# Albert Chau

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## Graduate Student Supervision

##### Doctoral Student Supervision

Dissertations completed in 2010 or later are listed below. Please note that there is a 6-12 month delay to add the latest dissertations.

We first study the general theory of Kähler-Ricci flow on non-compact complexmanifolds. By using a parabolic Schwarz lemma and a local scalar curvature estimate,we prove a general existence theorem for Kähler metrics lying in the C^\infty_\loc closure of complete bounded curvature Kähler metrics that are uniformly equivalentto a fixed background metric. In particular we do not assume any curvature bounds. Next, we compare the maximal existence time of two complete bounded curvature solutions by using the equivalence of the initial metrics and using this, we also estimate the maximal existence time of a complete bounded curvature solution in terms of the curvature bound of a background metric. We also prove a uniqueness theorem for Kähler-Ricci flow which slightly improves the result of[13] in the Kähler case.We apply the above results to study the Kähler-Ricci flow on some specific non-compact complex manifolds. We first study the Kähler-Ricci flow on C^n. By applying our general existence theorem and existence time estimate, we show that any complete non-negatively curved U(n)-invariant Kähler metric admits a longtime U(n)-invariant solution to the Kähler-Ricci flow, and the solution converges to the standard Euclidean metric after rescaling.Then we study the Kähler-Ricci flow on a quasi-projective manifold. By modifying the approximation theorem of [1] and applying a general existence theorem of Lott-Zhang [23], we construct a Kähler-Ricci flow solution starting from certain smooth Kähler metrics. In particular, if the metric is the restriction of a smooth Kähler metric in the ambient space, then the solution instantaneously becomes complete and has cusp singularity at the divisor. We also produce a solution starting from some complete metrics that may not have bounded curvature, and the solution is likewise complete with cusp singularity for positive time. On the other hand, if the initial data has bounded curvature and is asymptotic to the standard cusp model in a certain sense, we find the maximal existence time of the corresponding complete bounded curvature solution to the Kähler-Ricci flow.

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##### Master's Student Supervision

Theses completed in 2010 or later are listed below. Please note that there is a 6-12 month delay to add the latest theses.

We establish three new upper bounds on the Bartnik quasi-local mass of triples (S²,g,H) where S² is a topological two sphere, g is a Riemannian metric on S², and H ≥ 0 is a specified (constant) value for the initial mean curvature. We use the initial data set approach under the additional assumptions of time-symmetry (TS) and the dominant energy condition (DEC) in which one first constructs a collar with initial boundary sphere isometric to (S² g) and then extends to an asymptotically flat (AF) 3-manifold with non-negative scalar curvature (which is the DEC under the TS setting). The first bound extends the main result in [Mantoulidis-Schoen "On the Bartnik mass of apparent horizons"] to include the boundary case. Precisely, we show that any metric g with non-negative first eigenvalue of the operator -Δ_g + K_g appears as an apparent horizon (in the TS/DEC/AF setting) and that its Bartnik mass is precisely the corresponding Hawking mass.The second bound establishes that the Bartnik mass of the triple (S²,g,H) is bounded above by r/2 whenever g has non-negative Gaussian curvature K_g and H > 0. This result was known when K_g is assumed to be strictly positive (see [Miao-Xie "Bartnik mass via vacuum extensions"]) though the methods used there do not apply when min K_g =0.For the last bound, given any metric g with K_g ≥ 0 and any H > 0, we give an explicit constant C (depending only on g and H) such that the Bartnik mass of the triple (S²,g,H) is bounded above by a quantity involving C which approaches the Hawking mass as C → 0, which happens as either H → 0 or as g becomes round. Moreover, C remains bounded if H → ∞ or r² min K_g → 0. This result can be extended to arbitrary metrics (that do not necessarily satisfy K_g ≥ 0) although the resulting bound in this case is only finite if H is sufficiently large depending on g.

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In this article we study the Kähler Ricci flow on a class of ℂℙ¹ bundles over ℂℙⁿ⁻¹ known as Hirzebruch manifolds. These are defined by ℙ(Hⁱ⊕ℂ-1), where H is the canonical line bundle, ℂ is the trivial line bundle, and n,i∈ℕ. We follow the work by Song and Weinkove, who study solutions to the Kähler Ricci flow for a Calabi symmetric Kähler metrics on Hirzebruch manifolds. They were able to show that, depending on the initial Kähler class, the Ricci flow would reach a finite time singularity corresponding to the manifold either shrinking to a point, contracting the zero section to a point, or collapsing the fibres. In this paper, we investigate how the fibres collapse in the latter case with the further assumptions that the singularity is formed at a type I rate, and that the length of a generic vector does not decay too quickly in some sense. In this case we show that the fibres converge to round spheres after blowing up around a singular point on a fibre.

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