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Let (M , ω) be a compact connected Kähler manifold of complex dimension four and let [ χ ] ∈ H 1 , 1 (M ; R). We confirm the conjecture by Collins–Jacob–Yau [8] of the solvability of the deformed Hermitian–Yang–Mills equation, which is given by the following nonlinear elliptic equation ∑ i arctan ⁡ (λ i) = θ ˆ , where λ i are the eigenvalues of χ with respect to ω and θ ˆ is a topological constant. This conjecture was stated in [8] , wherein they proved that the existence of a supercritical C -subsolution or the existence of a C -subsolution when θ ˆ ∈ [ ((n − 2) + 2 / n) π / 2 , n π / 2) will give the solvability of the deformed Hermitian–Yang–Mills equation. Collins–Jacob–Yau conjectured that their existence theorem can be improved to θ ˆ > (n − 2) π / 2 , where n is the complex dimension of the manifold. In this paper, we confirm their conjecture that when the complex dimension equals four and θ ˆ is close to the supercritical phase π from the right, then the existence of a C -subsolution implies the solvability of the deformed Hermitian–Yang–Mills equation. [ABSTRACT FROM AUTHOR]

In this paper, we prove that any C 1 -regular Hamiltonian stationary Lagrangian submanifold in a symplectic manifold is smooth. More broadly, we develop a regularity theory for a class of fourth order nonlinear elliptic equations with two distributional derivatives. Our fourth order regularity theory originates in the geometrically motivated variational problem for the volume functional, but should have applications beyond. [ABSTRACT FROM AUTHOR]