The paper studies real hypersurfaces in complex space forms with Miao-Tam critical metrics.
problem Characterizing real hypersurfaces with Miao-Tam critical metrics in complex space forms.
method Analyzing the equation −(Δgλ)g+ablag2λ−λRic=g for real hypersurfaces in complex space forms. result Compact real hypersurfaces in complex Euclidean space with Miao-Tam critical metrics are spheres, and non-flat complex space forms do not admit such metrics.
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 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 proves rigidity for critical metrics of volume functional in specific spaces.
problem Proving rigidity for critical metrics of the volume functional in specific spaces.
method Using geodesic balls and Einstein hypersurfaces, the paper extends rigidity theorems to Miao-Tam critical metrics and static metrics.
result Geodesic balls in specific spaces have maximum boundary volume among Miao-Tam critical metrics with connected boundary.
Study characterizes specific almost Kenmotsu metrics meeting Miao-Tam equation.
problem Characterizing almost Kenmotsu metrics.
method Characterization through Miao-Tam equation.
result Characterized specific almost Kenmotsu metrics.
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
The paper proves conjectures and classifies metrics on 3D manifolds.
problem Proving conjectures and classifying metrics on 3D manifolds with specific curvature conditions.
method Analytical proofs and classification theorems.
result Critical metrics on 3D manifolds are isometric to geodesic balls in space forms.
Calculates mass of hyperbolic manifolds using Ricci tensor and second fundamental form.
problem Evaluating the mass of asymptotically hyperbolic manifolds with noncompact boundaries.
method Uses Ricci tensor and second fundamental form via coordinates, similar to Miao-Tam's approach for asymptotically flat manifolds.
result Mass can be evaluated for these manifolds.
In this article we study any 4-dimensional Riemannian manifold (M,g) with harmonic curvature which admits a smooth nonzero solution f to the following equation \begin{eqnarray} \label{0002bx} \nabla df = f(Rc -\frac{R}{n-1} g) + x Rc+ y(R) g. \end{eqnarray} where Rc is the Ricci tensor of g, x is a constant a…
In this note, we use Chern's magic form Φk in his famous proof of the Gauss-Bonnet theorem to define a mass for asymptotically flat manifolds. It turns out that the new defined mass is equivalent to the one that we introduced recently by using the Gauss-Bonnet-Chern curvature Lk. Moreover, this equivalence implie…
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.
Study sharp geometric estimates for critical metrics on compact manifolds.
problem Investigating critical metrics of the volume functional on compact manifolds.
method Establishing sharp estimates for mean curvature and area of boundary components.
result Sharp estimates for mean curvature and area of boundary components of critical metrics.
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.
The study classifies critical metrics on manifolds with specific curvature conditions.
problem Characterizing critical metrics for quadratic curvature functionals.
method Analyzing closed n-dimensional manifolds with Ricci, scalar curvature, and Riemannian curvature tensor.
result Critical metrics are Einstein under certain curvature conditions.
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.
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.
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.
The study finds inequalities and formulas for special metrics on manifolds.
problem Finding inequalities and formulas for critical metrics of volume.
method Isoperimetric inequality and Weitzenböck formula for critical metrics of volume.
result Classification of critical metrics on four-dimensional manifolds.
Critical metrics of volume functional on compact 4-manifolds with boundary are rigid.
problem Finding critical metrics of volume functional on compact 4-manifolds with boundary.
method Proved using critical point theory and integral curvature estimates.
result Critical metrics are isometric to geodesic balls in R4, H4 or S4. 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…
Study critical exponent for geodesic currents using quasi-metric spaces.
problem Understanding the critical exponent for geodesic currents.
method Associated a quasi-metric space to geodesic currents and defined a metric for filling currents, studying the critical exponent and its relation to curve intersection growth.
result The critical exponent equals the exponential growth rate of the intersection function for closed curves.
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.
Paper proves conjecture about critical metrics with divergence-free Bach tensor.
problem Proving conjecture about critical metrics with specific curvature properties.
method Used divergence-free Bach tensor to prove conjecture.
result Proved conjecture about critical metrics with divergence-free Bach tensor.
The paper classifies weakly Einstein critical metrics on compact manifolds with boundary.
problem Identifying weakly Einstein critical metrics on compact manifolds with boundary.
method Complete classification for 3D and 4D cases with nonnegative scalar curvature; similar result for higher dimensions with Weyl tensor constraint.
result Complete classification of weakly Einstein critical metrics on compact manifolds with boundary.
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 proves gap properties for critical metrics under specific conditions.
problem Proving gap properties for critical metrics under divergence-free Bach tensor condition.
method Analyzing critical point equation of total scalar curvature with divergence-free Bach tensor.
result Proves gap properties for n≥5 and a similar condition for n=4. 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.
Study rigidity of Einstein metrics as critical points of curvature functionals.
problem Characterize Einstein metrics as critical points of quadratic curvature functionals.
method Analyze pointwise inequalities involving Weyl curvature and traceless Ricci curvature.
result Provide rigidity results for Einstein metrics and locally conformally flat critical metrics.
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…
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 study finds infinitely many periodic orbits just above a critical value on a 2-sphere.
problem Finding periodic orbits just above a critical value on a 2-sphere.
method Introduced a new critical value c∞(L) and showed its strict inequality to the Mañé critical value c(L), proving the existence of infinitely many periodic orbits on energy levels e∈(c(L),c∞(L)). result Infinitely many periodic orbits exist on energy levels just above the Mañé critical value.
Critical metrics in Levy-Gromov inequality are rigid in 2D.
problem Understanding critical metrics in Levy-Gromov inequality.
method Analyzing critical points of the isoperimetric profile.
result Criticality condition characterizes only round spheres and projective planes in 2D.
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.
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.
The paper proves rigidity of Einstein metrics as critical points of quadratic curvature functionals.
problem Characterizing Einstein metrics as critical points of quadratic functionals.
method Analyzing Einstein metrics on closed manifolds using quadratic curvature functionals and point-wise inequalities.
result Rigidity results for Einstein metrics involving Weyl curvature, trace-less Ricci curvature, and Yamabe invariant.
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.
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.
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.
The paper studies critical metrics on compact manifolds with zero radial Weyl curvature.
problem Finding critical metrics on compact manifolds with specific curvature properties.
method Analyzing the critical point of the total scalar curvature functional under zero radial Weyl curvature condition.
result CPE metrics with nonnegative sectional curvature in 3-dimension are isometric to a standard 3-sphere.
The paper classifies special Riemannian manifolds with cyclic parallel Ricci tensor.
problem Classifying Riemannian manifolds with specific properties.
method Analyzing solutions to a specific partial differential equation under cyclic parallel Ricci tensor conditions.
result Classification of manifolds with positive static triples, critical metrics, and total scalar curvature.
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.