A Hamiltonian Stationary submanifold of complex space is a Lagrangian manifold whose volume is stationary under Hamiltonian variations.  We consider gradient graphs $(x,Du(x))$ for a function $u$.    For a smooth $u$, the Euler-Lagrange equation can be expressed as a fourth order nonlinear equation in $u$ that can be locally linearized (using a change of tangent plane) to the bi-Laplace.  The volume can be defined for lower regularity, however, and computing the Euler-Lagrange equation with less assumed regularity gives a "double divergence" equation of second order quantities.   We show several results.  First, there is a $c_n$ so that if the Hessian $D^2u$ is $c_n$-close to a continuous matrix-valued function, then the potential must be smooth.  Previously, Schoen and Wolfson showed that when the potential was $C^{2,\alpha}$, then the potential $u$ must be smooth.    We are also able to show full regularity when the Hessian is bounded within certain ranges.   This allows us to rule out conical solutions with mild singularities.

This is joint work with Jingyi Chen.

Conference website: https://www.math.ucsd.edu/~scgas/

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We generalize the classical Bochner formula for the heat flow on a manifold M to martingales on path space PM, and develop a formalism to compute evolution equations for martingales on path space. We see that our Bochner formula on PM is related to two sided bounds on Ricci curvature in much the same manner as the classical Bochner formula on M is related to lower bounds on Ricci curvature. This establishes a new link between geometry and stochastic analysis, and provides a crucial new tool for the study of Einstein metrics and Ricci flow in the smooth and non-smooth setting. Joint work with Aaron Naber.

I will discuss some problems arising in the study of toric Kaehler metrics, mostly focusing on studying the invariant spectrum of the Laplacian, explicit constructions of distinguished metrics (Einstein, Ricci soliton, and quasi-Einstein metrics) and connections between these topics. Time permitting, I will also outline numerical approaches to these problems.

I will discuss a Kuranishi-type theorem for deformations of complex structure on ALE Kahler surfaces, which will be used to prove that for any scalar-flat Kahler ALE surface, all small deformations of complex structure also admit scalar-flat Kahler ALE metrics. A local moduli space of scalar-flat Kahler ALE metrics can then be constructed, which is universal up to small diffeomorphisms. I will also discuss a formula for the dimension of the local moduli space in the case of a scalar-flat Kahler ALE surface which deforms to a minimal resolution of an isolated quotient singularity.  This is joint work with Jiyuan Han.