arXiv research

The paper introduces combinatorial Calabi flows to find hyperbolic metrics on surfaces with boundary.

For triangulated surfaces, we introduce the combinatorial Calabi flow which is an analogue of smooth Calabi flow. We prove that the solution of combinatorial Calabi flow exists for all time. Moreover, the solution converges if and only if Thurston's circle packing exists. As a consequence, combinatorial Calabi flow pro…

Fractional combinatorial flow improves surface conformal structures.

The paper studies the combinatorial p-th Calabi flow for finite and infinite circle patterns.

For triangulated surfaces and any $p>1$, we introduce the combinatorial $p$-th Calabi flow which precisely equals the combinatorial Calabi flows first introduced in H. Ge's thesis when $p=2$. The difficulties for the generalizations come from the nonlinearity of the $p$-th flow equation when $p\neq 2$. Adopting differe…

For triangulated surfaces locally embedded in the standard hyperbolic space, we introduce combinatorial Calabi flow as the negative gradient flow of combinatorial Calabi energy. We prove that the flow produces solutions which converge to ZCCP-metric (zero curvature circle packing metric) if the initial energy is small …

New method finds ideal circle patterns on spheres.

The paper extends circle pattern flows to hyperbolic and Euclidean geometry.

Study of combinatorial Calabi flow on ideal circle patterns.

Computing uniformization maps for surfaces has been a challenging problem and has many practical applications. In this paper, we provide a theoretically rigorous algorithm to compute such maps via combinatorial Calabi flow for vertex scaling of polyhedral metrics on surfaces, which is an analogue of the combinatorial Y…

Paper introduces new flows to find circle packings with specific curvature.

New method finds metrics on surfaces with prescribed curvatures using circle packings and surgery.

Paper improves estimates for discrete Laplace in hyperbolic geometry.

The paper develops algorithms for finding metrics with prescribed combinatorial curvature on polyhedral surfaces.

The paper studies deformation of discrete conformal structures on surfaces using combinatorial curvature flows.

Study on deforming discrete conformal structures on surfaces with boundaries.

The paper solves fractional combinatorial flows for prescribed hyperbolic bordered surfaces.

The paper finds hyperbolic metrics on surfaces with boundary using combinatorial curvature flows.

The paper studies rigid sphere packings on 3D manifolds with boundary.

The paper establishes a discrete uniformization theorem for surfaces with piecewise hyperbolic metrics.

Proves existence of circle patterns on surfaces with cusps.

The paper finds circle packings with specific curvatures in hyperbolic geometry.

New discrete conformal structures on surfaces with boundary, proving global rigidity and constructing hyperbolic metrics.

Study of twisted Calabi flow connecting J-flow and Calabi flow on Kähler manifolds.

The paper studies the twisted Calabi flow on Kähler manifolds.

We define regularity scales to study the behavior of the Calabi flow. Based on estimates of the regularity scales, we obtain convergence theorems of the Calabi flow on extremal Kahler surfaces, under the assumption of global existence of the Calabi flow solutions. Our results partially confirm Donaldson's conjectural p…

Calabi flow extended with bounded curvature integrals.

Calabi flow works well with bounded curvature on compact manifolds.

The paper improves the regularity and existence of pseudo Calabi flow.

Study on flows of G2-structures on contact Calabi-Yau 7-manifolds.

Let X be a toric surface with Delzant polygon P and u(t) be a solution of the Calabi flow equation on P. Suppose the Calabi flow exists in [0, T). By studying local estimates of the Riemann curvature and the geodesic distance under the Calabi flow, we prove a uniform interior estimate of u(t) for t < T.

In this paper, we obtain several a-priori estimates for the Calabi flow on projective bundles admitting the generalized Calabi constructions.

The modified J-flow with Calabi ansatz shows convergence or blow-up behavior based on topological constants.

The paper solves curvature problems on graphs using a special flow.

In this paper, we continue to study the Calabi flow on complex tori. We develop a new method to obtain an explicit bound of the curvature of the Calabi flow. As an application, we show that when $n=2$, the Calabi flow starting from a weak Kähler metric will become smooth immediately. It implies that in our settings, th…

Flow stabilizes on non-Kähler metrics near Calabi-Yau.

In this note, we study the long time existence of the Calabi flow on $X = \mathbb{C}^n/\mathbb{Z}^n + i\mathbb{Z}^n$. Assuming the uniform bound of the total energy, we establish the non-collapsing property of the Calabi flow by using Donaldson's estimates and Streets' regularity theorem. Next we show that the curvatur…

We prove that on a Kähler manifold admitting an extremal metric $ω$ and for any Kähler potential $\varphi_0$ close to $ω$, the Calabi flow starting at $\varphi_0$ exists for all time and the modified Calabi flow starting at $\varphi_0$ will always be close to $ω$. Furthermore, when the initial data is invariant under t…

We consider the local solution to the Calabi flow for C^αinitial metric. We also prove that the Calabi flow on compact Kaehler surfaces can be extended once the metrics along the flow are bounded in L^\infty sense. This can be viewed as obtaining higher order derivative estimates from second order derivatives for a fou…

Stability of Type IIA flow ensures Kähler properties of Calabi-Yau 3-folds.

Inspired by recent work of S. K. Donaldson on constant scalar curvature metrics on toric complex surfaces, we study obstructions to the extension of the Calabi flow on a polarized toric variety. Under some technical assumptions, we prove that the Calabi flow can be extended for all time.

We recast the Calabi flow in DeGiorgi's language of minimizing movements. We establish the long time existence of minimizing movements for K-energy with arbitrary initial condition. Furthermore we establish some a priori regularity of these solutions, and that sufficiently regular minimizing movements are smooth soluti…

We first proved a compactness theorem of the Kähler metrics, which confirms a prediction of Chen. Then we prove several eigenvalue estimates along the Calabi flow. Combining the compactness theorem and these eigenvalue estimates, we generalize the method developed by Chen-Li-Wang to prove the small energy theorems of t…

We first define Pseudo-Calabi flow, as {equation*} {{aligned}{{\partial \varphi}\over {\partial t}}&= -f(\varphi), \triangle_varphi f(\varphi) &= S(\varphi) - \ul S.{aligned}. \end{equation*} Then we prove the well-posedness of this flow including the short time existence, the regularity of the solution and the continu…

Study collapsing Calabi-Yau metrics and flows on fiber spaces.

In this short note we prove that if the curvature tensor is uniformly bounded along the Calabi flow and the Mabuchi energy is proper, then the flow converges to a constant scalar curvature metric.

Symplectic forms can be preserved under small deformations on Calabi-Yau manifolds.

Suppose $(X,ω)$ is a compact Kähler manifold. We introduce and explore the metric geometry of the $L^{p,q}$-Calabi Finsler structure on the space of Kähler metrics $\mathcal H$. After noticing that the $L^{p,q}$-Calabi and $L^{p'}$-Mabuchi path length topologies on $\mathcal H$ do not typically dominate each other, we …