Abstract: In their seminal work on the minimal surface system, Lawson and Osserman conjectured that Lipschitz graphs that are critical points of the area functional with respect to outer variations are also critical with respect to domain variations. We will discuss the proof of this conjecture for two-dimensional graphs of arbitrary codimension. This is joint work with J. Hirsch and R. Tione.

Abstract: In joint work with Hao Fang, I introduce and prove the existence of metrics on complex surfaces with split tangent bundle. These metrics are analogous to Calabi-Yau metrics, as they flatten certain holomorphically trivial line bundles adapted to the geometric structure, in this case the splitting. First, we will review the Calabi-Yau theorem in the Kahler setting and some issues with generalizing it to non-Kahler manifolds. Then, I will discuss some machinery — introduced by Streets -- that makes it possible to reduce this problem to the study of a family of non-concave full-nonlinear elliptic PDE. Finally, I will show that these PDE are smoothly solvable and draw some parallels to the twisted Monge-Ampere equation.

Abstract: Intermediate k^th Ricci curvatures are curvature conditions interpolating between Sectional curvature (k=1) and Ricci curvature(k=n-1). In this talk I will give a broad overview of what is and isn't known or expected about spaces admitting such metrics, on both sides of the apparent behavioral breakpoint of k=n/2. As an example, I will sketch the proof of an upcoming result that spaces with positive Ric_2 and some fixed degree of symmetry (say an action by T^10) must satisfy the Hopf conjecture, i.e. have positive Euler characteristic, and that the possible cohomology of the fixed points are very restricted. This is joint work with Lee Kennard and Lawrence Mouillé 

The Chern-Ricci flow is one of several proposed generalisations of the Kahler-Ricci flow in the Hermitian setting. The aim of this talk is twofold. We first outline the behaviour of the Chern-Ricci flow on complex minimal surfaces. Then, motivated by several results on minimal surfaces, we show that the curvature type is independent of the starting metric in its 'class' for long-time solutions.  This demonstrates a curvature type independence result that holds for the Kahler case.  

Since Calabi's original paper, the Calabi Ansatz has been central for constructions in Kähler geometry. Calabi himself used it to construct complete Ricci-flat metrics on the total space of the canonical bundle of a Kähler-Einstein Fano manifold (B, \omega_B), generalizing some well-known examples coming from physics. Over the years, work of Koiso, Cao, Feldman-Ilmanen-Knopf, Futaki-Wang, Chi Li, and others have shown that the Calabi Ansatz can be used to produce complete Kähler-Ricci solitons, important singularity models for the Kähler-Ricci flow, on certain line bundles over (B, \omega_B). In this talk, I'll explain a generalization of these results to the total space of some higher-rank direct-sum vector bundles over (B, \omega_B). In our case the Calabi Ansatz is not suitable, and we instead use the theory of hamiltonian 2-forms, introduced by Apostolov-Calderbank-Gauduchon-Tønneson-Friedman. The construction produces new examples of complete shrinkers, steadies, and Calabi-Yau metrics, as well as recovering some known ones.