Moment maps, nonlinear PDE and stability in mirror symmetry. I: Geodesics
type: Item
instance of: scholarly article
title: Moment maps, nonlinear PDE and stability in mirror symmetry. I: Geodesics
zbMATH Open document ID: 1469.58007
author: Tristan C. Collins
author: Shing-Tung Yau
published in: Annals of PDE
publication date: 2021-07-12T00:00:00Z
review text: The present paper is the first in a series regarding the deformed Hermitian-Yang-Mills equation. The goal is to discuss a mirror symmetry related to a geometric duality. Much of the work concerns an associated infinite dimensional symplectic manifold for which the geodesics and approximate geodesics are investigated. The geodesic structure of the space of Kähler metrics is also studied.
Mathematics Subject Classification ID: 58E15
Mathematics Subject Classification ID: 32Q25
Mathematics Subject Classification ID: 58E10
Mathematics Subject Classification ID: 53C22
Mathematics Subject Classification ID: 53D05
zbMATH DE Number: 7371001
zbMATH Keywords: mirror symmetry
zbMATH Keywords: stability
zbMATH Keywords: deformed Hermitian-Yang-Mills
zbMATH Keywords: special Lagrangian
zbMATH Keywords: Calabi-Yau
reviewed by: Dumitru Motreanu
MaRDI profile type: Publication
full work available at URL: s40818-021-00100-7
OpenAlex ID: W3156571785
cites work: Bridgeland-stable moduli spaces for \(K\)-trivial surfaces
cites work: Q3374325
cites work: The Yang-Mills equations over Riemann surfaces
cites work: Polynomial Bridgeland stability conditions and the large volume limit
cites work: Bridgeland stability conditions on threefolds I: Bogomolov-Gieseker type inequalities
cites work: The Dirichlet problem for a complex Monge-Ampère equation
cites work: A Brunn-Minkowski type inequality for Fano manifolds and some uniqueness theorems in Kähler geometry
cites work: Q5265950
cites work: A gradient estimate in the Calabi-Yau theorem
cites work: Uniform \(K\)-stability, Duistermaat-Heckman measures and singularities of pairs
cites work: Uniform K-stability and asymptotics of energy functionals in Kähler geometry
cites work: Stability conditions on triangulated categories
cites work: Q4855688
cites work: The Dirichlet problem for nonlinear second order elliptic equations. III: Functions of the eigenvalues of the Hessian
cites work: The dirichlet problem for nonlinear second-order elliptic equations. II. Complex monge-ampère, and uniformaly elliptic, equations
cites work: The space of Kähler metrics.
cites work: A new parabolic flow in Kähler manifolds
cites work: Kähler-Einstein metrics on Fano manifolds. I: Approximation of metrics with cone singularities
cites work: Kähler-Einstein metrics on Fano manifolds. II: Limits with cone angle less than \boldmath2𝜋
cites work: Kähler-Einstein metrics on Fano manifolds. III: Limits as cone angle approaches \boldmath2𝜋 and completion of the main proof
cites work: On the \(C^{1,1}\) regularity of geodesics in the space of Kähler metrics
cites work: \((1,1)\) forms with specified Lagrangian phase: \textit{a priori} estimates and algebraic obstructions
cites work: Concavity of the Lagrangian phase operator and applications
cites work: The deformed Hermitian-Yang-Mills equation in geometry and physics
cites work: Morse theory and geodesics in the space of Kähler metrics
cites work: Weak geodesics in the space of Kähler metrics
cites work: A minimum principle for Lagrangian graphs
cites work: Tian’s properness conjectures and Finsler geometry of the space of Kähler metrics
cites work: The largest eigenvalue of a convex function, duality, and a theorem of Slodkowski
cites work: Q4463918
cites work: Q4934663
cites work: Anti Self-Dual Yang-Mills Connections Over Complex Algebraic Surfaces and Stable Vector Bundles
cites work: The Ding Functional, Berndtsson Convexity and Moment Maps
cites work: Lower bounds on the Calabi functional
cites work: Q4549213
cites work: The Dirichlet problem for complex Monge-Ampère equations and regularity of the pluri-complex Green function
cites work: Second-order estimates and regularity for fully nonlinear elliptic equations on Riemannian manifolds
cites work: The dirichlet problem for a class of fully nonlinear elliptic equations
cites work: Flat surfaces and stability structures
cites work: Dirichlet duality and the nonlinear Dirichlet problem
cites work: Doubly Stochastic Matrices and the Diagonal of a Rotation Matrix
cites work: Uniqueness results for special Lagrangians and Lagrangian mean curvature flow expanders in \(\mathbb{C}^m\)
cites work: A special Lagrangian type equation for holomorphic line bundles
cites work: Conjectures on Bridgeland stability for Fukaya categories of Calabi-Yau manifolds, special Lagrangians, and Lagrangian mean curvature flow
cites work: Homological Algebra of Mirror Symmetry
cites work: Geodesics in the space of Kähler metrics
cites work: Mirror symmetry without corrections
cites work: Symplectic structures on gauge theory
cites work: From special Lagrangian to Hermitian-Yang-Mills via Fourier-Mukai transform.
cites work: Some symplectic geometry on compact Kähler manifolds. I
cites work: Lectures on Bridgeland Stability
cites work: Nonlinear instantons from supersymmetric \(p\)-branes
cites work: Geometric Invariant Theory
cites work: Finite time singularities for Lagrangian mean curvature flow
cites work: Complex Monge Ampere Equations
cites work: The Dirichlet problem for degenerate complex Monge-Ampère equations

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Moment maps, nonlinear PDE and stability in mirror symmetry. I: Geodesics (DOI)
DOI: 10.1007/S40818-021-00100-7