The Mabuchi geometry of finite energy classes

Darvas, Tamás

doi:10.1016/j.aim.2015.08.005

Cited by 139 publications

(292 citation statements)

References 45 publications

(140 reference statements)

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“…On the other hand, there is strong evidence to suggest that the L 2 geometry of Mabuchi-Semmes-Donaldson alluded to above does not have the right compactness properties to allow for a characterization of existence of KE metrics. In order to address this, one needs to introduce more general L p type Finsler metrics on H ω and compute the metric completion of the related path length metric spaces (H ω , d p ) [46,47].…”

Section: Ricω = λωmentioning

confidence: 99%

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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: Ricω = λωmentioning

confidence: 99%

“…Again, throughout this section we will assume the volume normalization condition (3.24) holds. The main result of this section is the Pythagorean formula of [47]:…”

Section: The Pythagorean Formula and Applicationsmentioning

confidence: 99%

Geometric pluripotential theory on Kähler manifolds

Darvas¹

2019

Advances in Complex Geometry

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106

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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%

“…This was done in [20] by establishing a relative L ∞ -estimate which is quite delicate in the Hessian setting due to a lack of integrability of ω-m-subharmonic functions. We overcome this by constructing ω-msubharmonic subextensions via a complete metric in the space E 1 , inspired by [15,17,19].…”

Section: Introductionmentioning

confidence: 99%

Degenerate complex Hessian equations on compact Kahler manifolds

Lu¹,

Nguyen²

2015

Indiana Univ. Math. J.

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“…Recently Darvas [23] provided a deep understanding of the metric completion of the space of Kähler potentials H ! with the Mabuchi metric.…”

Section: Introductionmentioning

confidence: 99%

Constant Scalar Curvature Equation and Regularity of Its Weak Solution

Zeng

2018

Comm Pure Appl Math

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In this paper we study the constant scalar curvature equation (CSCK), a nonlinear fourth‐order elliptic equation, and its weak solutions on Kähler manifolds. We first define the notion of a weak solution of CSCK for an L∞ Kähler metric. The main result is to show that such a weak solution (with uniform L∞ bound) is smooth. As an application, this answers in part a conjecture of Chen regarding the regularity of K‐energy minimizers. The new technical ingredient is a W2, 2 regularity result for the Laplacian equation Δgu=f on Kähler manifolds, where the metric has only L∞ coefficients. It is well‐known that such a W2, 2 regularity (W2, p regularity for any p > 1) fails in general (except for dimension 2) for uniform elliptic equations of the form aij∂ij2u=f for aij ∊ L∞ without certain smallness assumptions on the local oscillation of aij. We observe that the Kähler condition plays an essential role in obtaining a W2, 2 regularity for elliptic equations with only L∞ elliptic coefficients on compact manifolds. © 2017 Wiley Periodicals, Inc.

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The Mabuchi geometry of finite energy classes

Cited by 139 publications

References 45 publications

Geometric pluripotential theory on Kähler manifolds

Geometric pluripotential theory on Kähler manifolds

Degenerate complex Hessian equations on compact Kahler manifolds

Constant Scalar Curvature Equation and Regularity of Its Weak Solution

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