Study local properties of Chern-scalar curvature through linearization stability.
problem Local properties of Chern-scalar curvature function.
method Linearization analysis of the Chern-scalar curvature function.
result Stability of linearization and structure of metrics with prescribed curvature.
Paper investigates prescribing Chern scalar curvatures on specific manifolds.
problem Prescribing Chern scalar curvatures on noncompact Hermitian manifolds with nonpositive curvatures.
method Establishes existence results and sufficient conditions for negative curvature metrics.
result Obtains sufficient conditions for the existence of a constant negative Chern scalar curvature metric.
Estimates for metrics with constant Chern scalar curvature on complex manifolds.
problem Finding metrics with constant Chern scalar curvature on complex manifolds.
method Proving a priori estimates conditional on an upper bound on entropy.
result Extending a recent result by Chen-Cheng in the Kähler setting.
Paper explores prescribing Chern scalar curvatures on noncompact Hermitian manifolds.
problem Prescribing Chern scalar curvatures on complete noncompact Hermitian manifolds.
method Generalizes Aviles-McOwen's existence results to higher-dimensional Hermitian manifolds.
result Existence results for Chern scalar curvatures on Hermitian manifolds.
The paper solves a problem related to curvature in complex geometry.
problem Resolving the prescribed Chern scalar curvature problem.
method Divided into three cases based on the sign of the Gauduchon degree, analyzed separately.
result Proves that certain functions are Chern scalar curvatures of conformal metrics.
Unified flow approach to curvature problem on specific manifolds.
problem Prescribed Chern scalar curvature problem on compact Hermitian manifolds with negative Gauduchon degree.
method Unified flow approach with conditions on curvature function f. result Flow converges to a conformal Hermitian metric with specified curvature.
Study noncompact manifolds' Chern scalar curvatures, proving existence and multiplicity.
problem Prescribing Chern scalar curvatures on noncompact manifolds.
method Solving a Kazdan-Warner type equation on noncompact non-Kähler manifolds with an analytic condition.
result Established existence results and provided a new proof of multiplicity theorem.
The paper constructs metrics on Hirzebruch surfaces and ruled surfaces.
problem Existence of Hermitian metrics with constant Chern scalar curvature.
method Using Page--Bérard-Bergery's ansatz to construct metrics on Hirzebruch surfaces.
result Construction of Hermitian metrics of positive constant Chern scalar curvature on Hirzebruch surfaces.
Let X be a compact connected Riemann surface of genus g≥0, and let Symd(X), d≥1, denote the d-fold symmetric product of X. We show that Symd(X) admits a Hermitian metric with negative Chern scalar curvature if and only if g≥2, and positive Chern scalar curvature if and only if…
Study on prescribing curvature on specific manifolds with negative Gauduchon degree.
problem Prescribing Chern scalar curvatures on compact Hermitian manifolds with negative Gauduchon degree.
method Analysis of geometric flow convergence to obtain existence results.
result Existence results for curvature functions that are nonzero and nonpositive, and sign-changing cases.
The paper proves estimates for Hermitian metrics and shows curvature blow-up on complex manifolds.
problem Estimating curvature blow-up in Hermitian metrics.
method Local Calabi and higher order estimates for continuity equations.
result Chern scalar curvature blows up at a finite-time singularity on compact complex manifolds.
Study on compact Kähler surfaces for sign-changing curvatures.
problem Prescribing sign-changing Chern scalar curvatures on compact Kähler surfaces.
method Established a Chen-Li type existence theorem and provided an alternative proof.
result Alternative proof of Ding-Liu's theorem on sign-changing Gaussian curvatures.
The paper classifies flag manifolds with specific isotropy components and finds conditions for Kähler-like scalar curvature.
problem Classifying flag manifolds with specific isotropy components and finding conditions for Kähler-like scalar curvature.
method Investigating invariant almost Hermitian structures on generalized flag manifolds with two or three irreducible components.
result Classification of flag manifolds admitting Kähler-like scalar curvature and conditions for such structures.
Survey on metrics on non-Kähler complex manifolds.
problem Existence and properties of Hermitian metrics.
method Analytic study of Chern connection and related flows.
result Generalizations of Kähler-Einstein condition.
The study explores metrics with constant curvature on compact manifolds.
problem Finding Hermitian metrics with constant second scalar curvature on compact manifolds.
method Analyzes Yamabe-type and elliptic equations, derives geometric consequences, and proves existence under specific curvature conditions.
result Under certain curvature conditions, a pluriclosed Gauduchon Hermitian metric has constant second Chern scalar curvature, leading to the existence of Kähler-Einstein metrics.
On a closed balanced manifold, we show that if the Chern scalar curvature is small enough in a certain Sobolev norm then a slightly modified version of the Chern-Yamabe flow~\cite{Angella:2015aa} converges to a solution of the Chern-Yamabe problem. We also prove that if the Chern scalar curvature, on closed almost-Herm…
We initiate the study of an analogue of the Yamabe problem for complex manifolds. More precisely, fixed a conformal Hermitian structure on a compact complex manifold, we are concerned in the existence of metrics with constant Chern scalar curvature. In this note, we set the problem and we provide a positive answer when…
The various scalar curvatures on an almost Hermitian manifold are studied, in particular with respect to conformal variations. We show several integrability theorems, which state that two of these can only agree in the Kähler case. Our main question is the existence of almost Kähler metrics with conformally constant Ch…
For a Kahler metric, the Riemannian scalar curvature is equal to twice the Chern scalar curvature. The question we address here is whether this equivalence can hold for a non-Kahler Hermitian metric. For such metrics, if they exist, the Chern scalar curvature would have the same geometric meaning as the Riemannian sc…
Flow smooths Chern-Ricci-flat metrics on Hermitian manifolds.
problem Smooth Chern-Ricci-flat metrics on Hermitian manifolds.
method An analogue of the Calabi flow for compact Hermitian manifolds with vanishing first Bott-Chern class.
result The flow converges to the unique Chern-Ricci-flat metric under certain conditions.
Blowing up flat metrics yields balanced ones with constant curvature.
problem Constructing balanced metrics with constant curvature on orbifolds.
method Blowing up a compact orbifold with balanced Chern-Ricci flat metrics.
result Blown-up orbifolds admit balanced metrics with constant Chern scalar curvature.
In this note we study finite-time singularities in the Chern-Ricci flow. We show that finite-time singularities are characterized by the blow-up of the scalar curvature of the Chern connection.
Study on special Hermitian metrics on cohomogeneity one manifolds.
problem Characterizing and constructing Hermitian metrics on cohomogeneity one manifolds.
method Investigation of geometry of Hermitian manifolds with compact Lie group action by holomorphic isometries.
result Construction of new examples of cohomogeneity one Hermitian metrics solving specific equations.
Researchers find unique metrics solving complex PDEs for constant scalar curvature.
problem Finding metrics with constant scalar curvature in complex manifolds.
method Proving existence and uniqueness of smooth functions f that solve a fourth-order nonlinear PDE related to the Calabi functional. result Critical metrics minimize the Calabi functional and have constant Chern scalar curvature.
The paper explores Kähler-like metrics on generalized flag manifolds.
problem Finding invariant almost Hermitian structures with specific scalar curvature properties.
method Investigating invariant almost Hermitian geometry on generalized flag manifolds, focusing on Kähler-like metrics.
result Examples of Kähler-like metrics satisfying s=2smC are provided. 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, as balanced and Gaud…
On a Kahler manifold there is a clear connection between the complex geometry and underlying Riemannian geometry. In some ways, this can be used to characterize the Kahler condition. While such a link is not so obvious in the non-Kahler setting, one can seek to understand extensions of these characterizations to genera…
Balanced metrics found on Lie groups and their quotients.
problem Existence of balanced metrics on Lie groups and quotients.
method Proved existence of invariant complex structures and Hermitian balanced metrics on Lie groups and quotients.
result Existence of balanced metrics on Lie groups and quotients, and no pluriclosed metrics.
We define a partition of the space of projectively flat metrics in three classes according to the sign of the Chern scalar curvature; we prove that the class of negative projectively flat metrics is empty, and that the class of positive projectively flat metrics consists precisely of locally conformally flat-Kähler met…
Recently, J. Streets and G. Tian introduced a natural way to evolve an almost-Kähler manifold called the symplectic curvature flow, in which the metric, the symplectic structure and the almost-complex structure are all evolving. We study in this paper different aspects of the flow on locally homogeneous manifolds, incl…
The paper proves complex geometry results for manifolds of the form X × R², answering a 1994 conjecture.
problem Proving complex geometry results for manifolds of the form X × R².
method Using Riemannian and complex geometry techniques, the authors show the existence of metrics with positive scalar curvature.
result The paper answers a 1994 Rosenberg-Stolz conjecture for X × R², extending results to noncompact manifolds.
Uniform estimates lead to Gromov-Hausdorff limits for Hermitian minimal models.
problem Uniform diameter and volume estimates for Chern-Ricci flow on Hermitian minimal models.
method Uniform diameter and volume estimates, local Kähler assumption, Perelman's reduced length, almost monotonicity formula for reduced volume.
result Gromov-Hausdorff convergence of the Chern-Ricci flow on Hermitian minimal models.
Study on volumes and scalar curvature in complex geometry, proving new bounds and conditions.
problem Understanding volumes and scalar curvature in complex geometry.
method Asymptotic behavior of Bergman kernel, Kähler-Ricci flow, singular Kähler-Einstein metrics, and gluing techniques.
result New bounds and conditions for volumes and scalar curvature in complex manifolds.
We study different notions of Riemannian curvatures: The p-curvatures which interpolate between the scalar curvature and the sectional curvature, the Gauss-Bonnet-Weyl curvatures form another interpolation from the scalar curvature to the Gauss-Bonnet integrand. We bring out the (p,q)-curvatures, which incorporate …
Paper establishes a relation between Berwald scalar curvature and S-curvature.
problem Understanding the relationship between Finsler metrics' curvature properties.
method Proved conditions for isotropic Berwald scalar curvature and weakly isotropic S-curvature.
result Finsler metrics with isotropic Berwald scalar curvature have weakly isotropic S-curvature.
New scalar curvature defined from Ollivier-Ricci curvature for graphs.
problem Defining scalar curvature for graphs and point clouds.
method Defining a new scalar version of Ollivier-Ricci curvature and proving its convergence.
result The new scalar curvature converges to scalar curvature for sampled manifolds.
We show any Riemannian curvature model can be geometrically realized by a manifold with constant scalar curvature. We also show that any pseudo-Hermitian curvature model, para-Hermitian curvature model, hyper-pseudo-Hermitian curvature model, or hyper-para-Hermitian curvature model can be realized by a manifold with co…
New metric with negative curvature found near positive-curvature spaces.
problem Understanding spaces with positive curvature bounds.
method Analyzing metrics in relation to curvature bounds.
result Near any point in a positive-curvature space, there exists a metric with negative upper curvature.
Study examines preservation of curvature-adaptedness during mean curvature flow.
problem Preservation of curvature-adaptedness during mean curvature flow.
method Investigates curvature-adaptedness in locally symmetric spaces.
result Curvature-adaptedness is preserved along mean curvature flow.
Paper explores entropic curvature in Markov chains, comparing it to other curvatures.
problem Comparing entropic curvature to other curvatures in Markov chains.
method Adapted Γ-calculus for θ-curvatures, explicit lower bounds, curvature perturbation.
result Entropic curvature differs significantly from other curvature notions.
The paper studies Berwald scalar curvature properties in Finsler geometry.
problem Characterizing Finsler manifolds based on Berwald scalar curvature.
method Analyzes properties of Berwald scalar curvature and its implications for Finsler manifolds.
result Landsberg manifolds with vanishing Berwald scalar curvature are Berwald manifolds.
The paper studies Finsler manifolds with a new curvature concept.
problem Understanding Finsler manifolds with positive weighted flag curvature.
method Introducing a new curvature concept based on the flag curvature and a non-Riemannian quantity, T-curvature.
result Positive weighted flag curvature implies the manifold is diffeomorphic to Euclidean space.
New proof shows holomorphic sectional curvature fully determines curvature tensor.
problem Determining the curvature tensor from holomorphic sectional curvature.
method Representation-theoretic means to calculate L2-norm of holomorphic sectional curvature. result Holomorphic sectional curvature fully determines the curvature tensor.
Given a compact four dimensional smooth Riemannian manifold (M,g) with smooth boundary, we consider the evolution equation by Q-curvature in the interior keeping the T-curvature and the mean curvature to be zero and the evolution equation by T-curvature at the boundary with the condition that the Q-curvature …
The curvature-dimension condition implies a new weighted scalar curvature.
problem Studying the properties of the n-volumic scalar curvature. method Using the curvature-dimension condition mCD(κ,n) and smGH-convergence. result The stability of n-volumic scalar curvature ≥κ under smGH-convergence. Compact shrinkers with curvature pinching conditions proven.
problem Ensuring shrinkers are compact under curvature pinching conditions.
method Various curvature pinching conditions applied to shrinkers with positive Ricci curvature and asymptotically nonnegative sectional curvature.
result Shrinkers with curvature pinching conditions are proven to be compact.
Study geodesic curvature of logarithmic spirals on curved surfaces.
problem Understanding geodesic curvature on curved surfaces.
method Computed geodesic curvature of logarithmic spirals on surfaces of constant Gaussian curvature.
result Asymptotic behavior of geodesic curvature is independent of the ambient surface's curvature.
Study on singularity behavior of mean curvature flow with bounded curvature and index.
problem Understanding singularity formation in mean curvature flow with constraints.
method Analyzing flow with bounded mean curvature and Morse index.
result Either mean curvature or Morse index blows up at first singular time.