A special Lagrangian type equation for holomorphic line bundles
Adam Jacob,Shing-Tung Yau +1 more
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In this paper, the deformed Hermitian-Yang-Mills equation on a holomorphic line bundle over a compact Kahler manifold X is studied, and it is shown that this equation is the Euler-Lagrange equation for a positive functional and that solutions are unique global minimizers.Abstract:
Let L be a holomorphic line bundle over a compact Kahler manifold X. Motivated by mirror symmetry, we study the deformed Hermitian–Yang–Mills equation on L, which is the line bundle analogue of the special Lagrangian equation in the case that X is Calabi–Yau. We show that this equation is the Euler-Lagrange equation for a positive functional, and that solutions are unique global minimizers. We provide a necessary and sufficient criterion for existence in the case that X is a Kahler surface. For the higher dimensional cases, we introduce a line bundle version of the Lagrangian mean curvature flow, and prove convergence when L is ample and X has non-negative orthogonal bisectional curvature.read more
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(1,1) forms with specified Lagrangian phase: A priori estimates and algebraic obstructions
TL;DR: In this article, the authors studied the problem of specifying the Lagrangian phase of a Kahler manifold with respect to its mirror, and proved that the Hermitian-Yang-Mills (dHYM) equation admits a smooth solution in the supercritical phase case, whenever a subsolution exists.
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Moment maps, nonlinear PDE, and stability in mirror symmetry
TL;DR: In this article, the deformed Hermitian-Yang-Mills (dHYM) equation is studied from the variational point of view via an infinite dimensional GIT problem mirror to Thomas' GIT picture for special Lagrangians.
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The deformed Hermitian-Yang-Mills equation in geometry and physics
TL;DR: The Hermitian-Yang-Mills equation (HMM) is a fully nonlinear geometric PDE on Kahler manifolds which plays an important role in mirror symmetry.
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Concavity of the Lagrangian phase operator and applications
TL;DR: In this article, the authors studied the Dirichlet problem for the Lagrangian phase operator in both the real and complex setting, and they showed that there exists a solution with right-hand side h(x) satisfying (n-2) √ σ 2 σ σ ≥ (n − 2) ∞ and boundary data if and only if there is a subsolution.
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A class of curvature type equations
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Journal ArticleDOI
On The Ricci Curvature of a Compact Kahler Manifold and the Complex Monge-Ampere Equation, I*
TL;DR: In this paper, the Ricci form of some Kahler metric is shown to be closed and its cohomology class must represent the first Chern class of M. This conjecture of Calabi can be reduced to a problem in non-linear partial differential equation.
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Mirror symmetry is T duality
TL;DR: In this paper, it was argued that every Calabi-Yau manifold X with a mirror Y admits a family of supersymmetric toroidal 3-cycles and that the moduli space of such cycles together with their flat connections is precisely the space Y.
Book ChapterDOI
Homological Algebra of Mirror Symmetry
TL;DR: Mirror symmetry was discovered several years ago in string theory as a duality between families of 3-dimensional Calabi-Yau manifolds (more precisely, complex algebraic manifolds possessing holomorphic volume elements without zeros).
Journal ArticleDOI
Asymptotic-behavior for singularities of the mean-curvature flow
TL;DR: In this paper, the authors study the singularities of (1) which can occur for nonconvex initial data and characterize the asymptotic behavior of the hypersurface Mt near a singularity using rescaling techniques.