I am currently a fifth year graduate student at the Department of Mathematics at UC Berkeley. I am generally interested in differential geometry. I am glad to be advised by Song Sun.
Contact me via [email protected] or [email protected].
Research
Till now, my works mainly focus on the study of complete non-compact Ricci-flat 4-manifolds with certain special structure. It was a result of Derdzinski that an oriented Einstein 4-manifold must have one of the following three types, based on the number of eigenvalues of the self-dual Weyl curvature W^+:
Type I: W^+ vanishes identically. The Einstein metric is anti-self-dual. In Ricci-flat case this is equivalent to being locally hyperkahler.
Type II: W^+ has repeated eigenvalues. The Einstein metric is (locally) conformally Kahler, but is non-anti-self-dual. In Ricci-flat case this is equivalent to (passing to a double cover) being Hermitian non-Kahler, but conformally Kahler. The conformal Kahler metric actually is extremal Kahler in the sense of Calabi.
Type III: W^+ generically has three distinct eigenvalues. This is the generic case, in which case the Einstein metric has no special structure in this sense.
There are lots of examples for Type I and Type II. For example, hyperkahler metrics are Type I. Kahler-Einstein metrics with non-zero scalar curvature are Type II. The Riemannian version of the famous Kerr metrics from general relativity are Type II. The Chen-LeBrun-Weber metric is Type II. There is no known compact Type III Einstein metric with non-negative scalar curvature.
My previous works focus on the study of complete non-compact Ricci-flat 4-manifolds with a regularity assumption at infinity (namely the L^2 integral of the curvature tensor is finite). A fancier and widely used name for this is gravitational instantons.
This paper studied Type II ALE gravitational instantons. I proved they correspond to a special kind of Bach-flat Kahler orbifolds. Moreover, I showed there does not exist Type II ALE gravitational instantons with stucture groups in SU(2) at infinity. Note that there is a well-known conjecture due to Bando-Kasue-Nakajima that any ALE gravitational instanton is Type I.
2. Classification results for conformally Kahler gravitational instantons, arxiv preprint, 2023. Here for the submitted version.
This paper investigated the asymptotic geometry of Type II Ricci-flat metrics. I proved that any Type II Ricci-flat metric on an end with finite L^2 integral of the curvature tensor is asymptotic to an asymptotic model at infinity. There are five families of asymptotic models: ALE, ALF, AF, skewed special Kasner, ALH*, or their further Z2 quotients. I also classified all Type II gravitational instantons with non-Euclidean volume growth. As a consequence, it answers a conjecture of Aksteiner-Andersson.
The following work is about collapsing geometry of asymptotically flat 4-manifolds, which has a different flavor with the above works. It confirms a conjecture of Petrunin-Tuschmann.
3. On asymptotically flat 4-manifolds, arxiv preprint, 2024. Joint work with Song Sun. Submitted to a special issue in honor of Professor Xiaochun Rong's 70th birthday
It was proved by Petrunin-Tuschmann that for an asymptotically flat 4-manifold, who has simply-connected end and has unique asymptotic cone that is a metric cone, its asymptotic cone can only be R^4, R^3, or the half-plane H. They further conjectured the half-plane cannot be realized as the asymptotic cone. We confirm their conjecture in this paper.
With the previous work by Chen-Li, an interesting corollary is: any Ricci-flat metric on R^4, with faster than quadratic curvature decay, must be either the flat metric, or the Taub-NUT metric.
I am grateful for support from Simons Collaboration on Special Holonomy in Geometry, Analysis, and Physics, and support from IASM, ZJU.