Characterizes and proves isometries in Kähler potential spaces.
problem Understanding isometries in Kähler potential spaces.
method Characterization and proof of local isometries.
result Existence and uniqueness of isometries proved.
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48 results for “Kahler potential”
We consider the geodesic equation for the generalized Kahler potential with only mixed second derivatives bounded. We show that given such two generalized Kahler potentials, there is a unique geodesic segment such that for each point on the geodesic, the generalized Kahler potential has uniformly bounded mixed second d…
Study magnetic geodesics on Kähler potentials using variational methods.
problem Understanding magnetic geodesics on Kähler potentials.
method Variational method for a generalized Landau-Hall functional.
result Magnetic geodesic equation and its relation to a perturbed complex Monge-Ampère equation.
The unit ball is characterized by a Kähler-Einstein potential.
problem Characterizing the unit ball in complex geometry.
method Using a global potential function of the Kähler-Einstein metric.
result A compact Kähler manifold with an ample canonical bundle is the unit ball if it has a specific potential function.
Develops potential theory for WZW equation in Kähler potentials space.
problem Solving the Wess--Zumino--Witten equation in Kähler potentials.
method Introduces -harmonicity on graphs to characterize the WZW equation and uses subharmonic distance.
result Shows solvability of Dirichlet problem and approximation by finite-dimensional maps.
Upper bounds for Bergman kernels from smooth Kähler potentials.
problem Bounding Bergman kernels from smooth Kähler potentials.
method Using Taylor coefficients of the Kähler potential, we give upper bounds for Bergman kernels of tensor powers of a smooth positive line bundle.
result Improved off-diagonal rate of decay for analytic, quasi-analytic, and Gevrey potentials.
It is shown that geodesics in the space of Kähler potentials can be uniformly approximated by geodesics in the spaces of Bergman metrics. Two important tools in the proof are the Tian-Yau-Zelditch approximation theorem for Kähler potentials and the pluripotential theory of Bedford-Taylor, suitably adapted to Kähler man…
Continuity of Kähler-Einstein potentials at singularities proven.
problem Regularity of solutions to degenerate complex Monge-Ampère equations on singular spaces.
method Investigation of Dirichlet problem and global continuity of solutions.
result Kähler-Einstein potentials are continuous at isolated singularities.
Note on concavity of Perelman's W-functional near Kähler-Ricci solitons.
problem Concavity of Perelman's W-functional over Kähler potentials.
method Observation and proof based on previous work on Kähler-Ricci solitons.
result Direct consequence of variational stability problem for Kähler-Ricci solitons.
Continuity of solutions to complex Monge-Ampère equations on compact Kähler spaces proved.
problem Continuity of solutions to complex Monge-Ampère equations on compact Kähler spaces.
method Analyzing bounded solutions on reduced, locally irreducible compact Kähler spaces.
result Proves continuity of solutions, affirming conjectures and solving open problems.
Researchers found LCK manifolds can have any rank from 1 to b1(M).
problem Understanding the rank of LCK manifolds with potential.
method Analyzing quotient spaces of Kahler manifolds with positive potentials and character multiplication.
result LCK manifolds with potential can have any rank between 1 and b1(M).
New method constructs potential functions for Kähler-Einstein metrics.
problem Constructing potential functions for Kähler-Einstein metrics on pseudoconvex domains.
method Method of potential scaling.
result Existence of 1-parameter family of automorphisms for certain pseudoconvex domains.
Defines new extremal potentials and measures for Kähler forms.
problem No specific problem stated; dealing with Kähler forms and measures.
method Introduces new extremal potentials and measures for collections of Kähler forms.
result New extremal potentials and measures coincide with classical ones when the collection is a singleton.
Quantizes geodesics in Kähler and Sasaki geometry.
problem Quantize geodesics in Kähler and Sasaki spaces.
method Classical Fubini-Study map and quantization procedure.
result Proves conditions for geodesics in Kähler potentials.
Method finds approximate Ricci-flat metrics on Calabi-Yau manifolds.
problem Finding analytic Kähler potentials for Calabi-Yau manifolds.
method Numerically calculating Ricci-flat Kähler potentials via machine learning and fitting to Donaldson's Ansatz.
result Simple analytic expressions for approximately Ricci-flat Kähler potentials are found, including explicit dependence on complex structure parameter.
The paper finds geodesics in Kähler potentials with no degeneration.
problem Finding geodesics in Kähler potentials without degeneration.
method Establishing a lower bound estimate for eigenvalues of complex Hessian.
result Geodesics can connect close points in Kähler potentials without degeneration.
New equation approximates Kähler potentials using Hermitian-Yang-Mills metrics.
problem Approximating Kähler potentials in complex domains.
method New Wess-Zumino-Witten type equation and Berndtsson's theorem on direct image bundles.
result Approximation of Kähler potentials by Hermitian-Yang-Mills metrics.
We prove that locally any hyper-Kähler metric with torsion admits an HKT potential.
Study Kähler-Einstein potentials on stable varieties near singularities
problem Asymptotic behavior of Kähler-Einstein potentials on stable varieties near singularities
method Using iterated logarithmic functions and refined lower bounds
result Improved estimates for Kähler-Einstein potentials
A unique Kähler potential on the unit ball is identified with constant differential norm.
problem Finding a unique Kähler potential with constant differential norm on the unit ball.
method Analyzing the Kähler potential of the unit ball and its biholomorphic equivalence to the Siegel domain.
result The Kähler potential of the Siegel domain is unique up to automorphisms with constant differential norms.
New proof shows compact homogeneous LCK manifolds are Vaisman.
problem Proving compact homogeneous LCK manifolds are Vaisman.
method Using homogeneous LCK manifolds with potential and a new metric construction.
result Compact homogeneous LCK manifolds are Vaisman.
Entire self-shrinking solutions on complex plane are rigid and come from quadratic potentials.
problem Rigidity of entire self-shrinking solutions to Kähler-Ricci flow.
method Showed that all entire self-shrinking solutions must be generated by quadratic potentials.
result Entire self-shrinking solutions on complex plane are rigid and come from quadratic potentials.
The paper proves the existence of a complete holomorphic vector field on a complex manifold with a Kähler-Einstein metric.
problem Existence of complete holomorphic vector fields on complex manifolds with specific metrics.
method Method of potential scaling to find a potential function with constant length differential, then constructing a vector field from its gradient.
result A complete holomorphic vector field is constructed on a complex manifold with a Kähler-Einstein metric.
Paper proves integrability and entropy compactness for Kähler potentials with uniform log-log threshold.
problem Integrability and entropy compactness for Kähler potentials with specific density.
method Skoda-Zeriahi type integrability theorem and log-log threshold detection.
result Positivity of integrability threshold and entropy compactness for uniform log-log threshold.
Characterizes Kähler-hyperbolicity of bounded symmetric domains based on rank and genus.
problem Understanding the Kähler-hyperbolicity of bounded symmetric domains.
method Defines Kähler-hyperbolicity length by rank and genus, and characterizes it through a special Bergman potential.
result Establishes a unique constant for Kähler-hyperbolicity based on gradient length of a Bergman potential.
Analytic Kähler potentials yield analytic Bergman kernels.
problem Characterizing Bergman kernels for analytic Kähler potentials.
method Linear recursive formula for Bergman kernel coefficients, simplified from Charles's work.
result Bergman kernels are analytic symbols with phase determined by Kähler potential polarization.
Integrable systems and Kähler metrics linked via spectral curves.
problem Connecting integrable systems to Kähler geometry.
method Describing Special Kähler structure on spectral curve base.
result Simple formula for Kähler potential derived.
We classify compact Kähler surfaces with nonconstant Killing potentials such that all integral curves of their gradients are reparametrized geodesics.
Study variational problems in Kähler geometry to construct metrics.
problem Maximizing/minimizing Monge--Ampère energy on Kähler potentials.
method Prove existence and uniqueness of extremals, use them to construct metrics.
result Existence and uniqueness of extremals with simple characterization.
Kähler cones over Sasakian manifolds are flat if projectively induced.
problem Characterizing Kähler cones over Sasakian manifolds.
method Relating Kähler potentials and using Ricci-flatness.
result Kähler cones over regular Sasakian manifolds are flat if projectively induced.
In this note, we shall prove geodesic convexity of the space of Kähler potentials on an ALE Kähler manifold. This extends earlier results in the compact case proved in the fundamental work of X-X. Chen. We further prove the boundedness from below of the Mabuchi energy, and give an alternative proof for the uniqueness o…
Locally conformally Kahler (LCK) manifolds with potential are those which admit a Kahler covering with a proper, automorphic Kaehler potential. Existence of a potential can be characterized cohomologically as a vanishing of a certain cohomology class, called the Bott-Chern class. Compact LCK manifolds with potential ar…
A special Kähler-Ricci potential on a Kähler manifold is any nonconstant function such that is a Killing vector field and, at every point with , all nonzero tangent vectors orthogonal to and are eigenvectors of both and the Ricci tensor. For instan…
Quantizes Kähler metrics using Finsler structures.
problem Approximating Kähler metrics by algebraic metrics.
method Quantization of Finsler structures on Kähler potentials.
result Metric completions of Finsler structures on Kähler potentials are recovered.
Geodesics on Kähler manifold potentials are paths of least action.
problem Understanding geodesics on the space of Kähler potentials.
method Study Lagrangians and geodesics on the Fréchet manifold of Kähler potentials, showing geodesics are paths of least action.
result Geodesics on the space of Kähler potentials are paths of least action, and conversely under suitable conditions.
Solves the generalized Kähler structure problem for symplectic type.
problem Determine the fundamental degrees of freedom in generalized Kähler structures.
method Uses symplectic Morita equivalence and holomorphic Poisson manifolds.
result Generalized Kähler structure of symplectic type is determined by a pair of holomorphic Poisson manifolds, a holomorphic symplectic Morita equivalence, and a generalized Kähler potential.
Geodesics between Kähler potentials are C^{1,1} regular.
problem Regularity of geodesics in Kähler metrics.
method Interior real Hessian bound for complex Monge-Ampere equation.
result Geodesics are C^{1,1} regular.
Classifies holonomy algebras of Lorentz-Kähler manifolds.
problem Classifying holonomy algebras of Lorentz-Kähler manifolds.
method Simple construction of metrics and introduction of complex Walker coordinates.
result Characterization of complex pp-waves and classification of Lorentz-Kähler symmetric spaces.
We show that, locally, all geometric objects of Generalized Kahler Geometry can be derived from a function K, the "generalized Kahler potential''. The metric g and two-form B are determined as nonlinear functions of second derivatives of K. These nonlinearities are shown to arise via a quotient construction from an aux…
Study -minimal Lagrangian submanifolds in Kähler manifolds with real holomorphy potentials.
problem Variational properties of -minimal Lagrangian submanifolds.
method Derive second variation formula and study stability of solitons.
result Stability of expanding and translating solitons for LMCF.
A hyperKähler potential is a function rho that is a Kähler potential for each complex structure compatible with the hyperKähler structure. Nilpotent orbits in a complex simple Lie algebra are known to carry hyperKähler metrics admitting such potentials. In this paper, we explicitly calculate the hyperKähler potential w…
New metrics found on Hirzebruch surfaces solve complex geometry questions.
problem Finding new conformally Kähler, Einstein-Maxwell metrics on Hirzebruch surfaces.
method Used special Killing potentials found by Futaki and Ono.
result Proved existence of new conformally Kähler, Einstein-Maxwell metrics.
We consider the generalized Kahler structures (g,J_+,J_-) that arise on a hyperkahler manifold (M,g,I,J,K) when we choose J_+ and J_- from the twistor space of M. We find a relation between semichiral and arctic superfields which can be used to determine the generalized Kahler potential for hyperkahler manifolds whose …
New theorem on Lee classes for LCK manifolds with potential.
problem Determining Lee classes on LCK manifolds with potential.
method Analyzing cohomology classes of Lee forms and proving the result for Vaisman manifolds.
result The set of Lee classes on LCK manifolds with potential forms an open half-space in .
We extend the Weil-Petersson metric to a projective variety with continuous local potentials.
problem Continuity of the Weil-Petersson potential on moduli spaces of Kähler-Einstein manifolds and varieties.
method Proving the extension of the Weil-Petersson metric as a closed positive current with continuous local potentials.
result The Weil-Petersson metric extends uniquely to the projective variety as a closed positive current with continuous local potentials.
We prove that any shrinking Kahler Ricci soliton has only one end, and that any expanding Kahler Ricci soliton with proper potential has only one end.
Uniform estimates for Calabi-Yau degenerations proved.
problem Calabi-Yau degenerations of polarised algebraic manifolds.
method Uniform Skoda and -estimates for Kähler potentials.
result Uniform Skoda type estimate and -estimate for Calabi-Yau Kähler potentials proved.
Continuity of complex Monge-Ampère potentials on Kähler manifolds.
problem Continuity of solutions to complex Monge-Ampère equations on compact Kähler manifolds.
method Extending DiNezza-Lu's approach to big cohomology classes, proving continuity on Zariski open sets.
result Singular Kähler-Einstein metrics have continuous potentials on the ample locus outside of the non-klt part.