UC Davis

In this dissertation we study the deformed Hermitian-Yang-Mills equation, an equation that can be derived via mirror symmetry as the mirror of the special Lagrangian graph equation. In particular, we are interested in how certain notions of stability associated with the geometric setup relate to existence of a solution. We restrict our study to a certain class of manifolds with large symmetry, with special emphasis on the blowup of complex projective space. Using symmetry, we can rewrite the deformed Hermitian-Yang-Mills equation as an exact ODE with boundary values. This allows us to accomplish two things. First, it allows for a more simple setup in which one can compute the special Lagrangrian angle associated to the equation, and second it allows the stability condition we consider to be expressed in a simple combinatorial matter. With these two observations, we demonstrate that our stability condition forces the boundary values into a configuration where one can then solve the ODE, and thus the deformed Hermitian-Yang-Mills equation.