Upper diameter bound for manifolds with positive scalar curvature.
problem Estimating the maximum size of manifolds with positive scalar curvature.
method Proving an upper diameter bound using scalar curvature integral, Yamabe constant, and manifold dimension.
result The power of scalar curvature integral in diameter estimates is sharp and occurs at round spheres with canonical metric.
Study negative scalar curvature metrics with positive boundary mean curvature.
problem Bounding conformal metrics with specific curvature properties.
method Analyzing Riemannian manifolds with boundary conditions.
result A priori boundedness of metrics in specific cases.
Constructs metrics with negative constant scalar curvature.
problem Negative constant scalar curvature metrics.
method One-parameter family of complete metrics.
result Verifies positive energy conjecture for these metrics.
We study uniqueness of positive solutions to the conformal scalar curvature equation on complete Riemannian manifolds with constant negative scalar curvature. We apply the results to show that conformal transformations on certain complete Riemannian manifolds of constant negative scalar curvature are isometries. We als…
Paper finds conditions for non-Einstein relative Yamabe metrics.
problem Finding relative Yamabe metrics with positive scalar curvature.
method Sufficient condition for positive constant scalar curvature metrics on manifolds with boundary.
result Examples of non-Einstein relative Yamabe metrics with positive scalar curvature.
The study examines special properties of compact Riemannian manifolds with harmonic Weyl curvature.
problem Characterizing compact Riemannian manifolds with specific curvature properties.
method Rigidity theorems and geometric analysis on manifolds with positive scalar curvature and constant σ2. result Theorems proving the isometry of certain manifolds under specific curvature conditions.
The paper proves conditions for positive scalar curvature and Yamabe constant on noncompact cylinders.
problem Conditions for positive scalar curvature and Yamabe constant on noncompact cylinders.
method Analyzes complete metrics with positive scalar curvature and Yamabe constant on noncompact cylinders.
result Positive scalar curvature and Yamabe constant conditions are satisfied under specific geometric and conformal class constraints.
3-manifolds with positive scalar curvature have controlled foliations.
problem Understanding foliations in 3-manifolds with positive scalar curvature.
method Showed a singular foliation by surfaces with controlled area and diameter.
result 3-manifolds with positive scalar curvature admit controlled foliations.
Complete constant positive scalar curvature metrics on S^n - {p_1, ..., p_k} admit a definite asymptotic structure; i.e. the metric is asymptotic to a specific S^{n-1}-invariant metric near the puncture points. This allows one to glue together two such metrics near their puncture points, provided the asymptotic structu…
Sharp decay constant for positive scalar curvature metrics on manifolds.
problem Finding the optimal decay constant for positive scalar curvature metrics.
method New exhaustion result using μ-bubbles.
result Decay constant of 2/3 is sharp and necessary for complete metrics.
The paper proves uniformization for specific curvature types on manifolds.
problem Uniformizing fourth order conformal curvature on Riemannian manifolds.
method Proving existence of conformal deformations for specific curvature conditions.
result Existence of conformal deformations for positive Yamabe invariant and total Q-curvature.
Let (M,g) be a compact Riemannian manifold with dimension n > 2. The Yamabe problem is to find a metric with constant scalar curvature in the conformal class of g, by minimizing the total scalar curvature. The proof was completed in 1984. Suppose (M',g') and (M'',g'') are compact Riemannian n-manifolds with constant sc…
In this paper, we prove some rigidity theorems for compact Bach-flat n-manifold with the positive constant scalar curvature. In particular, our conditions in Theorem 1.4 have the additional properties of being sharp.
Unique conformal metrics found on certain manifolds.
problem Finding unique conformal metrics with constant Q-curvature.
method Proving uniqueness on manifolds with positive scalar curvature.
result Only metrics of the form λg with λ>0 are constant Q-curvature.
Study shows non-uniqueness and failure of compactness in constant curvature equations.
problem Non-uniqueness and failure of compactness in constant curvature equations.
method Warped product manifold construction and smooth counterexample creation.
result Compactness of solutions fails for dimensions ≥ 62.
Modified condition proves no positive scalar curvature for enlargeable manifolds.
problem Proving no positive scalar curvature for modified Λ2-enlargeable manifolds. method Replacing constant near infinity with locally constant near infinity and proving the result.
result Modified Λ2-enlargeable manifolds cannot carry a complete Riemannian metric of positive scalar curvature. We give some classifications of biharmonic hypersurfaces with constant scalar curvature. These include biharmonic Einstein hypersurfaces in space forms, compact biharmonic hypersurfaces with constant scalar curvature in a sphere, and some complete biharmonic hypersurfaces of constant scalar curvature in space forms and…
Bonnet-Myers theorem applied to Q-curvature on 4-manifolds.
problem Bounding Q-curvature on 4-manifolds.
method Scalar curvature and Q-curvature bounds.
result Diameter bound for Q-curvature quotient.
We present a conformal deformation involving a fully nonlinear equation in dimension 4, starting with positive scalar curvature. Assuming a certain conformal invariant is positive, one may deform from positive scalar curvature to a stronger condition involving the Ricci tensor. We also give a new conformally invariant …
New Kazdan-Warner problem for equivariant metrics on manifolds.
problem Equivariant scalar curvature functions on manifolds with group actions.
method Established equivariant analogue of Kazdan-Warner trichotomy.
result New class of totally G-positive pairs with positive constant scalar curvature.
The paper studies 3D manifolds with positive scalar curvature and volume growth.
problem Understanding the geometry of 3D manifolds with positive scalar curvature.
method Analyzes volume and geometric properties of 3D complete manifolds with positive scalar curvature, considering different curvature conditions.
result Volume growth estimates for 3D manifolds with positive scalar curvature, answering Gromov's question affirmatively.
Compact metrics found with specific curvature properties on 3D surfaces.
problem Finding compact metrics with constant curvature on 3D surfaces.
method Blow-up analysis of Yamabe equation with critical Sobolev exponents.
result Proved the compactness of conformal metrics with constant scalar curvature and boundary mean curvature.
Study on creating flat Kähler metrics on algebraic manifolds minus a hypersurface.
problem Existence of scalar-flat Kähler metrics on algebraic manifolds minus a hypersurface.
method Assumption of constant positive scalar curvature Kähler metric on the hypersurface.
result Existence of complete scalar-flat Kähler metrics on the manifold minus the hypersurface.
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.
Study finds conditions for conformal deformations to constant scalar curvature in conic metrics.
problem Finding conditions for conformal deformations to constant scalar curvature in conic metrics.
method Analyzes conformal deformations within a class of incomplete Riemannian metrics that generalize conic orbifold singularities.
result Determines sufficient conditions for the existence of a conformal deformation to a conic metric with constant scalar curvature -1.
We study the problem of deforming a Riemannian metric to a conformal one with nonzero constant scalar curvature and nonzero constant boundary mean curvature on a compact manifold of dimension n≥3. We prove the existence of such conformal metrics in the cases of n=6,7 or the manifold is spin and some other remai…
Classifies 1D exponential families with constant Hessian curvature.
problem Classifying 1D exponential families with constant Hessian curvature.
method Complete classification through mathematical analysis.
result If an exponential family has constant Hessian curvature, it must have curvature λ=k2 for some integer k≤m. The study preserves upper bounds of total scalar curvature in conformal classes.
problem Preserving upper bounds of total scalar curvature in conformal classes.
method Analyzing Yamabe constant and scalar curvature conditions.
result The upper bound condition of total scalar curvature is preserved in a conformal class.
Sharp bounds and parabolicity results for 3-manifolds with scalar curvature.
problem Understanding the spectrum and parabolicity of 3-manifolds with scalar curvature constraints.
method Established global results for complete three-dimensional manifolds under a topological assumption.
result Sharp upper bounds for the bottom spectrum and parabolicity results for manifolds with scalar curvature lower bounds.
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.
We classify compact conformally flat n-dimensional manifolds with constant positive scalar curvature and satisfying an optimal integral pinching condition: they are covered isometrically by either Sn with the round metric, S1×Sn−1 with the product metric or $\mathbb{S}^{1…
Paper studies metrics with constant Q-curvature near singular points.
problem Deriving properties of metrics with constant Q-curvature near singularities.
method Refined asymptotic expansion for metrics with constant Q-curvature and scalar curvature.
result Modelled results on similar metrics with scalar curvature, analyzing linearization about Delaunay metrics.
Sharp bound on scalar curvature integral in 3-manifolds.
problem Bounding the integral of scalar curvature on 3-manifolds.
method Geodesic ball analysis with nonnegative Ricci curvature.
result Integral of scalar curvature is bounded by 8πR for large radii. Proves a quantitative index theorem for positive scalar curvature metrics.
problem Studying conjectures and open questions on positive scalar curvature.
method Quantitative relative index theorem and λ-Lipschitz rigidity theorem. result Positive answers to Gromov's open questions on scalar curvature.
Sharp estimate for 2-systole on Kähler surfaces with positive scalar curvature.
problem Estimating the 2-systole on compact Kähler surfaces with positive scalar curvature.
method Combining classification of positive scalar curvature Kähler surfaces with Stern's level set method adapted to Kähler setting.
result Proved the sharp estimate minXS(ω)⋅sys2(ω)≤12π. In this note, we give a geometric characterization of the compact and totally umbilical hypersurfaces that carry a non trivial locally static Killing Initial Data (KID). More precisely, such compact hypersurfaces have constant mean curvature and are isometric to one of the following manifolds: (i) Sn the standard spher…
Constructs fill-ins with scalar curvature lower bounds for geometric applications.
problem Realizing (n−1)-dimensional manifolds as boundaries of higher-dimensional ones with controlled scalar curvature. method Variations of an argument by Miao and the author, constructing fill-ins with different scalar curvature lower bounds.
result Illustrates applications to geometric inequalities in general relativity, including mass bounds and Penrose inequalities.
The paper proves solutions for Yamabe equations on manifolds with boundary.
problem Existence and multiplicity of positive solutions for Yamabe equations.
method Use of isoparametric functions to prove existence and multiplicity results.
result Existence and multiplicity results for positive solutions of Yamabe equations.
Study classifies certain Einstein 4-manifolds with twistorial properties.
problem Classifying Einstein manifolds with positive scalar curvature.
method Proving properties of Einstein four-manifolds and their twistor spaces.
result Compact Einstein four-manifolds with positive scalar curvature and specific twistorial conditions are S4 and CP2. It has been showed by Byde that it is possible to attach a Delaunay-type end to a compact nondegenerate manifold of positive constant scalar curvature, provided it is locally conformally flat in a neighborhood of the attaching point. The resulting manifold is noncompact with the same constant scalar curvature. The main…
Let (Mn,g)(n≥3) be an n-dimensional complete Riemannian manifold with harmonic curvature and positive Yamabe constant. Denote by R and Rm˚ the scalar curvature and the trace-free Riemannian curvature tensor of M, respectively. The main result of this paper states that Rm˚ goes to ze…
In this paper, we consider the scalar curvature of Yamabe solitons. In particular we show that, with natural conditions and non positive Ricci curvature, any complete Yamabe soliton has constant scalar curvature, namely, it is a Yamabe metric. We also show that the quadratic decay at infinity of the Ricci curvature of …
Estimates for scalar curvature equations on Kähler manifolds with singularities.
problem Developing estimates for scalar curvature equations with singular metrics.
method Estimates and Laplacian estimates for scalar curvature equations of degenerate Kähler metrics.
result Derivation of estimates for singular constant scalar curvature Kähler metrics and singular Kähler-Einstein metrics.
In this paper we study sectional curvature of invariant hyper-Hermitian metrics on simply connected 4-dimensional real Lie groups admitting invariant hypercomplex structure. We give the Levi-Civita connections and explicit formulas for computing sectional curvatures of these metrics and show that all these spaces have …
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 paper proves a quantitative positive mass theorem for spin manifolds with distance estimates.
problem Proving a positive mass theorem for spin manifolds with arbitrary ends.
method Analyzing the scalar curvature and using distance estimates.
result Quantitative answer to Schoen and Yau's question on the positive mass theorem.
New approach linking CR Yamabe invariant to Sasaki structures.
problem Existence of constant transversal scalar curvature Sasaki structures.
method Drawing on CR Yamabe problem ideas, establishing link between invariant, Sasaki structures, and K-stability.
result CR Yamabe invariant value determines K-semistability of Sasaki manifolds.
In this note we study the conformal metrics of constant Q curvature on closed locally conformally flat manifolds. We prove that for a closed locally conformally flat manifold of dimension n≥5 and with Poincarë exponent less than 2n−4, the set of conformal metrics of positive constant Q and positive …