arXiv research

New maximal surfaces solve Bernstein problems.

Calabi flow works well with bounded curvature on compact manifolds.

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…

In this paper, we study the behavior of Ricci-flat Kähler metrics on Calabi-Yau manifolds under algebraic geometric surgeries: extremal transitions or flops. We prove a version of Candelas and de la Ossa's conjecture: Ricci-flat Calabi-Yau manifolds related by extremal transitions and flops can be connected by a path c…

Paper extends Calabi's extremal metric existence to compact Kähler manifolds.

In this paper, we discuss a Donaldson's version of the modified $K$-energy associated to the Calabi's extremal metrics on toric manifolds and prove the existence of the weak solution for extremal metrics in the sense of convex functions which minimizes the modified $K$-energy.

We apply a local differential geometric framework from Kähler toric geometry to (re)construct Calabi's extremal Kähler metrics on $\bbC\bbP^n$ blown-up at a point from data on the moment polytope.

Solves a general problem for toric manifolds in Kaehler-Ricci solitons.

In this note we give a characterization of Kaehler metrics which are both Calabi extremal and Kaehler-Ricci solitons in terms of complex Hessians and the Riemann curvature tensor. We apply it to prove that, under the assumption of positivity of the holomorphic sectional curvature, these metrics are Einstein.

New Kähler metrics generalize Calabi's and relate to Fano manifolds.

We define relative Gromov-Witten invariants and establish a general gluing theory of pseudo-holomorphic curves for symplectic cutting and contact surgery. Then, we use our general gluing theory to study the change of GW-invariants of Calabi-Yau 3-folds tranform under flops and extremal transitions. We prove a complete …

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…

Using the short time existence of the Calabi flow, we prove that any extremal Kaehler metric on a product toric variety is a product extremal Kaehler metric.

We study the Calabi functional on a ruled surface over a genus two curve. For polarisations which do not admit an extremal metric we describe the behaviour of a minimising sequence splitting the manifold into pieces. We also show that the Calabi flow starting from a metric with suitable symmetry gives such a minimising…

We prove a theorem which asserts that the Lie algebra of all holomorphic vector fields on a compact Kähler manifold with a perturbed extremal metric has the structure similar to the case of an unperturbed extremal Kähler metric proved by Calabi.

The paper studies conformally Einstein-Maxwell Kähler metrics and automorphism group structure.

We prove the longtime existence and convergence of the Calabi flow on toric Fano surfaces in a large family of Kahler classes where the class has positive extremal Hamiltonian potential and the initial Calabi energy is bounded by some constant. This is an extension of our previous work. We use the toric condition in a …

The paper finds minimal-area metrics on polygons with multiple crossing geodesics.

This paper is a survey of some recent progress on the study of Calabi's extremal Kähler metrics. We first discuss the Yau-Tian-Donaldson conjecture relating the existence of extremal metrics to an algebro-geometric stability notion and we give some example settings where this conjecture has been established. We then tu…

Study on Hermitian Calabi functional in complexified orbits of symplectic manifolds.

Extremal metrics lead to scalar-flat Kähler cones.

Three distinct 3D components found in local extremal Kähler metrics.

Paper explores new Kähler metrics from old, aiming to solve YTD conjecture.

Recently Guillemin gave an explicit combinatorial way of constructing "toric" Kahler metrics on (symplectic) toric varieties, using only data on the moment polytope. In this paper, differential geometric properties of these metrics are investigated using Guillemin's construction. In particular, a nice combinatorial for…

The Einstein-Maxwell equations on a smooth compact 4-manifold are reformulated as a purely Riemannian variational problem analogous to Calabi's variational problem for extremal Kahler metrics. Next, Seiberg-Witten theory is used to show that these two problems are in fact intimately related. Extremal Kahler metrics are…

In this paper, we prove the equivalence of the existence of extremal Kahler metrics and the properness of the modified K energy on projective bundles. Moreover, we discuss the relations of the lower boundedness of the K energy, the infimum of the Calabi energy and the extremal polynomials. In particular, we give an exa…

Counting HCMU sphere components using weighted trees.

Let $(X, P)$ be a toric variety. In this note, we show that the $C^0$-norm of the Calabi flow $\varphi(t)$ on $X$ is uniformly bounded in $[0, T)$ if the Sobolev constant of $\varphi(t)$ is uniformly bounded in $[0, T)$. We also show that if $(X, P)$ is uniform $K$-stable, then the modified Calabi flow converges expone…

Based on recent work of S. K. Donaldson and T. Mabuchi, we prove that any extremal Kaehler metric in the sense of E. Calabi, defined on the product of polarized compact complex projective manifolds is the product of extremal Kaehler metrics on each factor, provided that the integral Futaki invariants of the polarized m…

We derive a formula for the L^2 norm of the scalar curvature of any extremal Kaehler metric on a compact toric manifold, stated purely in terms of the geometry of the corresponding moment polytope. The main interest of this formula pertains to the case of complex dimension 2, where it plays a key role in construction o…

Recently, F. Balacheff proved that the Calabi-Croke sphere made of two flat 1-unit-side equilateral triangles glued along their boundaries is a local extremum for the length of the shortest closed geodesic among the Riemannian spheres with conical singularities of fixed area. We give an alternative proof of this theore…

In this note, we prove that on polarized toric manifolds the relative $K$-stability with respect to Donaldson's toric degenerations is a necessary condition for the existence of Calabi's extremal metrics, and also we show that the modified $K$-energy is proper in the space of $G_0$-invariant Kähler metrics for the case…

Let $X$ be a toric variety and $u$ be a normalized symplectic potential of the corresponding polytope $P$. Suppose that the Riemannian curvature is bounded by 1 and $\int_{\partial P} u ~ d σ< C_1,$ then there exists a constant $C_2$ depending only on $C_1$ and $P$ such that $\max_P u < C_2$. As an application, we sh…

We study Kähler geometry of Bott manifolds and find extremal metrics.

Let $X$ be a toric surface and $u$ be a normalized symplectic potential on the corresponding polygon $P$. Suppose that the Riemannian curvature is bounded by a constant $C_1$ and $\int_{\partial P} u ~ d σ< C_2,$ then there exists a constant $C_3$ depending only on $C_1, C_2$ and $P$ such that the diameter of $X$ is b…

Lecture notes on Kähler geometry in toric varieties.

Newly discovered Eguchi-Hanson metric arises from edge metrics.

We consider the formation of singularities along the Calabi flow with the assumption of the uniform Sobolev constant. In particular, on Kähler surface we show that any "maximal bubble" has to be a scalar flat ALE Kähler metric. In some certain classes on toric Fano surface, the Sobolev constant is a priori bounded alon…

We establish a regularity result for the metric on any 4-dimensional extremal Kähler manifold, and a weak compactness theorem on the space of such metrics. Specifically, the sectional curvature at a point is bounded when the quantity $L^2(|\Riem|)$ in a surrounding ball is sufficiently small compared to the pointwise n…

The paper classifies rotationally symmetric extremal Kähler metrics on complex manifolds.

The study finds conditions for the existence of extremal toric almost Kähler metrics.

In these notes, after an introduction to toric Kahler geometry, we present Calabi's family of U(n)-invariant extremal Kahler metrics in symplectic action-angle coordinates and show that it actually contains, as particular cases, many interesting cohomogeneity one examples of constant scalar curvature.

Equivalence found between certain Kahler and Sasaki metrics.

In this thesis we study the relationship between the existence of canonical metrics on a complex manifold and stability in the sense of geometric invariant theory. We introduce a modification of K-stability of a polarised variety which we conjecture to be equivalent to the existence of an extremal metric in the polaris…

The paper explores complex geometries of 3-forms on symplectic 6-manifolds.

The Hitchin-Kobayashi correspondence for vector bundles, established by Donaldson, Kobayashi, Luebke, Uhlenbeck and Yau, states that an indecomposable holomorphic vector bundle over a compact Kaehler manifold is stable in the sense of Takemoto-Mumford if and only if the vector bundle admits a Hermitian-Einstein metric.…

Classifies automorphisms of conformally Kähler, Einstein-Maxwell metrics.

We generalise a theorem of Engman and Abreu--Freitas on the first invariant eigenvalue of non-negatively curved $S^{1}$-invariant metrics on $\mathbb{CP}^{1}$ to general toric Kähler metrics with non-negative scalar curvature. In particular, a simple upper bound of the first non-zero invariant eigenvalue for such metri…