Explorations

This post aims to give an overview of a paper I recently uploaded to arxiv.

M. L. Michelsohn in the 80s introduced the notion of a balanced metric on a complex manifold. The definition can be simply written as a Hermitian metric $\omega$ that satisfies $d \omega^{n-1} = 0 $. These metrics are more abundant on complex manifolds as compared to Kähler metrics. Hence these might be more relevant in studying complex geometry outside of the Kähler world. The significance of non-Kähler geometry is a different topic and will not be touched upon here.

My motivation for working on this topic was to search for extensions of well-studied structures on Kähler manifolds to the balanced case. One such example is the set of volume forms introduced by Donaldson on a Riemannian manifold that is parametrized by the set of smooth functions $\{\phi \in C^{\infty}(M): 1+ \Delta \phi > 0\}$. This can be seen as an infinite dimensional manifold with the tangent space being the set of all smooth functions on $M$. The significance of this space comes in the context of the “Nahm’s equation” for studying a free boundary problem in physics. It can be given a Riemannian structure,

$$||\psi||^2 = \int_M \psi^2 (1 + \Delta \phi) dV$$

The equation of geodesic with respect to this Riemannian metric is given by $\phi_{tt}(1 + \Delta \phi) – |\nabla \phi_t|^2 = 0$. This was subsequently solved by Chen-He in 2008.

In the current paper we consider the set of all volume forms $\{\phi\in C^{\infty}(M):\omega_{\phi}^{p} \wedge \omega^{n-p} > 0 \}$ on a balanced manifold with an $L^2$ metric. The geodesic equation becomes the following in this case.

$$\phi_{tt} (n + nX\phi + \Delta \phi) – |\nabla \phi|^2 = -\frac{nX\phi_t^2}{2},$$

where $X \omega^{n} = \partial \bar{\partial} \omega^{p-1} \wedge \omega^{n-p}$. Here we need the balanced condition in the variational calculations. So this does not extend easily to all Hermitian manifolds.

A crucial condition for the PDE to be degenerate elliptic is to assume that the function $X \leq 0$ so that the RHS is non-negative. Our first main result of this paper is to show that this equation admits a weak $C^{1,1}$ solution for smooth boundary data. It is widely known that such equations cannot admit $C^2$ solutions. So this is the optimal outcome for this equation. But it is also left open to treat this equation in the case when $X \geq 0$. This would give a degenerate hyperbolic PDE that cannot be approached with the elliptic methods in this paper.