Calabi Flow Research Articles

Hermitian Calabi Functional in Complexified Orbits

Let $(M,\omega)$ be a compact symplectic manifold. We denote by $\ac$ the space of all almost complex structure compatible with $\omega$. $\ac$ has a natural foliation structure with the complexified orbit as leaf. We obtain an explicit formula of the Hessian of Hermitian Calabi functional at an extremal almost K\"ahler metric in $\ac$. We prove that the Hessian of Hermitian Calabi functional is semi-positive definite at critical point when restricted to a complexified orbit, as corollaries we obtain some results analogy to K\"ahler case. We also show weak parabolicity of the Hermitian Calabi flow.

Combinatorial Curvature Flows for Generalized Circle Packings on Surfaces with Boundary

Abstract In this paper, we investigate the deformation of generalized circle packings on ideally triangulated surfaces with boundary, which is the $(-1,-1,-1)$-type generalized circle packing metric introduced by Guo and Luo [ 16]. To find hyperbolic metrics on surfaces with totally geodesic boundaries of prescribed lengths, we introduce combinatorial Ricci flow and combinatorial Calabi flow for generalized circle packings on ideally triangulated surfaces with boundary. Then we prove the longtime existence and global convergence for the solutions of these combinatorial curvature flows, which provide effective algorithms for finding hyperbolic metrics on surfaces with totally geodesic boundaries of prescribed lengths.

Combinatorial Calabi flows on surfaces with boundary

Motivated by Luo's combinatorial Yamabe flow on closed surfaces \cite{L1} and Guo's combinatorial Yamabe flow on surfaces with boundary \cite{Guo}, we introduce combinatorial Calabi flow on ideally triangulated surfaces with boundary, aiming at finding hyperbolic metrics on surfaces with totally geodesic boundaries of given lengths. Then we prove the long time existence and global convergence of combinatorial Calabi flow on surfaces with boundary. We further introduce fractional combinatorial Calabi flow on surfaces with boundary, which unifies and generalizes the combinatorial Yamabe flow and the combinatorial Calabi flow on surfaces with boundary. The long time existence and global convergence of fractional combinatorial Calabi flow are also proved. These combinatorial curvature flows provide effective algorithms to construct hyperbolic surfaces with totally geodesic boundaries with prescribed lengths.

A Chern–Calabi Flow on Hermitian Manifolds

We study an analogue of the Calabi flow in the non-Kähler setting for compact Hermitian manifolds with vanishing first Bott–Chern class. We prove a priori estimates for the evolving metric along the flow given a uniform bound on the Chern scalar curvature. If the Chern scalar curvature remains uniformly bounded for all time, we show that the flow converges smoothly to the unique Chern–Ricci-flat metric in the \(\partial {\bar{\partial }}\)-class of the initial metric.

Parameterized discrete uniformization theorems and curvature flows for polyhedral surfaces, II

This paper investigates the combinatorial $\alpha$-curvature for vertex scaling of piecewise hyperbolic metrics on polyhedral surfaces, which is a parameterized generalization of the classical combinatorial curvature. A discrete uniformization theorem for combinatorial $\alpha$-curvature is established, which generalizes Gu-Guo-Luo-Sun-Wu's discrete uniformization theorem for classical combinatorial curvature. We further introduce combinatorial $\alpha$-Yamabe flow and combinatorial $\alpha$-Calabi flow for vertex scaling to find piecewise hyperbolic metrics with prescribed combinatorial $\alpha$-curvatures. To handle the potential singularities along the combinatorial curvature flows, we do surgery along the flows by edge flipping. Using the discrete conformal theory established by Gu-Guo-Luo-Sun-Wu, we prove the longtime existence and convergence of combinatorial $\alpha$-Yamabe flow and combinatorial $\alpha$-Calabi flow with surgery, which provide effective algorithms for finding piecewise hyperbolic metrics with prescribed combinatorial $\alpha$-curvatures.

Combinatorial Calabi flow on 3-manifolds with toroidal boundary

Motivated by Chow-Luo's combinatorial Ricci flow on surfaces and Luo's combinatorial Ricci flow on compact 3-manifolds with boundary, we introduce combinatorial Calabi flow for decorated hyperbolic polyhedral metrics on 3-manifolds with toroidal boundary to find complete hyperbolic metrics. Dual to Casson and Rivin's approach to solve Thurston's gluing equation by maximizing the volume function for angle structures, the combinatorial Calabi flow on 3-manifolds with toroidal boundary works on decorated hyperbolic polyhedral metrics to ensure that there is no shearing around the edges and finds the complete hyperbolic metric by minimizing the combinatorial Calabi energy. Basic properties of combinatorial Calabi flow on 3-manifolds with toroidal boundary are established. The most important ones include that the equilibrium points of the combinatorial Calabi flow correspond to complete hyperbolic metrics on 3-manifolds with toroidal boundary and the local convergence of the combinatorial Calabi flow. We also study the combinatorial Calabi flow on an ideal tetrahedron. It is shown that for any prescribed admissible dihedral angles, the solution of combinatorial Calabi flow on an ideal tetrahedron exists for all time and converges exponentially fast to a complete hyperbolic polyhedral metric with the prescribed dihedral angles.

A combinatorial Yamabe problem on two and three dimensional manifolds

In this paper, we define a new discrete curvature on two and three dimensional triangulated manifolds, which is a modification of the well-known discrete curvature on these manifolds. The new definition is more natural and respects the scaling exactly the same way as Gauss curvature does. Moreover, the new discrete curvature can be used to approximate the Gauss curvature on surfaces. Then we study the corresponding constant curvature problem, which is called the combinatorial Yamabe problem, by the corresponding combinatorial versions of Ricci flow and Calabi flow for surfaces and Yamabe flow for 3-dimensional manifolds. The basic tools are the discrete maximal principle and variational principle.

A scalar Calabi-type flow in Hermitian geometry: short-time existence and stability

We introduce a new geometric flow of Hermitian metrics which evolves an initial metric along the second derivative of the Chern scalar curvature. The flow depends on the choice of a background metric, it always reduces to a scalar equation and preserves some special classes of Hermitian structures, such as balanced and Gauduchon metrics. We show that the flow has always a unique short-time solution and we provide a stability result when the background metric is Ka¨hler with constant scalar curvature (cscK). The main theorem is obtained by proving a general result about stability of parabolic flows on Riemannian manifolds which is interesting in its own right and in particular implies the stability of the classical Calabi flow near cscK metrics.

Metric first reconstruction for interactive curvature-aware modeling

We present a novel curvature-aware modeling technique for interactively designing shapes. The basic principle we used is that an intrinsic metric of a mesh determines the Euclidean edge lengths and the angles, thereby defining the Laplacian matrix. Hence central to our approach is a metric first algorithm that interactively reconstructs a shape for matching the prescribed curvatures. Given Gaussian curvature, the Calabi flow, which implies a conformal deformation, is first used to compute the desired intrinsic metric. We accelerate the convergence of the Calabi flow using a progressive strategy. Then we reconstruct a shape by solving a new unconstrained problem to match the target Euclidean edge lengths defined by the computed metric and the input mean curvature computed by the Laplacian matrix. Since the obtained metric makes the target edge lengths and the Laplacian matrix fixed during the optimization, we can easily apply a local–global solver capable of interactively reconstructing shapes. Based on this reconstruction tool, we develop three curvature-aware modeling operations. A large number of experiments demonstrate the capability and feasibility of our method for interactively modeling complex shapes.

Combinatorial Calabi flow with surgery on surfaces

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 Yamabe flow introduced by Luo (Commun Contemp Math 6(5):765–780, 2004). To handle the singularies along the combinatorial Calabi flow, we do surgery on the flow by flipping. Using the discrete conformal theory established in Gu et al. (J Differ Geom 109(3):431–466, 2018; J Differ Geom 109(2):223–256, 2018), we prove that for any initial Euclidean or hyperbolic polyhedral metric on a closed surface, the combinatorial Calabi flow with surgery exists for all time and converges exponentially fast after finite number of surgeries. The convergence is independent of the combinatorial structure of the initial triangulation on the surface.

Discrete Calabi Flow: A Unified Conformal Parameterization Method

AbstractConformal parameterization for surfaces into various parameter domains is a fundamental task in computer graphics. Prior research on discrete Ricci flow provided us with promising inspirations from methods derived via Riemannian geometry, which is rigorous in theory and effective inpractice. In this paper, we propose a unified conformal parameterization approachfor turning triangle meshes into planar and spherical domains using discrete Calabi flow onpiecewise linear metric. We incorporate edge‐flipping surgery to guarantee convergence as well as other significant improvements including approximate Newton's method, optimal step‐lengths, priority embedding and boundary customizing, which achieve better performance and functionality with robustness and accuracy.

Compactness of Kähler metrics with bounds on Ricci curvature and $${\mathcal {I}}$$ functional

We prove a compactness theorem for Kahler metrics with various bounds on Ricci curvature and the $${\mathcal {I}}$$ functional. We explore applications of our result to the continuity method and the Calabi flow.

Combinatorial p-th Calabi Flows for Discrete Conformal Factors on Surfaces

For triangulated surfaces, we introduce the combinatorial p-th ( $$p>1$$ ) Calabi flow for Euclidean and hyperbolic polyhedral metrics on surfaces which precisely equals the combinatorial Calabi flows for discrete conformal factors first introduced in Ge’s Ph.D. thesis (Combinatorial methods and geometric equations, Peking University, Beijing, 2012) and then followed by Zhu and Xu (Combinatorial Calabi flow with surgery on surfaces, arXiv:1806.02166v1 , 2018) when $$p=2$$ . Adopting different approaches, we show that the solution to the combinatorial p-th Calabi flow for Euclidean (hyperbolic resp.) polyhedral metric exists for all time and converges to a piecewise linear (hyperbolic resp.) polyhedral metric with constant (zero resp.) combinatorial curvature. Our results generalize the work of Ge (2012) and Zhu and Xu (2018) on the combinatorial Calabi flow with surgery from $$p=2$$ to any $$p>1$$ .

The Calabi Flow with Rough Initial Data

Abstract In this paper, we prove that there exists a dimensional constant $\delta> 0$ such that given any background Kähler metric $\omega $, the Calabi flow with initial data $u_0$ satisfying $$\begin{equation*} \partial \bar \partial u_0 \in L^\infty (M)\ \textrm{and}\ (1- \delta )\omega < \omega_{u_0} < (1+\delta )\omega, \end{equation*}$$admits a unique short-time solution, and it becomes smooth immediately, where $\omega _{u_0}: = \omega +\sqrt{-1}\partial \bar \partial u_0$. The existence time depends on initial data $u_0$ and the metric $\omega $. As a corollary, we get that the Calabi flow has short-time existence for any initial data satisfying $$\begin{equation*} \partial \bar \partial u_0 \in C^0(M)\ \textrm{and}\ \omega_{u_0}> 0, \end{equation*}$$which should be interpreted as a “continuous Kähler metric”. A main technical ingredient is a new Schauder-type estimates for biharmonic heat equation on Riemannian manifolds with time-weighted Hölder norms.

Combinatorial p-th Calabi flows on surfaces

For triangulated surfaces and any p>1, we introduce the combinatorial p-th Calabi flows which precisely equal the combinatorial Calabi flows first introduced in H. Ge's thesis [9] (or see H. Ge [13]) when p=2. The difficulties for the generalizations come from the nonlinearity of the p-th flow equation when p≠2. Adopting different approaches, we show that the solution to the combinatorial p-th Calabi flow exists for all time and converges if and only if there exists a circle packing metric of constant (zero resp.) curvature in Euclidean (hyperbolic resp.) background geometry. Our results generalize the work of H. Ge [13], Ge–Xu [19] and Ge–Hua [14] on the combinatorial Calabi flow from p=2 to any p>1.

Kähler–Ricci flow, Kähler–Einstein metric, and K–stability

We prove the existence of Kahler-Einstein metric on a K-stable Fano manifold using the recent compactness result on Kahler-Ricci flows. The key ingredient is an algebro-geometric description of the asymptotic behavior of Kahler-Ricci flow on Fano manifolds. This is in turn based on a general finite dimensional discussion, which is interesting in its own and could potentially apply to other problems. As one application, we relate the asymptotics of the Calabi flow on a polarized Kahler manifold to K-stability assuming bounds on geometry.

Correction to: Regularity Scales and Convergence of the Calabi Flow

The authors would like to acknowledge additional funding. Corrected acknowledgments appear here.

Covariant Galilean versus Carrollian hydrodynamics from relativistic fluids

We provide the set of equations for non-relativistic fluid dynamics on arbitrary, possibly time-dependent spaces, in general coordinates. These equations are fully covariant under either local Galilean or local Carrollian transformations, and are obtained from standard relativistic hydrodynamics in the limit of infinite or vanishing velocity of light. All dissipative phenomena such as friction and heat conduction are included in our description. Part of our work consists in designing the appropriate coordinate frames for relativistic spacetimes, invariant under Galilean or Carrollian diffeomorphisms. The guide for the former is the dynamics of relativistic point particles, and leads to the Zermelo frame. For the latter, the relevant objects are relativistic instantonic space-filling branes in Randers–Papapetrou backgrounds. We apply our results for obtaining the general first-derivative-order Galilean fluid equations, in particular for incompressible fluids (Navier–Stokes equations) and further illustrate our findings with two applications: Galilean fluids in rotating frames or inflating surfaces and Carrollian conformal fluids on two-dimensional time-dependent geometries. The first is useful in atmospheric physics, while the dynamics emerging in the second is governed by the Robinson–Trautman equation, describing a Calabi flow on the surface, and known to appear when solving Einstein’s equations for algebraically special Ricci-flat or Einstein spacetimes.

On combinatorial Calabi flow with hyperbolic circle patterns

In this paper, we prove the long time existence of the combinatorial Calabi flow in hyperbolic background geometry and the convergence of the flow to a smooth hyperbolic surface under Thurston's combinatorial condition in full generality, which improves the result in [14]. This flow provides a natural algorithm to find Thurston's hyperbolic circle patterns.

Conformal mesh parameterization using discrete Calabi flow

In this paper, we introduce discrete Calabi flow to the graphics research community and present a novel conformal mesh parameterization algorithm. Calabi energy has a succinct and explicit format. Its corresponding flow is conformal and convergent under certain conditions. Our method is based on the Calabi energy and Calabi flow with solid theoretical and mathematical base. We demonstrate our approach on dozens of models and compare it with other related flow based methods, such as the well-known Ricci flow and conformal equivalence of triangle meshes (CETM). Our experiments show that the performance of our algorithm is comparably the same with other methods. The discrete Calabi flow in our method provides another perspective on conformal flow and conformal parameterization.