Related papers: The Chern-Ricci flow on complex surfaces
We study the limiting behavior of the Kahler-Ricci flow on $\mathbb{P}(\mathcal{O}_{\mathbb{P}^n} \oplus \mathcal{O}_{\mathbb{P}^n}(-1)^{\oplus (m+1)})$, assuming the initial metric satisfies the Calabi symmetry. We show that the flow…
This book gives an introduction to fundamental aspects of generalized Riemannian, complex, and K\"ahler geometry. This leads to an extension of the classical Einstein-Hilbert action, which yields natural extensions of Einstein and…
The existence of K\"ahler-Einstein metrics on a compact K\"ahler manifold has been the subject of intensive study over the last few decades, following Yau's solution to Calabi's conjecture. The Ricci flow, introduced by Richard Hamilton has…
We use the momentum construction of Calabi to study the conical K\"ahler-Ricci flow on Hirzebruch surfaces with cone angle along the exceptional curve, and show that either the flow Gromov-Hausdorff converges to the Riemann sphere or a…
The Ricci flow is an evolution system on metrics. For a given metric as initial data, its local existence and uniqueness on compact manifolds was first established by Hamilton \cite{Ha1}. Later on, De Turck \cite{De} gave a simplified…
In this paper, the author has considered the hyperbolic Kahler-Ricci flow introduced by Kong and Liu [11], that is, the hyperbolic version of the famous Kahler-Ricci flow. The author has explained the derivation of the equation and…
We develop a theory of Ricci flow for metrics on Courant algebroids which unifies and extends the analytic theory of various geometric flows, yielding a general tool for constructing solutions to supergravity equations. We prove short time…
In this paper we reconcile several different approaches to Ricci flow through singularities that have been proposed over the last few years by Kleiner-Lott, Haslhofer-Naber and Bamler. Specifically, we prove that every noncollapsed limit of…
We extend some results known for the K\"ahler-Ricci flow to the Chern-Ricci flow regarding the independence of singularity types for long-time solutions. Specifically, we show that if a solution to the Chern-Ricci flow exists with uniformly…
We study the Hermitian curvature flow of locally homogeneous non-K\"ahler metrics on compact complex surfaces. In particular, we characterize the long-time behavior of the solutions to the flow. We also provide the first example of a…
We study the generalized K\"ahler-Ricci flow on complex surfaces with nondegenerate Poisson structure, proving long time existence and convergence of the flow to a weak hyperK\"ahler structure.
We study the Ricci flow on complete Kaehler metrics that live on the complement of a divisor in a compact complex manifold. In earlier work, we considered finite-volume metrics which, at spatial infinity, are transversely hyperbolic. In…
Identifying any conformally round metric on the $2$-sphere with a unique cross section on the standard lightcone in the $3+1$-Minkowski spacetime, we gain a new perspective on $2d$-Ricci flow on topological spheres. It turns out that in…
In this note we study finite-time singularities in the Chern-Ricci flow. We show that finite-time singularities are characterized by the blow-up of the scalar curvature of the Chern connection.
We investigate Riemannian (non-Kahler) Ricci flow solutions that develop finite-time Type-I singularities and present evidence in favor of a conjecture that parabolic rescalings at the singularities converge to singularity models that are…
We establish the existence of K\"ahler-Ricci flow on pseudoconvex domains with general initial metric without curvature bounds. Moreover we prove that this flow is simultaneously complete, and its normalized version converge to the complete…
In this paper we investigate a kind of generalized Ricci flow which possesses a gradient form. We study the monotonicity of the given function under the generalized Ricci flow and prove that the related system of partial differential…
In this paper we study the evolution of almost non-negatively curved (possibly singular) three dimensional metric spaces by Ricci flow. The non-negatively curved metric spaces which we consider arise as limits of smooth Riemannian manifolds…
We show that any non-collapsed finite time singularity of the Ricci flow on a compact K\"ahler surface is of Type I. Combined with a previous result of the first author, Cifarelli, and Deruelle, it follows that any such singularity is…
In this paper we survey the recent developments of the Ricci flows on complete noncompact K\"{a}hler manifolds and their applications in geometry.