We present three large classes of examples of conformal structures whose Fefferman–Graham ambient metrics can be found explicitly. Our method for constructing these examples rests upon a set of sufficiency conditions under which the Fefferman–Graham equations are assured to reduce to a system of inhomogeneous linear partial differential equations. Our examples include conformal pp-waves and, more importantly, conformal structures that are defined by generic co-rank $3$ distributions in dimensions $5$ and $6$. Our examples illustrate various aspects of the ambient metric construction.

The holonomy algebras of our ambient metrics are studied in detail. In particular, we exhibit a large class of metrics with holonomy equal to the exceptional non-compact Lie group $\operatorname{G}_2$ as well as ambient metrics with holonomy contained in $\operatorname{Spin}(4, 3)$.

In this paper, we discuss the Weyl problem in warped product spaces. We apply the method of continuity and prove the openness of the Weyl problem. A counterexample is constructed to show that the isometric embedding of the sphere with canonical metric is not unique up to an isometry if the ambient warped product space is not a space form. Then, we study the rigidity of the standard sphere if we fixed its geometric center in the ambient space. Finally, we discuss a Shi–Tam type of inequality for the Schwarzschild manifold as an application of our findings.

We study Le Potier’s strange duality conjecture on a rational surface. We focus on the case involving the moduli space of rank $2$ sheaves with trivial first Chern class and second Chern class $2$, and the moduli space of $1$-dimensional sheaves with determinant $L$ and Euler characteristic $0$. We show the conjecture for this case is true under some suitable conditions on $L$, which applies to $L$ ample on any Hirzebruch surface $\Sigma_e := \mathbb{P} (\mathcal{O}_{\mathbb{P}^1} \oplus \mathcal{O}_{\mathbb{P}^1} (e))$ except for $e = 1$. When $e = 1$, our result applies to $L = aG + bF$ with $b \geq a + [a/2]$, where $F$ is the fiber class, $G$ is the section class with $G^2 = -1$ and $[a/2]$ is the integral part of $a/2$.

It is well known that the PDE (Partial Differential Equation) system arising from the infinitesimal Plücker formulas is a particular case of the Toda system of $\mathbf{A}_n$ type. In this paper, we prove an a priori estimate of solutions of the Toda systems associated with the simple Lie algebras $\mathbf{A}_n$, $\mathbf{B}_n$, $\mathbf{C}_n$ and $\mathbf{G}_2$. Previous results in this direction have been done only for the case of Lie algebras of rank two. Our result for $n \geq 3$ is new. The proof of this fundamental theorem is to combine techniques from PDE and the monodromy theory. One of the key steps is to calculate the local mass of a sequence of blowup solutions near each blowup point. At each blowup point, a sequence of bubbling steps (via scaling) are performed, and the local mass of the present step could be computed from the previous step. We find out that this transformation of the local mass of each step is related to the action of an element in the Weyl group of the Lie algebra. The correspondence of the Pohozaev identity and the Weyl group could reduce the complicated calculation of the local mass into a simpler one.