I am a Research Assistant Professor (postdoc) at Simons Center for Geometry and Physics, Stony Brook University.
I am interested in differential geometry.
I obtained my Ph.D. from the Department of Mathematics at UC Berkeley. I was glad to be advised by Song Sun.
Contact me via
I still have access to my old Berkeley email address mingyang_li@berkeley.edu.
My office is at Simons Center for Geometry and Physics (SCGP) 510.
I am teaching MAT 131 (Calculus I) in Fall 2025.
Research
My previous 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. According to my knowledge, there is no known compact Type III Einstein metric with non-negative scalar curvature so far.
From 2023 to 2024, my 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, 42 pages, arXiv preprint, 2023. Here for the submitted version (which fixes some impreciseness in the arXiv 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 finite quotients. I also classified all Type II gravitational instantons with non-Euclidean volume growth. As a consequence, it answers a conjecture of Aksteiner-Andersson.
Continuing the study of Einstein metrics with special structures, the following work concerns Type I/II Poincare–Einstein manifolds. A central problem in this area is to determine which conformal infinities can be filled in by Poincare–Einstein metrics, with most previous results being perturbative in nature. In joint work with Hongyi Liu, we construct infinite-dimensional families of Poincare–Einstein manifolds. A remarkable feature of our construction is that it is non-perturbative, and the resulting conformal infinities lie far from the classical known examples.
3. Poincare-Einstein 4-manifolds with conformally Kahler geometry, 32 pages, arXiv preprint, 2025. Joint work with Hongyi Liu.
In this paper, we study Poincare–Einstein 4-manifolds whose conformal class contains a Kahler metric. It turns out that such manifolds are either of Type I or Type II in the above terminology, and they always admit a symmetry. With this special structure, the Einstein equation reduces to a Toda-type equation. We formulate a suitable nonlinear Dirichlet boundary value problem and establish existence and uniqueness results, leading to infinite-dimensional families of Poincare–Einstein manifolds whose underlying spaces are complex line bundles over Riemann surfaces of genus greater than zero. The main technical difficulty lies in the degeneracy of the PDE along the inner boundary.
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.
4. On asymptotically flat 4-manifolds, 16 pages, 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.
In the recent year, I have been working on a project relating gravitational instantons and axisymmetric harmonic maps. Previous works mostly focused on constructing gravitational instantons (or proving certain uniqueness) using harmonic maps, but there are some essential difficulties to rule out possible conical singularities. We overcome this difficulty for some situation in the following work, resulting in infinite many new gravitational instantons (most of them are Type III). It will be interesting to see what is the relation between this and integrable systems.
There is the related work in the setting of general relativity [HKWX] for black holes.
5. Gravitational instantons and harmonic maps, 81 pages, arXiv preprint, 2025. Joint work with Song Sun.
It is known from general relativity that axisymmetric stationary black holes can be reduced to axisymmetric harmonic map into the hyperbolic plane H^2. In the Riemannian setting there is the similar reduction for toric Ricci-flat metrics, which in particular was applied to construct gravitational instantons (complete Ricci-flat 4-manifolds with finite energy). It has been a problem to rule out conical singularity for a very long time.
In this work we design suitable underlying topological space and successfully rule out all possible conical singularities, resulting in infinite many new asymptotically flat, simply-connected, gravitational instantons, with arbitrarily large Betti number b2. This relies on a very delicate bubbling analysis for such harmonic maps. The construction is non-perturbative. The asymptotic model at infinity is R^4 quotient by an isometric Z-action. The constructed gravitational instantons particularly recover Kerr and Chen-Teo gravitational instantons, while other gravitational instantons are Type III in our above language.
A byproduct is an interesting PDE classification result for axisymmetric harmonic maps. When an axisymmetric harmonic map into H^2 is tamed with degree less than or equal to two, we proved it must be explicitly given by Gibbons-Hawking ansatz or LeBrun-Tod ansatz.
I am grateful for support from Simons Collaboration on Special Holonomy in Geometry, Analysis, and Physics, and support from IASM, ZJU.