Extends Einstein-Hilbert functional definition for stable manifolds.
problem Stability of Einstein manifolds on Riemannian manifolds.
method Second variation of generalized Einstein-Hilbert functional.
result Properties of stable Einstein manifolds presented.
Defines spectral Einstein functional for manifolds with boundary.
problem Calculating the spectral Einstein functional for manifolds with boundaries.
method Defined spectral Einstein functional associated with the Dirac operator and proved a theorem for 4D manifolds.
result Proof of Kastler-Kalau-Walze type theorem for spectral Einstein functional.
Defines metrics and Einstein tensors on Riemannian manifolds, proving vanishing for non-commutative two-torus.
problem Defining metrics and Einstein tensors on Riemannian manifolds.
method Defines bilinear functionals of vector fields and differential forms, generalizing to non-commutative geometry.
result Proves the vanishing of the Einstein functional for the conformally rescaled geometry of the noncommutative two-torus.
Proves Kastler-Kalau-Walze theorem for spectral Einstein functional on low-dimensional manifolds.
problem Proving Kastler-Kalau-Walze type theorems for spectral Einstein functional.
method Defining spectral Einstein functional associated with Dirac operator and proving theorem for low-dimensional manifolds.
result Proves Kastler-Kalau-Walze type theorem for spectral Einstein functional on low-dimensional manifolds with boundary.
Defines spectral Einstein functionals for sub-Dirac operators on manifolds with boundary.
problem Calculating Einstein-like functionals for sub-Dirac operators.
method Introduced spectral Einstein functional for sub-Dirac operators on manifolds with boundary.
result Proved a theorem for spectral Einstein functions on four-dimensional manifolds.
Computes spectral Einstein functional for Witten deformation on even-dimensional spin manifolds.
problem Calculating the spectral Einstein functional for a specific deformation.
method Computes the spectral Einstein functional for the Witten deformation on even-dimensional spin manifolds.
result Computed the spectral Einstein functional for the Witten deformation on even-dimensional spin manifolds.
The paper introduces spectral Einstein functionals for Dirac operators on manifolds with boundary.
problem Analyzing perturbations of Dirac operators on manifolds with boundary.
method Introduction of spectral Einstein functionals and proof of Dabrowski-Sitarz-Zalecki type theorems.
result Proof of theorems associated with spectral Einstein functionals for perturbations of Dirac operators.
The study of potential functions on noncompact quasi-Einstein manifolds, focusing on dimensions and flatness.
problem Characterizing potential functions on noncompact quasi-Einstein manifolds.
method Analyzing the dimensionality and structure of potential functions on specific manifolds.
result The dimension of positive potential functions on a three-dimensional noncompact quasi-Einstein manifold is at most two, with equality if and only if the manifold is a product of a λ-Einstein surface and R. New bound on partition function proves Kähler-Einstein stability.
problem Proving Kähler-Einstein metrics on complex manifolds.
method Quantitative bound on partition function, connecting probabilistic and quantization approaches.
result Direct analytic proof of Kähler-Einstein stability for uniformly Gibbs stable manifolds.
Weakly Einstein Kähler surfaces are characterized and classified.
problem Characterizing and classifying weakly Einstein Kähler surfaces.
method Several conditions and constructions to characterize and classify weakly Einstein Kähler surfaces.
result Classification of weakly Einstein Kähler surfaces with specific properties and construction of new examples.
The paper proves compactness for Dirac-Einstein spin manifolds.
problem Compactness of Dirac-Einstein spin manifolds under specific conditions.
method Study of the Hilbert-Einstein-Dirac functional, proving compactness for critical points.
result Compactness result for Dirac-Einstein spin manifolds in dimensions three and four.
Paper calculates third coefficient in Kaehler-Einstein metric expansion.
problem Understanding Kaehler-Einstein metrics and their epsilon functions.
method Computes the third coefficient in the TYCZ-expansion of the epsilon function.
result Discovers the vanishing of the third coefficient's significance.
Extends spectral Einstein functionals computation to 4D spin manifolds with boundary.
problem Computing spectral Einstein functionals for 4D spin manifolds with boundary.
method Generalizes Dabrowski's results to 4D spin manifolds with boundary using noncommutative residue.
result Generalized spectral Einstein functionals computation for 4D spin manifolds with boundary.
The paper computes metrics and Einstein tensors on even-dimensional manifolds.
problem Computing metrics and Einstein tensors on even-dimensional Riemannian manifolds.
method Development of spectral Einstein functionals and equivariant Bismut Laplacian.
result Explicit computation of the equivariant noncommutative residue density.
The paper calculates a functional for a specific Dirac operator.
problem Computing a spectral Einstein functional for a Dirac operator with torsion.
method Computing the spectral Einstein functional for even-dimensional spin manifolds without boundary.
result The spectral Einstein functional for the Dirac operator with torsion is computed.
Paper defines spectral triple and computes functional for nonminimal de Rham-Hodge operator.
problem Computing spectral functions for nonminimal de Rham-Hodge operators.
method Definitions and computations of spectral triple and functional.
result Computed spectral Einstein functional for even-dimensional compact manifolds.
New spectral functionals for Dirac operators with inner fluctuations computed.
problem Spectral functionals and Dirac operators with inner fluctuations.
method Extension of spectral functionals for Dirac operators with inner fluctuations.
result Computed spectral Einstein functional for Dirac operator with inner fluctuations on even-dimensional spin manifolds.
Ricci solitons as critical points of quadratic curvature functionals
problem Einstein metrics and Ricci solitons as critical points of quadratic Riemannian functionals
method Study of Ricci solitons as critical points of a special quadratic curvature functional
result Ricci solitons are non-Einstein critical points of these functionals
The paper proves conditions for Einstein solitons to split into line and manifold.
problem Conditions for Einstein solitons to split into line and manifold.
method Weighted Laplacian comparison of distance function and bounded integral condition on Ricci curvature.
result Gradient ρ-Einstein solitons split off a line isometrically under certain conditions.
Analyzes Kähler-Einstein metrics on families of Fano varieties.
problem Establishing Kähler-Einstein metrics on Fano varieties in families.
method Analytic method to show unique Kähler-Einstein metrics on neighboring fibers.
result Uniform a priori estimates and continuous variation of Kähler-Einstein potentials.
The paper proves uniqueness and existence of Kähler-Einstein metrics on certain compactifications.
problem Existence and uniqueness of Kähler-Einstein metrics on Q-Fano group compactifications. method Analyzes Q-Fano group compactifications, proving uniqueness and existence of Kähler-Einstein metrics. result Proves the existence and uniqueness of Kähler-Einstein metrics on Q-Fano group compactifications. Characterizes warping functions in Einstein Poisson warped spaces.
problem Existence and nonexistence of warping functions with constant scalar curvature.
method Analyzes various dimensions of base space and constant scalar curvature conditions.
result Characterizes warping functions for different dimensions of base space.
Study on Einstein solitons with bounds and asymptotic behavior.
problem Understanding the properties of Einstein solitons.
method Computed lower bounds for scalar curvature, established asymptotic behavior, proved finiteness of fundamental group and weighted volume.
result Established finiteness of fundamental group and weighted volume for gradient shrinking Einstein solitons.
Study Einstein warped products with Einstein base and fiber.
problem Characterize Einstein warped products with Einstein base and fiber.
method Investigate necessary and sufficient conditions for a warped product to be Einstein.
result Explicitly determine the warping function when the base is hyperbolic space.
Hermitian-Einstein metrics linked to stability of bundles on orbifolds.
problem Existence of Hermitian-Einstein metrics on stable vector bundles over compact Kähler orbifolds.
method Equivalence of slope stability to the existence of Hermitian-Einstein metrics and properness of a functional.
result Equivalence of Hermitian-Einstein metrics and slope stability for stable vector bundles.
The double tetrahedron is the triangulation of the three-sphere gotten by gluing together two congruent tetrahedra along their boundaries. As a piecewise flat manifold, its geometry is determined by its six edge lengths, giving a notion of a metric on the double tetrahedron. We study notions of Einstein metrics, consta…
In this paper, we introduce notions of nonlinear stabilities for a relative ample line bundle over a holomorphic fibration and define the notion of a geodesic-Einstein metric on this line bundle, which generalize the classical stabilities and Hermitian-Einstein metrics of holomorphic vector bundles. We introduce a Dona…
We show that homogeneous Einstein metrics on Euclidean spaces are Einstein solvmanifolds, using that they admit periodic, integrally minimal foliations by homogeneous hypersurfaces. For the geometric flow induced by the orbit-Einstein condition, we construct a Lyapunov function based on curvature estimates which come f…
The paper defines a new functional and proves related theorems for manifolds with boundary.
problem Defining and proving theorems for manifolds with boundary.
method Defining the spectral Einstein functional and relating it to the noncommutative residue.
result Proof of Dabrowski-Sitarz-Zalecki type theorems for spectral Einstein functional on 4D manifolds with boundary.
The paper studies m-quasi Einstein manifolds with convex potential and finds constant scalar curvature.
problem Investigating m-quasi Einstein manifolds with a convex potential function. method Analyzing integral conditions and properties of the potential vector field.
result An m-quasi Einstein manifold with a convex potential function has constant scalar curvature. Study on stability of Einstein metrics on symmetric spaces.
problem Stability of Einstein-Hilbert functional on compact symmetric spaces.
method Classification of irreducible representations and use of Casimir eigenvalues.
result Proves stability of Einstein metrics on quaternionic and Cayley projective plane, instability on other quaternionic Grassmannians.
New flow deforms Riemannian metrics smoothly.
problem Deforming Riemannian metrics on spin manifolds.
method Parabolic flow based on Dirac-Einstein functional.
result Local well-posedness of smooth solutions proved.
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
The goal of this paper is to study weakly Einstein critical metrics of the volume functional on a compact manifold M with smooth boundary ∂M. Here, we will give the complete classification for an n-dimensional, n=3 or 4, weakly Einstein critical metric of the volume functional with nonnegative scalar …
We prove that complete warped product Einstein metrics with isometric bases, simply connected space form fibers, and the same Ricci curvature and dimension are isometric. In the compact case we also prove that the warping functions must be the same up to scaling, while in the non-compact case there are simple examples …
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.
Study on pseudo-Einstein 3-manifolds, calculating determinant changes under conformal transformations.
problem Prescribing the Q'-curvature on pseudo-Einstein 3-manifolds.
method Established an expression for the difference of determinants of Paneitz type operators under conformal changes.
result Generalized the expression of functional determinant from four to three dimensions.
The identity map of certain Einstein manifolds is stable in both energy and bienergy.
problem Stability of the identity map in Einstein manifolds.
method Investigation of conformal-biharmonic stability compared to harmonic stability.
result The conformal-biharmonic index coincides with the harmonic index, except for the 4D Euclidean sphere.
We introduce the coupled Ricci-Calabi functional and the coupled H-functional which measure how far from a coupled Kähler-Einstein metric in the sense of Hultgren-Witt Nyström. We first give corresponding moment weight type inequalities which estimate each functional in terms of algebraic invariants. Secondly, we give …
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.
Study generalizes Hermitian-Einstein equation for cyclic Higgs bundles, proving existence and inequality.
problem Addressing Hermitian-Einstein equation for cyclic Higgs bundles.
method Introducing generalizations using subharmonic functions and proving existence, uniqueness, and convergence of heat equations.
result Existence, uniqueness, and convergence of solutions for heat equations.
Smooth solutions and classification of Dirac-Einstein equations on 3-manifolds.
problem Analyzing solutions of Dirac-Einstein equations on R3. method Proving smoothness and asymptotic behavior, classifying ground state solutions.
result Scalar part is given by Aubin-Talenti functions, spinorial part is conformal image of −21-Killing spinors on S3. The study examines Einstein-Finsler spaces using Minkowskian products.
problem Characterizing Einstein-Finsler spaces through Minkowskian products.
method Proving conditions for Einstein-Finsler spaces in terms of Minkowskian products.
result If a Minkowskian product Finsler manifold is Einstein, then either the product manifold is Ricci flat or both quotient manifolds are Einstein with same scalar functions.
The paper studies Einstein metrics on specific manifolds and their rigidity properties.
problem Investigating rigidity of Einstein metrics on homogeneous Gray manifolds.
method Computing coindex and analyzing infinitesimal deformations of Einstein metrics.
result Infinitesimal Einstein deformations on F1,2=SU(3)/T2 are not integrable. Study on compact Quasi-Einstein manifolds yields diameter estimates and Hitchin-Thorpe inequality conditions.
problem Estimating diameters and verifying Hitchin-Thorpe inequality for compact Quasi-Einstein manifolds.
method Derive geometric estimates relating potential function oscillation to manifold diameter; derive lower bounds for diameter.
result Diameter conditions ensure compact Quasi-Einstein manifolds satisfy Hitchin-Thorpe inequality in dimension four.
The study proves rigidity for Poincaré-Einstein manifolds with flat Euclidean conformal infinity.
problem Proving rigidity for Poincaré-Einstein manifolds with specific conformal infinity.
method Analyzing curvature tensors over level sets of a boundary defining function.
result Rigidity theorem for Poincaré-Einstein manifolds with flat Euclidean conformal infinity.
Proves new theorems for manifolds with boundary.
problem No boundary conditions for manifolds.
method Generalizes Dabrowski-Sitarz-Zalecki theorems to manifolds with boundary.
result Proof of new theorems for even and odd dimensional manifolds with boundary.
In this article we give a classification of three dimensional m-quasi Einstein manifolds with two distinct Ricci-eigen values. Our study provides explicit description of local and complete metrics and potential functions. We also describe the associated warped product Einstein manifolds in detail. For the proof we pres…