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 to the special Lagrangian equation, and our infinite dimensional GIT problem is mirror to Thomas' GIT picture for special Lagrangians. This gives rise to infinite dimensional manifold $\mathcal{H}$ closely related to Solomon's space of positive Lagrangians. In the hypercritical phase case we prove the existence of smooth approximate geodesics, and weak geodesics with $C^{1,\alpha}$ regularity. This is accomplished by proving sharp with respect to scale estimates for the Lagrangian phase operator on collapsing manifolds with boundary. As an application of our techniques we give a simplified proof of Chen's theorem on the existence of $C^{1,\alpha}$ geodesics in the space of K\"ahler metrics. In two follow up papers, these results will be used to examine algebraic obstructions to the existence of solutions to dHYM and special Lagrangians in Landau-Ginzburg models.

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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 \[ \sum_{i=1}^{n} \arctan(\la_i)= h(x) \] where $\la_i$ 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 related by Mirror Symmetry to the existence of special Lagrangian submanifolds of the mirror. We introduce a notion of subsolution for this equation, and prove a priori $C^{2,\beta}$ estimates when $|h|>(n-2)\frac{\pi}{2}$ and a subsolution exists. Using the method of continuity we show that the dHYM equation admits a smooth solution in the supercritical phase case, whenever a subsolution exists. Finally, we discover some stability-type cohomological obstructions to the existence of solutions to the dHYM equation and we conjecture that when these obstructions vanish the dHYM equation admits a solution. We confirm this conjecture for complex surfaces.

We identify a strong stability condition on minimal submanifolds that implies uniqueness and dynamical stability properties. In particular, we prove a uniqueness theorem and a dynamical stability theorem of the mean curvature flow for minimal submanifolds that satisfy this condition. The latter theorem states that the mean curvature flow of any other submanifold in a neighborhood of such a minimal submanifold exists for all time, and converges exponentially to the minimal one. This extends our previous uniqueness and stability theorem which applies only to calibrated submanifolds of special holonomy ambient manifolds.

What is the shape of a uniformly massive object that generates a gravitational potential equivalent to that of two equal point-masses? If the weight of each point-mass is sufficiently small compared to the distance between the points then the answer is a pair of balls of equal radius, one centered at each of the two points, but otherwise it is a certain domain of revolution about the axis passing through the two points. The existence and uniqueness of such a domain is known, but an explicit parameterization is known only in the plane where the region is referred to as a Neumann oval. We construct a four-dimensional “Neumann ovaloid”, solving explicitly this inverse potential problem.