相关论文: Flows of $G_2$-Structures associated to Calabi-Yau…

In this article we study the K\"ahler Ricci flow, the corresponding parabolic Monge Amp\`{e}re equation and complete non-compact K\"ahler Ricci flat manifolds. In our main result Theorem \ref{mainthm} we prove that if $(M, g)$ is…

We study flows of $G_2$-structures guided by the principle of dimensional reduction: natural geometric flows in $G_2$-geometry reduce to natural flows in complex geometry. Our main examples are the $G_2$-Laplacian coflow, which lifts the…

We explore three versions of the Laplacian coflow of $G_2$-structures on circle fibrations over Calabi--Yau 3-folds, interpreting their dimensional reductions to the K\"ahler geometry of the base. Precisely, we reduce Ans\"atze for the…

We consider $G_{2}$-structures on $7$-manifolds that are warped products of an interval and a six-manifold, which is either a Calabi-Yau manifold, or a nearly K\"{a}hler manifold. We show that in these cases the $G_{2}$-structures are…

We consider the Laplacian "co-flow" of $G_2$-structures: $\frac{d}{dt} \psi = - \Delta_d \psi$ where $\psi$ is the dual 4-form of a $G_2$-structure $\phi$ and $\Delta_d$ is the Hodge Laplacian on forms. This flow preserves the condition of…

We investigate how to obtain various flows of K\"ahler metrics on a fixed manifold as variations of K\"ahler reductions of a metric satisfying a given static equation on a higher dimensional manifold. We identify static equations that…

We study the existence of left invariant closed $G_2$-structures defining a Ricci soliton metric on simply connected nonabelian nilpotent Lie groups. For each one of these $G_2$-structures, we show long time existence and uniqueness of…

In this paper, we introduce a new parabolic equation on K\"ahler manifolds. The static point of this flow is related to the existence of a lower bound of the Mabuchi energy. In this paper, we prove the flow always exists for all times for…

We study the Laplacian flow and coflow on contact Calabi-Yau $7$-manifolds. We show that the natural initial condition leads to an ancient solution of the Laplacian flow with a finite time Type I singularity which is not a soliton, whereas…

We formulate an extension of the Calabi conjecture to the setting of generalized K\"ahler geometry. We show a transgression formula for the Bismut Ricci curvature in this setting, which requires a new local Goto/Kodaira-Spencer deformation…

Following the recent development by Guo-Phong-Tong and Chen-Cheng, we derived the $L^{\infty}$ estimate for K\"ahler-Ricci flows under a weaker assumption. The technique also extends to more general cases coming from different geometric…

We prove the hypersymplectic flow of simple type on standard torus $\mathbb{T}^4$ exists for all time and converges to the standard flat structure modulo diffeomorphisms. This result in particular gives the first example of a…

A notion of asymptotically conical K\"ahler orbifold is introduced, and, following previous existence results in the setting of asymptotically conical manifolds, it is shown that a certain complex Monge-Amp\'ere equation admits a rapidly…

We develop a powerful new analytic method to construct complete non-compact G2-manifolds, i.e. Riemannian 7-manifolds (M,g) whose holonomy group is the compact exceptional Lie group G2. Our construction starts with a complete non-compact…

A complex Monge-Amp\`ere equation for differential $(p,p)$-forms is introduced on compact K\"ahler manifolds. For any $1 \leq p < n$, we show the existence of smooth solutions unique up to adding constants. For $p=1$, this corresponds to…

This article consists of some loosely related remarks about the geometry of G_2-structures on 7-manifolds and is partly based on old unpublished joint work with two other people: F. Reese Harvey and Steven Altschuler. Much of this work has…

In this paper, we study the global K\"ahler-Ricci flow on a complete non-compact K\"ahler manifold. We prove the following result. Assume that $(M,g_0)$ is a complete non-compact K\"ahler manifold such that there is a potential function $f$…

Geometric flows have proved to be a powerful geometric analysis tool, perhaps most notably in the study of 3-manifold topology, the differentiable sphere theorem, Hermitian-Yang-Mills connections and canonical Kaehler metrics. In the…

We find explicit solutions and singularities of the Ricci-harmonic flow of $\mathrm{G_2}$-structures, the Ricci-like flows of $\mathrm{G_2}$-structures studied by Gianniotis-Zacharopoulos in arXiv:2505.06872 (J. Geom. Anal. 36.2 (2026)) and…

We introduce a geometric flow of conformally coclosed $G_2$-structures, whose fixed points are large volume solutions of the heterotic $G_2$ system, with vanishing scalar torsion class $\tau_0 = 0$. After conformal rescaling, it becomes a…