Huy The Nguyen
Mathematics Institute
Zeeman Building
University of Warwick
Coventry CV4 7AL UK
Telephone: +44 (0)24 7615 0774
E-mail:
I am currently a postdoctoral fellow at the University of Warwick. I completed my phd at the Australian National University under the supervision of Dr Ben Andrews. My research interests are geometric analysis and differential geometry. My research focusses on geometric flows, particularly the Ricci flow, mean curvature flow and the Willmore flow, as well as conformal immersions of surfaces.
Papers
Isotropic Curvature and the Ricci flow, International Mathematical Research Notices, 2010, no. 3, pp 536-558
Abstract:
We prove that non-negative isotropic curvature is preserved by the Ricci flow in dimensions greater than or equal to four. In order do do so we introduce a new technique to prove that curvature functions defined on the orthonormal frame bundle are preserved by the Ricci flow. At a minimum of such a function, we compute the first and second derivatives in the frame bundle. Using an algebraic construction we can use these expressions to show that the nonlinearity is positive at a minimum. Finally using the maximum principle, we can show that the Ricci flow preserves the cone of curvature operators with non-negative isotropic curvature.
Four-manifolds with $1/4$-pinched flag curvatures, Asian Journal of Mathematics, 13, (2009), no. 2, pp 251-270 (joint with Ben Andrews)
Abstract:
The Ricci flow on a compact four-manifold preserves the condition of pointwise $1/4$-pinching of flag curvatures. Any compact Riemannian four-manifold with $1/4$-pinched flag curvatures is either isometric to ${\mathbb C}{\mathbb P}^2$ or diffeomorphic to a space-form, $ \mathbb{S}^{4} \backslash \Gamma $.
Geometric Rigidity for Analytic Estimates of Müller-Šverák . Also published online on Mathematische Zeitschrift
Abstract:
In a paper by Müller-Šverák, "On Surfaces of finite total curvature", JDG, (42) 1995, conformally immersed surfaces with finite total curvature were studied. In particular it was shown that surfaces with total curvature $\int_{\Sigma} |A|^2< 8 \pi$ in dimension three were embedded and conformal to the plane with one end. Here, using techniques from a a recent paper of Kuwert-Li, " $W^{2,2}$-conformal immersions of a closed Riemann surface into $ \mathbb{R}^n$" , we will show that if the total curvature $ \int_{\Sigma}|A|^2\leq8\pi$, then we are either embedded and conformal to the plane, isometric to a catenoid or isometric to Enneper's minimal surface. In fact the technique of our proof shows that if we are conformal to the plane, then if $n\geq 3 $ and $ \int_{\Sigma} | A|^{2} \leq 16 \pi $ then $\Sigma$ is embedded or $\Sigma$ is the image of a generalized catenoid inverted at a point on the catenoid. In order to prove these theorems, we prove a Gauss-Bonnet theorem for surfaces with complete ends and isolated finite area singularities which extends a theorem of Jorge-Meeks. Using this theorem, we then prove an inversion formula for the Willmore energy.
Preprints
Convexity and Cylindrical Estimates for Mean Curvature Flow in the Sphere
Abstract:
We study mean curvature flow in the sphere with the quadratic curvature condition $ |A|^{2} \leq \frac{ 1}{n-2} H^{2} + 4 K$ which is related but different to two-convexity for submanifolds of the sphere. We classify type $I$ singularities with no further hypotheses. If $H> 0$ then we extend the Huisken-Sinestrari convexity estimates to this situation and show that we classify type $II$ singularities. We prove cylindrical estimates for the mean curvature flow and a pointwise gradient estimate.
Branched Willmore Spheres (joint with Tobias Lamm)
Abstract :
In this paper we classify branched Willmore spheres with at most three branch points (including multiplicity), showing that they may be obtained from complete minimal surfaces in $\mathbb{R} ^ 3$ with ends of multiplicity at most three. This extends the classification result of Bryant. We then show that this may be applied to the analysis of singularities of the Willmore flow of non-Willmore spheres with Willmore energy $ \mathcal {W} ( f ) \leq 16\pi$.
Geometric Rigidity for Sequences of $ W^{2,2}$ Conformal Immersions
Abstract :
We analyse sequences of disks conformally immersed in $ \mathbb{R}^ n$ with energy $ \int _{ D} |A_k |^ 2 \leq \gamma_n$, where $ \gamma_n = 8\pi $ if $ n=3$ and $ \gamma_n = 4 \pi$ when $n\geq4$. We show that if such sequences do not weakly converge to a conformal immersion, then by rescaling we obtain a complete minimal surface with bounded total curvature, either Enneper's minimal surface or Chen's minimal graph.
In the papers by Kuwert-Li and Rivière it was shown that if a sequence of immersed tori diverges in moduli space then $\liminf_ {k\rightarrow \infty} \mathcal{W}( f_k )\geq 8\pi$. We apply the above analysis to show that in $ \mathbb{R}^3$ if the sequence diverges so that $ \lim _{ k \rightarrow \infty} \mathcal {W}(f_k) =8\pi$ then there exists a sequence of Möbius transforms $ \sigma_{k}$ such that $ \sigma_k\circ f _k$ converges weakly to a catenoid.
Orthogonal Bisectional Curvature and the Generalised Frankel Conjecture
Abstract :
In a paper by Siu-Yau it was shown that a Kahler manifold with strictly positive bisectional curvature was biholomorphic to $ \mathbb {CP }^ m$. In this paper, we use the harmonic map techniques developed by Siu-Yau, to prove that a compact Kahler manifold with positive orthogonal bisectional curvature is biholomorphic to $ \mathbb{CP}^m $, a condition strictly weaker than positive bisectional curvature. This gives a direct elliptic proof of this theorem, which was proved by Chen by applying the K\"ahler Ricci flow and the Siu-Yau theorem.