The Ward equation, also called the modified 2+1 chiral model,is obtained by a dimension reduction and a gauge fixing from the self-dual Yang-Mills field equation on R2,2. It has a Lax pair and is an integrable system. Ward constructed solitons whose ex- tended solutions have distinct simple poles. He also used a limiting method to construct 2-solitons whose extended solutions have a double pole. Ioannidou and Zakrzewski, and Anand constructed more soliton solutions whose extended solutions have a double or triple pole. Some of the main results of this paper are: (i) We construct algebraic B¨acklund transformations (BTs) that generate new solutions of the Ward equation from a given one by an algebraic method. (ii) We use an order k limiting method and algebraic BTs to construct explicit Ward solitons, whose extended solutions have arbitrary poles and multiplicities. (iii) We prove that our construction gives all solitons of the Ward equation explicitly and the entries of Ward solitons must be rational functions in x, y and t. (iv) Since stationary Ward solitons are unitons, our method also gives an explicit construction of all k-unitons from finite sequences of rational maps from C to Cn.

In this paper, we prove a general existence theorem for properly embedded minimal surfaces with free boundary in any compact Riemannian 3‐manifold M with boundary ∂M. These minimal surfaces are either disjoint from ∂M or meet ∂M orthogonally. The main feature of our result is that there is no assumptions on the curvature of M or convexity of ∂M. We prove the boundary regularity of the minimal surfaces at their free boundaries. Furthermore, we define a topological invariant, the filling genus, for compact 3‐manifolds with boundary and show that we can bound the genus of the minimal surface constructed above in terms of the filling genus of the ambient manifold M. Our proof employs a variant of the min‐max construction used by Colding and De Lellis on closed embedded minimal surfaces, which were first developed by Almgren and Pitts.

Let $M^n$ be a complete noncompact K\"ahler manifold with nonnegative bisectional curvature and maximal volume growth, we prove that $M$ is biholomorphic to $\mathbb{C}^n$. This confirms the uniformization conjecture of Yau under the assumption $M$ has maximal volume growth.

In this paper, we study smooth complex projective varieties X such that some exterior power ⋀^rT_X of the tangent bundle is strictly nef. We prove that such varieties are rationally connected. We also classify the following two cases. If T_X is strictly nef, then X isomorphic to the projective space P^n. If ⋀^2T_X is strictly nef and if X has dimension at least 3, then X is either isomorphic to P^n or a quadric Q^n.

We prove pinching estimates for dual flows provided the curvature function used in the inverse flow in de Sitter space is convex.