My research focuses on **differential geometry** and **geometric analysis** (partially supported by NSF grant **DMS-1906265**), including Ricci curvature, Calabi-Yau spaces, sectional curvature and conformal invariants

Here is the outline of my current research.**(A) Regularity and structure theory of Ricci curvature **

The $\epsilon$-regularity theorem of collapsing Einstein spaces:
Joint with Aaron Naber, we proved in [NZ16] the first $\epsilon$-regularity theorems for higher dimensional collapsing Einstein spaces. It relates
a priori curvature estimates with Gromov-Hausdorff behavior, which is entirely new in the collapsed context.
Mainly it states that, if a ball $B_2(p)$ is Gromov-Hausdorff close to a $k$-dimensional metric space,
then the group generated by *short loops*
\begin{equation}\Gamma_{\epsilon}(p)=Image[\pi_1(B_{\epsilon}(p))\longrightarrow\pi_1(B_2(p))]\end{equation} is virtually nilpotent with
$Nilrank(\Gamma_{\epsilon}(p))\leq n-k$. More importantly, $Nilrank(\Gamma_{\epsilon}(p))= n-k$ implies gives uniform curvature estimates in $B_1(p)$.
In particular, if $B_2(p)$
is collapsing to a ball in $\mathbb{R}^k$, then an $\epsilon$-regularity holds iff $Nilrank(\Gamma_{\epsilon}(p))= n-k$.
The crucially new nature of this regularity result is that uniform curvature estimates can be reduced to simple topological detections which can be conveniently applied in many different contexts.
Geometric structures of collapsed Ricci limits:
*nilpotency* can be strengthened as follows: $X_{\infty}^k$
is locally isometric to $Y^n/\mathcal{N}^k$ for some metric space $Y^n$ and nilpotent Lie group $\mathcal{N}^k\leq Isom(Y^n)$.
**(B) Collapsing and degenerations in complex geometry **

Nilpotent structures on K3 surfaces:
In [HSVZ18] (joint with Hans-Joachim Hein, Song Sun and Jeff Viaclovsky), we exhibit a family of Ricci-flat Kähler metrics on K3 surfaces which collapse to a closed interval $[0,1]$ with Tian-Yau space $X_{TY}^4$ and Taub-NUT space $\mathbb{C}_{TN}^2$ occurring as bubbles.
There is a singular nilpotent fibration over the interval such that nilpotent fibers change topologies when crossing the singularities.
Degenerations and metric geometry of collapsed Calabi-Yau spaces:
In the joint work [SZ19] with Song Sun,
we make progress on understanding the collapsing behavior of Calabi-Yau metrics on a degenerating family of polarized Calabi-Yau manifolds.
An especially
intriguing phenomenon is that Calabi-Yau metrics may collapse with highly non-algebraic features.
In the case of a family of smooth Calabi-Yau hypersurfaces $(X_t^{2n},g_t)\subset \mathbb{P}^{n+1}$ degenerating into the transversal union of two smooth Fano hypersurfaces in a generic way:
$$f_1(x)\cdot f_2(x) + t\cdot f(x) = 0, \quad t\to 0,\quad x\in \mathbb{P}^{n+1}.$$
In this setting, we are able to precisely characterize the delicate Riemannian and complex geometry of $g_t$:

**Typical examples:**
Type II degeneration of quartic K3 surfaces in $\mathbb{P}^3$
higher dimensional degenerations of quintic Calabi-Yau threefolds in $\mathbb{P}^4$.

**Remark:**
The degeneration theory developed in [HSVZ18] and [SZ19] is related to various * duality phenomena * and *domain walls* in superstring theory. Such relations have been recently
established by in
[CH19a, CH19b]),
New bubbling behaviors of collapsing Calabi-Yau manifolds:
New bubbling phenomena have been discovered in our recent studies.
For instance,
in [HSVZ18],
the Tian-Yau spaces, for the first time, were realized as bubbles of collapsing K3 surfaces.
With technically much more involved, this type of collapsing scenario was extended to higher dimensions in [SZ19]. In another direction, together with Gao Chen and Jeff Viaclovsky, our work in [CVZ19] is to investigate the collapsing hyperkähler metrics on an elliptic K3 surface which are collapsing to a singular metric on $\mathbb{P}^1$.
We managed to understand the singularity behavior around each type of singular fibers (in Kodaira's classification). It is worth mentioning that
we obtained the first degeneration with complete ALG spaces (satisfying $Vol(B_R)\sim R^2$ and $|Rm|\sim R^{-2}$) occurring as bubble limits of collapsing K3 surfaces.
In the generic case of our construction, a deepest bubble is asymptotic to $\mathbb{T}^k$-bundle over an ALE space for $k\in\{0,1,2\}$, which may occur in the meanwhile.
#

Moduli space of K3 manifolds:
We have made progress in understanding the moduli space of K3 surfaces $\mathfrak{M}$. Using Satake's compactification, Odaka-Oshima ([OO18]) have identified the boundary $\partial \mathfrak{M}$ as having $6$ strata.
The degeneration
analysis established in our recent work help understand the structure of $\mathfrak{M}$ near the boundary strata.
For instance, Constructions in [CVZ19] give an open subset in $\mathfrak{M}$ which contains the 36-dimensional boundary stratum. As a comparison, the hyperkähler metrics obtained in [HSVZ18]
constitute an open set in $\mathfrak{M}$ containing a $2$-dimensional boundary stratum.
**(C) Poincaré-Einstein manifolds and conformal invariants **

For a Poincaré-Einstein space $(X^{n+1}, g)$
with a conformal infinity $(M^n,[h])$, a central topic is to explore their connections.
A way in understanding this is to implement *nonlocal analysis* with the Dirichlet-to-Neumann operators $P_{2\gamma}$ with a leading term $(-\Delta)^{\gamma}$
and associated curvatures $Q_{2\gamma}$,
which originates from geometric scattering theory and effectively unifies
conformal invariants of different orders.
For instance, *scalar curvature* and the *Branson's Q curvature* occur as $\gamma=1$ and $\gamma=2$ respectively.
In [Zhang16], we obtain
a sharp estimate for the complexity of Kleinian group structure
which obstructs the existence of metrics $Q_{2\gamma}\geq 0$ of a Poincé-Einstein space.
This result can be viewed as a nonlocal version of Schoen-Yau's fundamental result in the case $\gamma=1$ ([SY88]).
As applications, we also obtained topological rigidity and classification theorems for the manifolds with $Q_{2\gamma}\geq 0$.
fractional curvatures
Recently in [CZ19] (joint with Wenxiong Chen), we obtained several regularity and isometric rigidity theorems for the conformally flat metrics with constant curvature $Q_{2\gamma}$ as $\gamma=\frac{n}{2}$. This requires delicate regularity and geometric analysis
for a new type of nonliearity, which arises from the scattering behavior
in the limiting case.

Here is the outline of my current research.

(i) *Nilpotent structures:*

(ii) * Regularity:*

(iii) *Canonical affine structure:*

(iv) * Limiting metric-measure geometry: *

(a) Letting $diam_{g_t}(X_t)=1$, then the limit space is $[0,1]$ and singularities occur only at rational points. In addition, it is an $RCD(0,\frac{2n}{n+1})$ space with optimal dimension $\frac{2n}{n+1}$.

(b) There is a singular fibration $\mathscr{F}_t:X_t \to [0,1]$ with graded collapsing fibers.(c) Bubble limits can be explicitly claissified. For example, different bubbles such as the Tian-Yau space and the product gravitational instanton $\mathbb{C}_{TN}^2\times \mathbb{C}^{n-2}$ appear in the collapsing sequence.

We also exhibit an effective way to produce both complete and incomplete Calabi-Yau metrics, which is of independent interest.

The following is my recent research in this direction: