# $(1,1)$ forms with specified Lagrangian phase: a priori estimates and algebraic obstructions

@article{Collins201511FW, title={\$(1,1)\$ forms with specified Lagrangian phase: a priori estimates and algebraic obstructions}, author={Tristan C. Collins and Adam Jacob and Shing-Tung Yau}, journal={arXiv: Differential Geometry}, year={2015} }

Let $(X,\alpha)$ be a K\"ahler manifold of dimension n, and let $[\omega] \in H^{1,1}(X,\mathbb{R})$. We study the problem of specifying the Lagrangian phase of $\omega$ with respect to $\alpha$, which is described by the nonlinear elliptic equation \[ \sum_{i=1}^{n} \arctan(\lambda_i)= h(x) \] where $\lambda_i$ are the eigenvalues of $\omega$ with respect to $\alpha$. When $h(x)$ is a topological constant, this equation corresponds to the deformed Hermitian-Yang-Mills (dHYM) equation, and is… Expand

#### 53 Citations

The space of almost calibrated $(1,1)$ forms on a compact K\"ahler manifold

- Mathematics
- 2020

The space $\mathcal{H}$ of "almost calibrated" $(1,1)$ forms on a compact Kahler manifold plays an important role in the study of the deformed Hermitian-Yang-Mills equation of mirror symmetry as… Expand

On J-equation

- Mathematics, Physics
- 2019

In this paper, we prove that for any Kahler metrics $\omega_0$ and $\chi$ on $M$, there exists $\omega_\varphi=\omega_0+\sqrt{-1}\partial\bar\partial\varphi>0$ satisfying the J-equation… Expand

Pseudoconvexity for the special Lagrangian potential equation

- Mathematics, Physics
- 2020

The Special Lagrangian Potential Equation for a function $u$ on a domain $\Omega\subset {\bf R}^n$ is given by ${\rm tr}\{\arctan(D^2 \,u) \} = \theta$ for a contant $\theta \in (-n {\pi\over 2}, n… Expand

Moment Maps, Nonlinear PDE and Stability in Mirror Symmetry, I: Geodesics

- Mathematics
- 2018

In this paper, the first in a series, we study the deformed Hermitian–Yang–Mills (dHYM) equation from the variational point of view as an infinite dimensional GIT problem. The dHYM equation is mirror… Expand

A rigid theorem for deformed Hermitian-Yang-Mills equation.

- Mathematics
- 2019

In this paper, we study the deformed Hermitian-Yang-Mills equation on compact K\"ahler manifold with non-negative orthogonal bisectional curvature. We prove that the curvatures of deformed… Expand

Concavity of the Lagrangian phase operator and applications

- Mathematics
- 2016

We study the Dirichlet problem for the Lagrangian phase operator, in both the real and complex setting. Our main result states that if $$\Omega $$Ω is a compact domain in $${\mathbb {R}}^{n}$$Rn or… Expand

Stability of line bundle mean curvature flow

- Mathematics
- 2020

Let $(X,\omega)$ be a compact Kahler manifold of complex dimension $n$ and $(L,h)$ be a holomorphic line bundle over $X$. The line bundle mean curvature flow was introduced in \cite{JY} in order to… Expand

Limit Behavior of Complex Special Lagrangian Equations With Neumann Boundary-Value Conditions

- Mathematics
- 2021

We consider the following complex special Lagrangian equations with Neumann boundary conditions on a domain $\Omega \subset \mathbb{C}^{n} $: $$\begin{align*}& \left\{\begin{array}{ll}… Expand

The deformed Hermitian Yang-Mills equation on three-folds

- Mathematics, Physics
- 2019

We prove an existence result for the deformed Hermitian Yang-Mills equation for the full admissible range of the phase parameter, i.e., $\hat{\theta} \in (\frac{\pi}{2},\frac{3\pi}{2})$, on compact… Expand

The inverse Monge-Ampere flow and applications to Kahler-Einstein metrics

- Mathematics
- 2017

We introduce the inverse Monge-Ampere flow as the gradient flow of the Ding energy functional on the space of Kahler metrics in $2 \pi \lambda c_1(X)$ for $\lambda=\pm 1$. We prove the long-time… Expand

#### References

SHOWING 1-10 OF 63 REFERENCES

The Calabi homomorphism, Lagrangian paths and special Lagrangians

- Mathematics, Physics
- 2013

Let $$\mathcal{O }$$O be an orbit of the group of Hamiltonian symplectomorphisms acting on the space of Lagrangian submanifolds of a symplectic manifold $$(X,\omega ).$$(X,ω). We define a functional… Expand

Numerical characterization of the K\"ahler cone of a compact K\"ahler manifold

- Mathematics
- 2001

The goal of this work is give a precise numerical description of the K\"ahler cone of a compact K\"ahler manifold. Our main result states that the K\"ahler cone depends only on the intersection form… Expand

Stability conditions on triangulated categories

- Mathematics
- 2002

This paper introduces the notion of a stability condition on a triangulated category. The motivation comes from the study of Dirichlet branes in string theory, and especially from M.R. Douglas's… Expand

A priori estimates for a generalised Monge-Amp\`ere PDE on some compact K\"ahler manifolds

- Mathematics, Physics
- 2015

We study a fully nonlinear PDE involving a linear combination of symmetric polynomials of the K\"ahler form on a K\"ahler manifold. A $C^0$ \emph{a priori} estimate is proven in general and a… Expand

A New Parabolic Flow in Kahler Manifolds

- Mathematics
- 2000

This is a follow-up work of my earlier paper [8]. In [8], we study the lower bound of the K energy on the Kahler manifold when the first Chern class is negative. This is an important problem in… Expand

Numerical characterization of the Kahler cone of a compact Kahler manifold

- Mathematics, Physics
- 2001

The goal of this work is to give a precise numerical description of the Kahler cone of a compact Kahler manifold. Our main result states that the Kahler cone depends only on the intersection form of… Expand

The degenerate special Lagrangian equation

- Mathematics
- 2015

Abstract This article introduces the degenerate special Lagrangian equation (DSL) and develops the basic analytic tools to construct and study its solutions. The DSL governs geodesics in the space of… Expand

Stability conditions and the braid group

- Mathematics
- 2002

We find stability conditions [6, 3] on some derived categories of differential graded modules over a graded algebra studied in [12, 10]. This category arises in both derived Fukaya categories and… Expand

On The Ricci Curvature of a Compact Kahler Manifold and the Complex Monge-Ampere Equation, I*

- Mathematics
- 1978

Therefore a necessary condition for a (1,l) form ( G I a ' r r ) I,,, Rlr dz' A d? to be the Ricci form of some Kahler metric is that it must be closed and its cohomology class must represent the… Expand

$$C^{2,\alpha }$$C2,α estimates for nonlinear elliptic equations of twisted type

- Mathematics
- 2015

We prove a priori interior $$C^{2,\alpha }$$C2,α estimates for solutions of fully nonlinear elliptic equations of twisted type. For example, our estimates apply to equations of the type convex +… Expand