Outline of Research 

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:
  • (i) Nilpotent structures:

    For a sequence of collapsing manifolds with uniformly bounded curvatures, if they collapse to a metric space $(X_{\infty}^k,d_{\infty})$ with maximal nilpotent rank $Nilrank(\Gamma_{\epsilon}(p))= n-k$, then the 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 $\mathcal{N}^k\leq Isom(Y^n)$.

    (ii) Regularity:

    In the above context, by [NZ16], the full measure regular set in $X_{\infty}^k$ is a smooth orbifold. In a special case, if $g_j$ are a hyperkähler sequence on a K3 manifold $\mathcal{K}$, then away from finite points, $g_j$ are close to the Gibbons-Hawking metric with nilpotent symmetries. More precise picture is characterized in [HSZ19] (joint with Shouhei Honda and Song Sun).

    (iii) Canonical affine structure:

    It is shown in [HSZ19], we discovered canonical affine structures for hyperkähler limits. If hyperkähler manifolds $(\mathcal{K},g_j)$ are collapsing to $[0,1]$ with $\frac{dvol_{g_j}}{Vol_{g_j}(\mathcal{K})}\to \nu_{\infty}$, then there is a canonical affine coordinate system $\{z\}$ on $[0,1]$ so that the limit volume density $V_{\infty}(z)$ is piecewise linear with non-smoothing loci precisely corresponding to the singularities of $g_j$. This is analogous to large complex structure limits in mirror symmetry theory, where one expects integral affine structure with singularities (see [Gr12]).

    (iv) Limiting metric-measure geometry:

    By [HSZ19], the hyperkähler limit $([0,1], dt^2, \nu_{\infty})$ is a $RCD(0,\frac{4}{3})$ metric-measure space with $Ric\geq 0$ and $\dim\leq \frac{4}{3}$, and the optimal dimension either $1$ or $\frac{4}{3}$. In fact, $\frac{4}{3}$-case is realized in [HSVZ18].

    (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 $g_t$:

    (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.
    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.

    The following is my recent research in this direction:

  • 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.