Research

Research interests: I am primarily interested in using methods of differential geometry and partial differential equations to address problems in complex analysis, algebraic geometry, and topology. In particular, understanding the curvature and algebro-geometric properties of Kobayashi hyperbolic manifolds is a main focus of my research.

Further details of my research can be found in my Marie Skłodowska-Curie fellowship application (scheduled to commence in September, 2025).

Articles and Preprints

Twisted Kähler--Einstein metrics and collapsing, (arXiv)
Second-Order Estimates for Collapsed Limits of Ricci-flat Kähler Metrics, Canad. Math. Bull., (arXiv)
The Schwarz lemma in Kähler and non-Kähler geometry, AJM, (arXiv)
The Schwarz lemma: an odyssey, RMJ, (arXiv)
On the weighted orthogonal Ricci curvature, with Kai Tang, J. Geom. Phys., (arXiv)
On the altered curvature of Hermitian manifolds, with Kai Tang, (arXiv)
On the nonnegativity of the Dirichlet energy of a weighted graph, Bull. Aust. Math., Soc., (arXiv)
An eigenvalue characterization of the dual EDM cone, Bull. Aust. Math. Soc., (arXiv)
Remarks on the quadratic orthogonal bisectional curvature, J. Geom., (arXiv)
On the Gauduchon curvature of Hermitian manifolds, with James Stanfield, Int. J. Math., (arXiv)
Some remarks on the Wu--Yau theorem, (arXiv)
(epsilon,delta)-quasi-negativity and positivity of the canonical bundle, with Kai Tang, J. Geom. Analysis, (arXiv)
A general Schwarz lemma for Hermitian manifolds, with James Stanfield, (arXiv)
Hermitian metrics with vanishing second Chern Ricci curvature, with Artem Pulemotov, (arXiv)

Kobayashi hyperbolic manifolds

One of the most fundamental notions in mathematics is convexity---a geometrically connectedness property of a space. The antithetical notion is hyperbolicity---a geometrically separatedness property.

For a compact set K in Euclidean space, the familiar notion of convexity may be articulated as any two points of K being contained in the image of some affine map from [0,1] to K. This formulation makes it straightforward to describe a convexity property for complex manifolds.

A complex manifold X is rationally connected if any two points are contained in the image of a holomorphic map from the projective line to X. The corresponding hyperbolicity notion would be that every holomorphic map from the complex line to X must be constant. Manifolds defined by this latter property are said to be Brody hyperbolic. When X is compact, Brody hyperbolicity coincides with the a priori stronger notion of hyperbolicity introduced by Kobayashi.

A long-standing conjecture of Kobayashi (1970) and Lang (1986) predicts that compact Kobayashi hyperbolic Kähler manifolds are projective and admit a Kähler--Einstein metric with negative Ricci curvature. In particular, the canonical bundle of such manifolds is ample.

I am interested in the Kobayashi--Lang conjecture, and in particular, in addressing via methods from differential geometry. In more detail, it is known that a complex manifold with a (not necessarily complete) Hermitian metric of holomorphic curvature bounded above by a negative constant is Kobayashi hyperbolic (Grauert--Reckziegel, Greene--Wu). The Wu--Yau theorem (Heier--Lu--Wong, Wu--Yau, Tosatti--Yang) shows that a compact Kähler manifold with a Kähler metric of negative holomorphic curvature is projective with a Kähler--Einstein metric of negative Ricci curvature.

In collaboration with James Stanfield, we established the most general evidence for the Kobayashi--Lang conjecture, showing that compact Kähler manifolds with a pluriclosed metric of negative holomorphic curvature admit a Kähler--Einstein metric with negative Ricci curvature.

Kähler--Ricci Flow

The Ricci flow has firmly established itself as an effective means of using methods from differential geometry and partial differential equations to address problems in geometry and topology. The most famous example being Perelman's resolution of the Poincaré conjecture.

The Ricci flow is a partial differential equation which deforms a (Riemannian) metric by a heat-type equation. It has been known for a long time that the Ricci flow preserves the Kähler condition in the sense that if the starting metric is Kähler, the metrics will remain Kähler (so long as the metrics remain smooth and non-degenerate). The Ricci flow starting from a Kähler metric is therefore referred to as the Kähler--Ricci flow.

Remarkably, the Kähler--Ricci flow performs canonical geometric surgeries on the manifold, precisely those that appear in birational algebraic geometry (i.e., in the minimal model program). The Kähler--Ricci flow (KRF) is known to exist for all time if and only if the canonical bundle is nef (Tian--Zhang, 2006). Motivated by the Kähler extension of the abundance conjecture, in the study of the long-time solutions of the KRF, it is natural to assume, in addition, that the canonical bundle is semi-ample.

If X is compact Kähler with semi-ample canonical bundle, then X is the total space of a surjective holomorphic map with connected fibers over a normal (irreducible and reduced) projective variety. The smooth fibers of this holomorphic map are Calabi--Yau in the sense that they have holomorphically torsion canonical bundle (this assumes abundance). The KRF, starting from a Kähler metric on X, exhibits volume collapsing and converges to a twisted Kähler--Einstein metric on the base.

A number of important conjectures involve understanding the singular behavior of this twisted Kähler--Einstein metric. For instance, showing that (modulo some logarithmic poles) the metric has conical singularities when pulled back to a log smooth resolution of the base.