Abstract. Let L be a holomorphic line bundle over a compact Kähler 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 Kähler 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.

Let (X, α) be a Kähler manifold of dimension n, and let [ω] ∈ H 1,1 (X, R). We study the problem of specifying the Lagrangian phase of ω with respect to α, which is described by the nonlinear elliptic equationwhere λi are the eigenvalues of ω with respect to α. 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,β estimates when |h| > (n − 2) π 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 discuss the isoperimetric problem in planes with density. In particular, we examine planes with generalized curvature zero. We solve the isoperimetric problem on the plane with density e x , as well as on the plane with density r p for p < 0. The Appendix provides a proof by Robert Bryant that the Gauss plane has a unique closed geodesic.2000 Mathematics subject classification: 53C42.

The purpose of this paper is to investigate canonical metrics on a semistable vector bundle E over a compact Kähler manifold X. It is shown that if E is semi-stable, then Donaldson's functional is bounded from below. This implies that E admits an approximate Hermitian-Einstein structure, generalizing a classic result of Kobayashi for projective manifolds to the Kähler case. As an application some basic properties of semi-stable vector bundles over compact Kähler manifolds are established, such as the fact that semi-stability is preserved under certain exterior and symmetric products.

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