The study shows conditions for Kähler manifolds to have rational cohomology of complex projective space.
problem Conditions for Kähler manifolds to have rational cohomology of complex projective space.
method Analyzing the Calabi curvature operator and its positivity conditions.
result Compact Kähler manifolds with specific curvature conditions have rational cohomology of complex projective space.
Extends Calabi operator to Riemannian locally symmetric spaces.
problem Local integrability conditions on Riemannian locally symmetric spaces.
method Generalizes Calabi operator to Riemannian locally symmetric spaces.
result Generalised operator works in irreducible case and fails in products.
Identifies Lorentzian locally symmetric spaces where Calabi operator suffices to determine Killing operator range.
problem Determining when the Calabi operator can identify the range of the Killing operator in Lorentzian locally symmetric spaces.
method Developed criteria for a connection to be in the range of a connection, applied to the Killing connection.
result For indecomposable spaces, the Calabi operator suffices to identify the range of the Killing operator; for products, it fails.
Linearized Einstein equations simplified via Calabi operator.
problem Linearizing Einstein equations for cosmological applications.
method Using the Calabi operator from projective differential geometry.
result Linearized Einstein equations simplified.
Solves Calabi-Yau equation on complex manifolds with non-positive astheno-Ricci curvature.
problem Solving the form type Calabi-Yau equation on complex manifolds.
method Defined astheno-Ricci curvature and proved existence of solution under non-positive curvature condition.
result Existence of solution for Calabi-Yau equation with non-positive astheno-Ricci curvature.
Calabi flow extended with bounded curvature integrals.
problem Extending Calabi flow on compact Kähler manifolds.
method Using bounded Lp scalar curvature integrals. result Calabi flow can be extended under certain curvature conditions.
Algorithm computes eigenvalues and eigenforms on Calabi-Yau threefolds.
problem Computing spectra of Laplace-de Rham operator on Calabi-Yau manifolds.
method Numerical algorithm using approximate Calabi-Yau metric.
result Computed eigenvalues and eigenforms for the Fermat quintic threefold.
Calabi flow works well with bounded curvature on compact manifolds.
problem Stability of extremal Kähler metrics under the Calabi flow.
method Extending the Calabi flow with Lp scalar curvature bounds. result Calabi flow converges exponentially to extremal Kähler metrics.
The paper solves curvature problems on graphs using a special flow.
problem Solving curvature problems on finite graphs.
method Defined the Calabi flow for a specific curvature type and established its global existence and convergence.
result The solution to the Calabi flow exists globally and converges under certain conditions.
Revisits zero modes of Dirac operator on Eguchi-Hanson space.
problem Determining zero modes of the Dirac operator on Eguchi-Hanson space.
method Uses spin-c spinors and formalism of differential forms to simplify calculations. result Reproduces known normalisable zero modes of the twisted Eguchi-Hanson Dirac operator.
Study on geometrically formal metrics on complex manifolds.
problem Existence and properties of geometrically formal metrics on complex manifolds.
method Topological and cohomological obstructions, detailed analysis for specific manifolds, and metric constructions.
result Existence and non-existence conditions for geometrically formal metrics on various complex manifolds.
The paper studies the twisted Calabi flow on Kähler manifolds.
problem Analyzing the behavior of the twisted Calabi flow on compact Kähler manifolds.
method Establishing convexity, proving short-time existence, and demonstrating stability of the flow.
result The stability of the twisted Calabi flow near twisted constant scalar curvature Kähler metrics.
Proves Calabi-Yau theorem for certain nonnegative curvature manifolds.
problem Proving a Calabi-Yau type theorem for specific manifolds.
method Existence result for bounded regions with weakly mean-concave boundary.
result Proves contractibility of certain manifolds with positive scalar curvature.
New invariant extends curvature estimates to noncompact manifolds.
problem Estimating scalar curvature in noncompact manifolds.
method Introducing vertical \(\widehat{A}\)-cowaist and using deformed Dirac operators.
result Sharp inequality relating scalar curvature, spectrum, and new invariant.
Symplectic forms can be preserved under small deformations on Calabi-Yau manifolds.
problem Preserving symplectic forms under deformations on Calabi-Yau manifolds.
method Dynamical stability of symplectic curvature flow.
result Any small symplectic deformation of a Kähler form remains Kähler on a compact Calabi-Yau manifold.
New complete hypersurfaces found in affine geometry.
problem Finding new complete hypersurfaces in affine geometry.
method Classifying special Calabi hypersurfaces and solving equations.
result Found a class of new Euclidean and Calabi complete affine hypersurfaces.
Study of spacelike hypersurfaces in twisted product spacetimes with specific conditions.
problem Characterizing and understanding spacelike hypersurfaces in twisted product spacetimes.
method Analysis of conditions ensuring global hyperbolicity, use of conformal geometry results.
result Non-existence results for constant mean curvature hypersurfaces under certain conditions.
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.
Study finds Calabi-Yau models' operator spectra match random matrix theory.
problem Understanding spectra of Calabi-Yau sigma models.
method Numerical methods for Ricci-flat metrics, averaging over complex structure moduli space.
result Spectrum matches Gaussian orthogonal ensemble of random matrix theory.
We study the Rarita-Schwinger operator on compact Riemannian spin manifolds. In particular, we find examples of compact Einstein manifolds with positive scalar curvature where the Rarita-Schwinger operator has a non-trivial kernel. For positive quaternion Kähler manifolds and symmetric spaces with spin structure we giv…
In this paper, we construct simply connected symplectic Calabi-Yau 6-manifolds by applying Gompf's symplectic fiber sum operation along T4. Using our construction, we also produce symplectic non-Kähler Calabi-Yau 6-manifolds with fundamental group Z. In this paper, we also produce the first examples of simply con…
The paper proves properties of Kähler surfaces with zero scalar curvature.
problem Characterizing Kähler surfaces with zero scalar curvature.
method Analyzing families of generalized Taub-Nut Kähler surfaces and Burn's metric.
result Proves that certain Kähler surfaces are QCH and of specific types.
Paper introduces new flows to find circle packings with specific curvature.
problem Finding circle packings with prescribed total geodesic curvatures.
method Introduces combinatorial Calabi flow, fractional combinatorial Calabi flow, and combinatorial p-th Calabi flow.
result Establishes conditions for the longtime behaviors of these flows.
Assuming local uniform bounds on the metric for a solution of the Chern-Ricci flow, we establish local Calabi and curvature estimates using the maximum principle.
New method finds ideal circle patterns on spheres.
problem Finding ideal circle patterns on spheres with prescribed curvatures.
method Combinatorial Calabi flow in spherical geometry.
result Existence and convergence of the flow for ideal circle patterns.
In this paper we prove the existence and uniqueness of the form-type Calabi-Yau equation on Kähler manifolds of nonnegative orthogonal bisectional curvature.
The paper describes a specific type of Kähler surfaces.
problem Understanding QCH Kähler surfaces with quasi-constant holomorphic sectional curvature.
method Detailed description of generalized Calabi type QCH Kähler surfaces.
result Detailed description of QCH Kähler surfaces of generalized Calabi type.
Suppose there is a constant scalar curvature metric on a compact Kahler manifold without holomorphic vector field. We prove that the Calabi flow, if it is assumed to exist for all time with bounded Ricci curvature, will converge to the constant scalar curvature metric.
Study Bismut connection curvatures and solve Yamabe and Calabi-Yau problems.
problem Yamabe problem and Calabi-Yau with torsion metrics for Bismut connection.
method Analysis of Bismut scalar and Ricci curvatures, construction of examples.
result Existence of metrics with constant Bismut scalar curvature.
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 …
The paper studies the combinatorial p-th Calabi flow for finite and infinite circle patterns.
problem Establishing convergence and long-time existence of the combinatorial p-th Calabi flow.
method Combinatorial p-th Calabi flow for finite and infinite ideal circle patterns.
result Sharp criterion for convergence in finite case and long-time existence in infinite case for p≥2. The paper connects Bergman-Calabi diastasis to Kähler metrics with constant holomorphic sectional curvature.
problem Characterizing domains with Bergman metrics of constant holomorphic sectional curvature.
method Using the Bergman-Calabi diastasis and its connection to the Bergman representative coordinate, the paper derives explicit formulas and proves properties of these metrics.
result Domains with Bergman metrics of constant holomorphic sectional curvature are hyperconvex or exhaustively biholomorphic to a ball.
The curvature properties of a specific type of 6-manifold are explored.
problem Curvature identities on almost Calabi-Yau 6-manifolds with torsion.
method Analysis of the Nijenhuis tensor, torsion connection, and Ricci solitons.
result The curvature of the torsion connection is symmetric and vanishes if the norm of the torsion or scalar curvature is constant.
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…
Complete Calabi-Yau metrics made on special 3D spaces.
problem Creating complete Calabi-Yau metrics on complex 3D spaces.
method Used gluing construction and perturbation argument.
result Produced complete Calabi-Yau metrics with unbounded curvature.
In this paper, we give an expression and some estimates of the curvature tensor of the Hodge metric over the moduli space of a polarized Calabi-Yau threefold. The symmetricity of the Yukawa coupling is also studied. In the last section of this paper, an extra restriction of the limiting Hodge structure for the degenera…
In this note, we study the long time existence of the Calabi flow on X=Cn/Zn+iZn. 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…
This is a survey of our work on spacelike graphic submanifolds in pseudo-Riemannian products, namely on Heinz-Chern and Bernstein-Calabi results and on the mean curvature flow, with applications to the homotopy of maps between Riemannian manifolds.
In this paper we generalize examples of Hamiltonian stationary Lagrangian submanifolds constructed by Lee and Wang in Cm to toric almost Calabi-Yau manifolds. We construct examples of weighted Hamiltonian stationary Lagrangian submanifolds in toric almost Calabi-Yau manifolds and solutions of generalized La…
Study shows continuity of non-Kähler Calabi-Yau conifold transitions.
problem Understanding the geometry of Calabi-Yau conifold transitions.
method Use of balanced and Hermitian-Yang-Mills metrics to analyze conifold transitions.
result The conifold transition is continuous in the Gromov-Hausdorff topology.
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…
Inspired by the recent work of Physicists Hertog-Horowitz-Maeda, we prove two stability results for compact Riemannian manifolds with nonzero parallel spinors. Our first result says that Ricci flat metrics which also admits nonzero parallel spinors are stable (in the direction of changes in conformal structures) as the…
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 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.
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 ∫∂Pu dσ<C1, then there exists a constant C2 depending only on C1 and P such that maxPu<C2. As an application, we sh…
Study of twisted Calabi flow connecting J-flow and Calabi flow on Kähler manifolds.
problem Existence and convergence of twisted Calabi flow on compact Kähler manifolds.
method Analysis of a family of twisted Calabi flows connecting J-flow and Calabi flow, showing long-time existence and convergence to cscK metrics.
result Long-time existence and convergence of twisted Calabi flow to cscK metrics, implying openness of continuity method.
Solves generalized Kähler Calabi-Yau problem on compact manifolds.
problem Calabi conjecture in generalized Kähler geometry.
method New local deformation result, Bismut Ricci curvature transgression formula, generalized Kähler-Ricci flow.
result Global existence and convergence of flow for initial data in generalized Kähler class of Kähler Calabi-Yau structure.
The paper finds circle packings with specific curvatures in hyperbolic geometry.
problem Finding circle packings with prescribed total geodesic curvatures and discrete Gaussian curvatures.
method Established existence and rigidity via variational principle, introduced combinatorial p-th Calabi flows.
result Introduced combinatorial p-th Calabi flows to find circle packings with prescribed curvatures.