Study vacuum static spaces with vanishing Bach and Weyl tensors, proving harmonicity and rigidity results.
problem Characterizing vacuum static spaces with specific tensor properties.
method Analyzing the complete divergence of Bach and Weyl tensors, proving conditions for harmonicity and rigidity.
result Proves the vanishing of complete divergence of Bach and Weyl tensors implies harmonicity of the metric.
Study on gradient ρ-Einstein solitons with radially nonnegative Bach tensor.
problem Characterizing gradient ρ-Einstein solitons with specific tensor properties.
method Analyzing the properties of Bach tensor and using local warping to classify solitons.
result Gradient ρ-Einstein solitons with radially nonnegative Bach tensor are locally warped products of an interval and an Einstein manifold.
The paper extends results on Bach-flat solitons to new types.
problem Analyzing Bach-like tensors on complete gradient Ricci solitons.
method Extending previous results to new types of solitons.
result Results on Bach-flat solitons extended to new types.
New findings on shrinking Ricci solitons with vanishing Bach-like tensors.
problem Characterizing gradient shrinking Ricci solitons with vanishing Bach-like tensors.
method Defining and analyzing Bach-like tensors, proving rigidity results, and deriving variational formulas.
result Vanishing Bach-like tensors force solitons to be either Einstein or isometric to the Gaussian soliton.
The paper introduces and characterizes almost ω-Bach solitons on various product manifolds.
problem Characterizing almost ω-Bach solitons on different manifolds.
method Introducing ω-Bach tensor and defining almost ω-Bach solitons; characterizing them under specific conditions. result Explicitly found gradient almost ω-Bach solitons on specific product manifolds.
The paper proves rigidity of certain solitons with specific properties.
problem Classifying and understanding Bach-flat gradient Schouten solitons.
method Analyzing the properties of Schouten solitons and their Ricci tensors.
result Rigidity of certain Schouten solitons under specific conditions.
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 generalizes Bach and Einstein equations with a field.
problem Generalizing classical equations in presence of a field.
method Introducing and characterizing new tensors and manifolds.
result Variational characterization of new flat and harmonic-Einstein manifolds.
Electrostatic systems with specific tensors are locally conformally flat.
problem Understanding the geometry of electrostatic systems with special tensors.
method Proving local conformal flatness for electrostatic manifolds with divergence-free Bach tensor.
result Three-dimensional electrostatic manifolds with divergence-free Bach tensor are locally conformally flat.
Rigidity theorem for manifolds with specific curvature properties.
problem Characterizing complete Riemannian manifolds with vanishing Bach tensor and positive scalar curvature.
method Pointwise inequalities and Sobolev constant inequalities.
result Rigidity results under specific curvature norms.
New metrics found with specific curvature properties on 4D manifolds.
problem Constructing metrics with specific curvature properties on closed manifolds.
method Using Aubin's deformation method to find metrics with pinched Bach tensor and scalar curvature.
result Existence of metrics with Bach tensor pinched by scalar curvature on 4D manifolds.
Study parallel tensors on affine surfaces to characterize Ricci recurrence.
problem Characterizing affine surfaces with specific geometric properties.
method Analyzing parallel trace-free tensors and Ricci recurrence.
result Existence of parallel tensors is linked to Ricci recurrence.
The paper constructs metrics on compact manifolds using Aubin's deformations.
problem Existence of metrics with non-vanishing Weyl tensor on compact manifolds.
method Special metric deformations introduced by Aubin.
result Existence of metrics with non-vanishing Weyl tensor on compact manifolds, no topological obstructions in dimension four.
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. We establish the existence of solvable Lie groups of dimension 4 and left-invariant Riemannian metrics with zero Bach tensor which are neither conformally Einstein nor half conformally flat.
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.
Researchers study solitons on homogeneous manifolds, proving properties of specific types of solitons.
problem Examining solitons on homogeneous manifolds to understand their properties and constraints.
method Analyzing ambient obstruction flow and specific solitons in homogeneous spaces, proving properties and constructing examples.
result Proved that any compact ambient obstruction soliton with constant scalar curvature is trivial, and characterized specific types of solitons in 4-dimensional homogeneous spaces.
The purpose of this article is to investigate Bach-flat critical metrics of the volume functional on a compact manifold M with boundary ∂M. Here, we prove that a Bach-flat critical metric of the volume functional on a simply connected 4-dimensional manifold with boundary isometric to a standard sphere must …
In this paper we show that a compact warped product Einstein manifold with vanishing Bach tensor of dimension n≥4 is a finite quotient of a warped product with (n−1)-dimensional Einstein fiber. The fiber has constant curvature if n=4.
This paper presents conformal invariants for Riemannian manifolds of dimension greater than or equal to four whose vanishing is necessary for a Riemannian manifold to be conformally related to an Einstein space. One of the invariants is a modification of the Cotton tensor, the other is a n--dimensional version of the…
Let (M,g) be a noncompact complete Bach-flat manifold with positive Yamabe constant. We prove that (M,g) is flat if (M,g) has zero scalar curvature and sufficiently small L2 bound of curvature tensor. When (M,g) has nonconstant scalar curvature, we prove that (M,g) is conformal to the flat space if $(…
In this paper we prove that any n-dimensional (n≥4) complete Bach-flat gradient steady Ricci soliton with positive Ricci curvature is isometric to the Bryant soliton. We also show that a three-dimensional gradient steady Ricci soliton with divergence-free Bach tensor is either flat or isometric to the Bryant sol…
The paper finds power series for Bach-flat metrics from spacetimes, including Einstein and constant curvature cases.
problem Extracting asymptotically anti-de Sitter Einstein 4-metrics from Bach-flat spacetimes.
method Using conformally compact Riemannian setting and formal power series, the paper finds expansions about conformal infinity.
result The mass is part of the free data at conformal infinity, leading to Einstein metrics.
We compute the evolution equation of the Cotton and the Bach tensor under the Ricci flow of a Riemannian manifold, with particular attention to the three dimensional case, and we discuss some applications.
New theorems in 2D and 4D for metrics with curvature or singularity.
problem Proving new versions of Huber theorem in dimensions 2 and 4.
method Using Coulomb frames and Bach tensor conditions to construct conformal metrics.
result Constructs conformal metrics with regularity across singularities in 4D.
We solve the classifying problem raised by Fischer and Marsden for Bach flat static spaces. We also prove the conjecture about critical point equations proposed by Besse for Bach flat manifolds. Particularly in dimension 3, we derive an integral identity that allows us to obtain conformal flatness from the vanish of th…
We obtain a local volume growth for complete, noncompact Riemannian manifolds with small integral bounds and with Bach tensor having finite L2 norm in dimension 4.
We show a closed Bach-flat Riemannian manifold with a fixed positive constant scalar curvature has to be locally spherical if its Weyl and traceless Ricci tensors are small in the sense of either L∞ or L2n-norm. Compared with the complete non-compact case done by Kim, we apply a different method t…
In this article we introduce local gauge conditions under which many curvature tensors appearing in conformal geometry, such as the Weyl, Cotton, Bach, and Fefferman-Graham obstruction tensors, become elliptic operators. The gauge conditions amount to fixing an n-harmonic coordinate system and normalizing the determi…
We establish short-time existence and regularity for higher-order flows generated by a class of polynomial natural tensors that, after an adjustment by the Lie derivative of the metric with respect to a suitable vector field, have strongly parabolic linearizations. We apply this theorem to flows by powers of the Laplac…
New classification of gradient steady Ricci solitons with vanishing D-tensor.
problem Classifying gradient steady Ricci solitons with specific properties.
method Extending Cao-Chen's work on Bach-flat gradient Ricci solitons, proving properties for D-flat solitons. result Any n-dimensional complete noncompact gradient steady Ricci soliton with vanishing D-tensor is either Ricci-flat or isometric to the Bryant soliton. New invariant connects boundary PDEs and conformal geometry.
problem Global conformal invariants of boundary PDEs.
method Variational considerations and conformal invariants.
result Compact Bach-flat manifolds with umbilic boundary admit Poincaré-Einstein metrics.
Study of Bach flow on specific nilmanifolds, converging to a soliton.
problem Analyzing the Bach flow on specific nilmanifolds.
method Fourth order geometric flow on four-dimensional simply connected nilmanifolds.
result The Bach flow converges to an expanding Bach soliton on these manifolds.
In this paper we introduce the notion of Einstein-type structure on a Riemannian manifold $\varrg$, unifying various particular cases recently studied in the literature, such as gradient Ricci solitons, Yamabe solitons and quasi-Einstein manifolds. We show that these general structures can be locally classified when th…
We classify complete gradient Ricci solitons satisfying a fourth-order vanishing condition on the Weyl tensor, improving previously known results. More precisely, we show that any n-dimensional (n≥4) gradient shrinking Ricci soliton with fourth order divergence-free Weyl tensor is either Einstein, or a finite q…
We define pure radiation metrics with parallel rays to be n-dimensional pseudo-Riemannian metrics that admit a parallel null line bundle K and whose Ricci tensor vanishes on vectors that are orthogonal to K. We give necessary conditions in terms of the Weyl, Cotton and Bach tensors for a pseudo-Riemannian metric to be …
Compact Bach-flat manifolds with positive σ2 are Einstein if curvature pinches.
problem Characterizing compact Bach-flat manifolds with positive σ2. method Proving compact Bach-flat manifolds with positive σ2 are Einstein under curvature pinching conditions. result Compact Bach-flat manifolds with positive σ2 are Einstein if curvature pinches. The paper extends gap theorems for Bach-flat 4-manifolds.
problem Proving gap theorems for specific Bach-flat 4-manifolds.
method Iteration argument and convergence theory of Bach-flat metrics.
result Conformally invariant gap theorems for (CP2,gFS) and (S2imesS2,gprod). Study on conformal harmonic coordinates on manifolds, proving existence and properties.
problem Existence and properties of conformal harmonic coordinates on Riemannian manifolds.
method Solutions to the conformal Laplace equation, proving up to boundary regularity results, elliptic regularity, and unique continuation results.
result Proves conformal harmonic coordinates are a close conformal analogue of harmonic coordinates.
We prove a removal of singularities result for Bach-flat metrics in dimension 4 under the assumption of bounded L^2 norm of curvature, bounded Sobolev constant and a volume growth bound. This result extends the removal of singularities result for special classes of Bach-flat metrics obtained in \cite{TVMOD}. For the pr…
Constructs Bach flat manifolds using modified Riemannian extension.
problem Constructing Bach flat manifolds of signature (2,2).
method Modified Riemannian extension of affine surfaces.
result Constructs scalar invariants not of Weyl type.
Introduces conformal Bach flow and proves its well-posedness and backward uniqueness.
problem Analyzing the long-time behavior of conformal Bach flow.
method Establishes well-posedness and backward uniqueness; derives L2-estimates of curvatures. result Derives Shi's pointwise-estimate of derivatives of curvatures without assuming Sobolev constant bound.
Grading function evaluates Bach-style chorales, outperforming human experts.
problem Difficulty in automatically evaluating musical style correctness.
method Introduces a grading function for evaluating four-part chorales in the style of J.S. Bach.
result Transformer model output is outperformed by the grading function at discriminating Bach chorales.
The study characterizes and classifies specific types of manifolds using conformal and quasi-Einstein properties.
problem Characterizing and classifying manifolds with specific geometric properties.
method Analyzing warped products, contact manifolds, and semi-Riemannian manifolds.
result Characterizations and classifications of weakly conformally flat and quasi-Einstein manifolds.
This paper classifies critical metrics on compact manifolds with boundary and harmonic Weyl tensor.
problem Classifying critical metrics on compact manifolds with boundary and harmonic Weyl tensor.
method Complete classification through geometric conditions and equivalence of assumptions.
result Critical metrics with harmonic Weyl tensor on simply connected compact manifolds with boundary are isometric to geodesic balls in simply connected space forms.
The paper proves stability for a modified Bach flow on various manifolds.
problem Stability of gauge-modified Bach flow on manifolds.
method Linear stability proved via spectral bounds and Koiso identity generalization. Nonlinear stability for hyperbolic and Poincaré-Einstein spaces.
result Linear and nonlinear stability results for the Bach flow on specific manifolds.
We construct examples of Bach-flat gradient Ricci solitons which are neither half conformally flat nor conformally Einstein.
In this paper, we classify n-dimensional (n>3) complete Bach-flat gradient shrinking Ricci solitons. More precisely, we prove that any 4-dimensional Bach-flat gradient shrinking Ricci soliton is either Einstein, or locally conformally flat hence a finite quotient of the Gaussian shrinking soliton R4 or the round cyl…