite dimensional convex optimization, yielding existence results for solutions of the underlying complex Monge-Ampère equations. The purpose of this survey is to describe these developments from basic principles. Smith, V. Tosatti and the anonymous referees for their suggested corrections, careful remarks, and precisions. Also, I thank the students of MATH868D at the University of Maryland for their intriguing questions and relentless interest throughout the Fall of 2016. Chapter 3 is partly based on work done as a graduate student at Purdue University, and I am indebted to L. Lempert for encouragement and guidance. Chapter 4 surveys to some extent joint work with Y. Rubinstein, and I am grateful for his mentorship over the years.

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Section: The Weak Geodesic Segments Of Psh(x ω)mentioning

confidence: 74%

Geometric pluripotential theory on Kähler manifolds

Darvas¹

2019

Advances in Complex Geometry

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106

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“…Remark 2.7. Condition (12) implies that H(u s , ·) : P → R vanishes on F s and in particular is constant. Then for all y ∈F s , we have…”

Section: 3mentioning

confidence: 99%

“…Fixing an affine invariant volume form dx = dx 1 ∧ · · · ∧ dx n , the labelling n ∈ N(P ) corresponds to a measure σ ∈ M(P ) as defined in §2.1.1. Observe that the Boundary Condition above (i.e condition (ii) namely (12),and (13)) implies that 2…”

Section: 3mentioning

confidence: 99%

A Note on Extremal Toric Almost Kähler Metrics

Legendre

2019

Springer INdAM Series

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An almost Kähler structure is extremal if the Hermitian scalar curvature is a Killing potential [29]. When the almost complex structure is integrable it coincides with extremal Kähler metric in the sense of Calabi [8]. We observe that the existence of an extremal toric almost Kähler structure of involutive type implies uniform K-stability and we point out the existence of a formal solution of the Abreu equation for any angle along the invariant divisor. Applying the recent result of Chen-Cheng [10] and He [27], we conclude that the existence of a compatible extremal toric almost Kähler structure of involutive type on a compact symplectic toric manifold is equivalent to its relative uniform K-stability (in a toric sense). As an application, using [5], we get the existence of an extremal toric Kähler metric in each Kähler class of P(O ⊕ O(k 1 ) ⊕ O(k 2 )).

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“…Proof. The argument is due to Darvas [15,16], see also [19,Theorem 3.10]. We can assume that d(u j , u j+1 ) ≤ 2 −j , j ≥ 1.…”

Section: Using the Definition Of D This Amounts To Showing Thatmentioning

confidence: 99%