New findings on Chern flat metrics and their criticality.
problem Understanding critical Hermitian metrics on Chern flat manifolds.
method Analyzing Chern flat manifolds as compact quotients of complex Lie groups and studying their criticality.
result Chern flat metrics on semi-simple Lie groups are torsion-critical and vice versa.
Paper finds critical metrics with pinched curvature are geodesic balls.
problem Identifying critical metrics with specific curvature constraints.
method Proved isometry to geodesic balls in S^n and provided conditions for the gradient of the potential function.
result Critical metrics with pinched curvature are isometric to geodesic balls in S^n.
In this paper we investigate complete critical metrics of the L2-norm of the scalar curvature. We prove that any complete critical metric with positive scalar curvature has constant scalar curvature and we characterize critical metrics with nonnegative scalar curvature in dimension three and four.
New rigidity results for critical metrics of a quadratic curvature functional.
problem Proving uniqueness of critical metrics for a specific curvature functional.
method Analyzing complete, possibly non-compact, critical metrics of the quadratic curvature functional.
result Critical metrics with finite energy are scalar flat (global minima) for dimensions n≥10.
Researchers found all special metrics in 4D for certain curvature functionals.
problem Identifying special metrics in 4D for quadratic curvature functionals.
method Determined all homogeneous metrics that are critical for quadratic curvature functionals.
result All homogeneous metrics in 4D for some quadratic curvature functionals have been identified.
Study critical metrics on manifolds with boundary using integral and boundary estimates.
problem Investigate geometry of critical metrics on compact manifolds with boundary.
method Use generalized Reilly's formula to derive integral and boundary estimates.
result Establish new boundary estimates for critical metrics of the volume functional.
The paper proves conditions under which critical point metrics are Einstein.
problem Proving that critical point metrics are Einstein under specific curvature constraints.
method Analyzing the traceless Ricci operator and scalar curvature constraints.
result The conjecture that critical point metrics are Einstein is proven under certain curvature conditions.
We disprove the generalized Chern-Hamilton conjecture on the existence of critical compatible metrics on contact 3-manifolds. More precisely, we show that a contact 3-manifold (M,α) admits a critical compatible metric for the Chern-Hamilton energy functional if and only if it is Sasakian or its associated Reeb fl…
Rigidity theorem for special metrics on 4-manifolds.
problem Rigidity of Bach-flat metrics on manifolds with boundary.
method Critical point analysis of Weyl energy with boundary conditions.
result Rigidity of critical metrics on upper hemisphere.
In this article, we investigate the geometry of critical metrics of the volume functional on an n-dimensional compact manifold with (possibly disconnected) boundary. We establish sharp estimates to the mean curvature and area of the boundary components of critical metrics of the volume functional on a compact manifol…
In this paper we prove rigidity results on critical metrics for quadratic curvature functionals, involving the Ricci and the scalar curvature, on the space of Riemannian metrics with unit volume. It is well-known that Einstein metrics are always critical points. The purpose of this article is to show that, under some c…
The paper studies critical metrics on a specific type of manifold.
problem Investigating critical metrics on almost Kenmotsu manifolds.
method Introducing and studying the ∗-Miao-Tam critical equation on (2n+1)-dimensional (k,μ)′-almost Kenmotsu manifolds. result If a (2n+1)-dimensional (k,μ)′-almost Kenmotsu manifold satisfies the ∗-Miao-Tam critical equation, it is ∗-Ricci flat and locally isometric to a specific product of manifolds. The paper classifies cosymplectic manifolds with critical metrics in dimension 3.
problem Classifying cosymplectic manifolds with critical metrics.
method Study of Chern-Hamilton energy functional on compact cosymplectic manifolds.
result Classification of manifolds admitting critical compatible metrics in dimension 3.
We study the space of smooth Riemannian structures on compact three-manifolds with boundary that satisfies a critical point equation associated with a boundary value problem, for simplicity, Miao-Tam critical metrics. We provide an estimate to the area of the boundary of Miao-Tam critical metrics on compact three-manif…
The paper classifies contact 3-manifolds with critical metrics and connects entropy to optimization.
problem Classifying contact 3-manifolds with critical metrics and understanding their entropy.
method Critical metrics optimization and entropy analysis.
result Anosov contact metrics' optimization is linked to Reeb dynamics and entropy.
In this paper, we investigate critical points of the Laplacian's eigenvalues considered as functionals on the space of Riemmannian metrics or a conformal class of metrics on a compact manifold. We obtain necessary and sufficient conditions for a metric to be a critical point of such a functional. We derive specific con…
We provide an isoperimetric inequality for critical metrics of the volume functional with nonnegative scalar curvature on compact manifolds with boundary. In addition, we establish a Weitzenböck type formula for critical metrics of the volume functional on four-dimensional manifolds. As an application, we obtain a clas…
Study critical metrics on manifolds, proving specific isometries.
problem Investigating critical metrics on complete manifolds.
method Analyzing volume functional and proving isometries.
result Critical metrics on specific manifolds are isometric to standard models.
Extends Gauduchon's result to higher dimensions, showing balanced metrics.
problem Understanding critical metrics in higher-dimensional Hermitian manifolds.
method Analyzes the functional of L2-norm of torsion 1-form and full Chern torsion. result Critical metrics are balanced in all dimensions.
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 …
The paper studies metrics on manifolds with scalar curvature properties.
problem Finding metrics with specific scalar curvature properties.
method Analyzing the squared L2-norm of the scalar curvature over constant volume metrics. result Critical points of the functional correspond to Einstein or scalar flat metrics.
We study closed n-dimensional manifolds of which the metrics are critical for quadratic curvature functionals involving the Ricci curvature, the scalar curvature and the Riemannian curvature tensor on the space of Riemannian metrics with unit volume. Under some additional integral conditions, we classify such manifol…
New metrics found in hyperbolic manifolds as volume-minimizers.
problem Finding critical points of volume-renormalized mass.
method Critical points of the volume-renormalized mass over asymptotically hyperbolic manifolds.
result V-static metrics are critical points of volume-renormalized mass.
New framework for studying eigenvalue functionals of metrics.
problem Understanding critical points of eigenvalue functionals.
method Clarke subdifferential theory to unify previous research.
result Unified understanding of critical metrics and new examples.
Let M be a real hypersurface of a complex space form with constant curvature c. In this paper, we study the hypersurface M admitting Miao-Tam critical metric, i.e. the induced metric g on M satisfies the equation:−(Δgλ)g+∇g2λ−λRic=g, where λ is a smooth function on M. At first, for the case wher…
Critical metrics on four-dimensional manifolds are either Einstein or product of two-dimensional manifolds.
problem Classifying critical metrics of a curvature functional on complete four-dimensional manifolds.
method Analyzing the curvature operator and energy condition to prove metric properties.
result Complete four-dimensional manifolds with finite energy are either Einstein or product of two-dimensional 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 new rigidity results for critical metrics of quadratic curvature functionals.
problem Proving rigidity of critical metrics for specific quadratic curvature functionals.
method Rigidity results for conformal vector fields, ODE argument, and new pointwise and integral estimates.
result Critical metrics are rigid under specific conditions.
Researchers confirm scalar-flatness for critical metrics in 5-9 dimensions.
problem Verifying scalar-flatness for critical metrics in specific dimensions.
method Analyzing complete Riemannian manifolds with critical metrics of the L2-scalar curvature functional. result The conjecture that all complete noncompact critical metrics with finite energy are scalar-flat is confirmed for dimensions 5 to 9.
Solves Besse conjecture on 3D manifolds, proving metric rigidity.
problem Besse conjecture on 3D compact manifolds
method Analytical proof of critical point equation
result Proves rigidity of Miao-Tam metric
New rigidity results for critical metrics with curvature pinching.
problem Understanding critical metrics with curvature pinching conditions.
method Proving rigidity for metrics defined on closed smooth manifolds that are critical for a quadratic functional.
result Bach-flat metrics with constant scalar curvature satisfying Sec > 1/48 R are Einstein and isometric to specific spaces.
We introduce a new critical value c∞(L) for Tonelli Lagrangians L on the tangent bundle of the 2-sphere without minimizing measures supported on a point. We show that c∞(L) is strictly larger than the Mañé critical value c(L), and on every energy level e∈(c(L),c∞(L)) there exist infinitely…
We study rigidity results for the Einstein metrics as the critical points of a family of known quadratic curvature functionals involving the scalar curvature, the Ricci curvature and the Riemannian curvature tensor, characterized by some pointwise inequalities involving the Weyl curvature and the traceless Ricci curvat…
Study stability of non-Kähler Calabi-Yau metrics using critical points of generalized Einstein Hilbert action.
problem Stability of critical points of the generalized Einstein Hilbert action in non-Kähler Calabi-Yau theory.
method Analysis of Bismut Hermitian Einstein manifolds and Bismut flat pluriclosed steady solitons, proving stability conditions.
result All Bismut Hermitian Einstein manifolds are linearly stable, and all Bismut flat pluriclosed steady solitons with positive Ricci curvature are linearly strictly stable.
We develop a gluing procedure designed to obtain canonical metrics on connected sums of Einstein four-manifolds. The main application is an existence result, using two well-known Einstein manifolds as building blocks: the Fubini-Study metric on CP2 and the product metric on S2×S2. Using these met…
Study on metrics maximizing eigenvalues of Paneitz operator on 4-manifolds.
problem Investigating metrics maximizing eigenvalues of Paneitz operator.
method Critical points of eigenvalues of Paneitz operator on Riemannian metrics with fixed volume.
result Critical metrics associated with extrinsic conformal-harmonic maps into round spheres.
Paper proves conjecture about Einstein metrics on manifolds with positive isotropic curvature.
problem Proving the Besse conjecture for metrics with positive isotropic curvature.
method Analyzing the critical point equation and using properties of metrics with positive isotropic curvature.
result The Besse conjecture is true for metrics with positive isotropic curvature.
We investigate rigidity and stability properties of critical points of quadratic curvature functionals on the space of Riemannian metrics. We show it is possible to "gauge" the Euler-Lagrange equations, in a self-adjoint fashion, to become elliptic. Fredholm theory may then be used to describe local properties of the m…
On a compact n-dimensional manifold M, it is well known that a critical metric of the total scalar curvature, restricted to the space of metrics with unit volume, is Einstein. It has been conjectured that a critical metric of the total scalar curvature, restricted to the space of metrics with constant scalar curvat…
We prove that a critical metric of the volume functional on a 4-dimensional compact manifold with boundary satisfying a second-order vanishing condition on the Weyl tensor must be isometric to a geodesic ball in a simply connected space form R4, H4 or S4. Moreover, we provide…
Proves mass-capacity inequalities for critical area-normalized capacitors, improving Schwarzschild metric uniqueness.
problem Proving mass-capacity inequalities for critical area-normalized capacitors.
method Analyzes asymptotically flat manifolds with boundary capacity potential satisfying an overdetermined problem.
result Improves Schwarzschild metric uniqueness and results for spin asymptotically flat spacetimes.
New approach to Z-stability and critical metrics on Kähler manifolds.
problem Determining Z-stability and existence of Z-critical metrics on Kähler manifolds. method Equivariant localisation applied to integrals over test configurations.
result Existence of Z-critical metrics is equivalent to Z-stability. The study examines metrics with unit volume or area on manifolds with boundaries, finding critical points and solving curvature problems.
problem Finding metrics with prescribed curvature on manifolds with boundaries.
method Variational properties of volume and boundary area functionals, using critical metrics and curvature conditions.
result Sufficient and necessary conditions for metrics to be critical points and for scalar/mean curvature functions.
Paper derives second variation formula for eigenvalue functionals on surfaces.
problem Determine if a critical metric is a local maximizer for eigenvalue functionals.
method Derive second variation formula for critical metrics and apply to specific cases.
result Flat metric on non-rhombic torus cannot be a conformal maximizer for first eigenvalue.
The aim of this paper is to classify three dimensional compact Riemannian manifolds (M3,g) that admits a non-constant solution to the equation −Δfg+Hessf−fRic=μRic+λg, for some special constants (μ,λ), under assumption that the manifold has cyclic parallel Ricci tensor. Namely, the structures that we will…
New approach finds Kähler metrics on compact complex manifolds.
problem Finding Kähler metrics on compact complex manifolds.
method Defining a new functional whose critical points are Kähler metrics.
result Critical points of the new functional are precisely the Kähler metrics.
The article studies critical points of a new energy functional in higher dimensions.
problem Investigating critical points of a new energy functional in higher dimensions.
method Holomorphic deformations, closed and open properties, differential of the functional.
result Properties of critical points under holomorphic deformations are closed and open.
The purpose of this paper is to investigate the critical points of the total scalar curvature functional restricted to space of metrics with constant scalar curvature of unitary volume, for simplicity CPE metrics. It was conjectured in 1980's that every CPE metric must be Einstein. We prove that a 4-dimensional CPE…