The paper establishes bounds on scalar curvature on asymptotically flat manifolds.
problem Establishing scalar curvature bounds on asymptotically flat manifolds.
method Using Ricci-DeTurck flow and distributional scalar curvature, the paper derives bounds on scalar curvature.
result The scalar curvature lower bound under Ricci-DeTurck flow depends on the scalar curvature lower bound in the β-weak sense and time.
Surveying Ricci flow for weak lower scalar curvature bounds.
problem Creating local definitions for weak lower scalar curvature bounds for C0 metrics. method Using Ricci flow to define and analyze weak lower scalar curvature bounds.
result Properties and applications of Ricci flow in defining weak lower scalar curvature bounds.
Study connects curvature bounds to map existence and flow solutions.
problem Existence of lower scalar curvature bounds and measures.
method Relates curvature bounds to map existence and backward limit of Ricci flow solutions.
result Sufficient condition for existence of limiting scalar curvature measure.
New examples of manifolds with lower scalar curvature bounds and submanifold collapse.
problem Stability of scalar curvature rigidity phenomena.
method Constructing Riemannian manifolds with specific curvature and collapse properties.
result Examples demonstrating stability and rigidity of scalar curvature.
The paper proves macroscopic versions of conjectures about scalar curvature and volume bounds.
problem Bounding simplicial volume and L2-Betti numbers with scalar curvature constraints. method Using upper bounds on volumes of 1-balls in universal covers.
result Macroscopic versions of conjectures about scalar curvature and volume bounds are proven.
Uniformly positive scalar curvature implies a lower bound on injectivity radius.
problem Bounding curvature and scalar curvature in three-manifolds.
method Analyzing bounded sectional curvature and uniformly positive scalar curvature properties.
result Uniform lower bound on injectivity radius.
Study on scalar curvature bounds and manifold topological complexity.
problem Understanding the topological complexity of manifolds with scalar curvature constraints.
method Introduced a small scale index theorem to establish bounds for Gromov's simplicial norm.
result Upper bound for Gromov's simplicial norm established in terms of scalar curvature, volume, and injectivity radius.
New bounds on scalar curvature for metric sequences.
problem Bounding scalar curvature in metric sequences.
method Integral convergence of scalar curvature; point-wise scalar curvature lower bound.
result Limiting metric has scalar curvature lower bound.
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.
Paper bounds the A-hat genus using curvature and isoperimetric constants.
problem Bounding the A-hat genus of Riemannian manifolds.
method Spectral analysis of the Dirac operator, scalar curvature lower bounds, and isoperimetric constants.
result Proves an upper bound on the A-hat genus using manifold properties.
Lower bounds on curvature integral for manifolds with curvature constraints.
problem Bounding curvature integrals under curvature constraints.
method Proving a lower bound for the curvature integral using dimension, upper curvature bounds, and injectivity radius.
result Uniformly bounded below integral of scalar curvature.
The study finds the maximum spectrum of 3D manifolds with lower scalar curvature.
problem Finding the maximum spectrum of 3D manifolds with lower scalar curvature.
method Establishing an analogous result to Cheng's theorem for 3D manifolds with scalar curvature lower bound.
result A splitting theorem for 3D manifolds with the maximal bottom spectrum.
This paper sets lower bounds for scalar curvatures in Ricci flow singularity models.
problem Understanding scalar curvatures in Ricci flow singularity models.
method Developed high-dimensional theory of Hamilton's Ricci flow, including new monotonicity formulas, compactness theorem, and partial regularity theory.
result Obtained a quadratic decay lower bound for the scalar curvature in 4-dimensional non-Ricci-flat steady soliton singularity models.
The paper proves scalar curvature lower bounds along Ricci flow on compact manifolds.
problem Preserving scalar curvature lower bounds along Ricci flow.
method Analyzing Ricci flow on compact manifolds with initial metrics having scalar curvature lower bounds.
result If initial metric has scalar curvature lower bound, it is preserved along Ricci flow.
The study preserves lower bounds of total scalar curvature under specific metric convergence.
problem Preserving lower bounds of total scalar curvature on smooth manifolds.
method Used stability of Ricci flow and heat flow with Ricci flow background.
result Lower bound of weighted total scalar curvature is preserved under specified convergence conditions.
We explore to what extent one may hope to preserve geometric properties of three dimensional manifolds with lower scalar curvature bounds under Gromov-Hausdorff and Intrinsic Flat limits. We introduce a new construction, called sewing, of three dimensional manifolds that preserves positive scalar curvature. We then use…
In this very short note we prove a lower bound for the scalar curvature of certain steady gradient Ricci solitons.
Sharp bounds on scalar curvature spectrum and rigidity theorems.
problem Understanding scalar curvature bounds and rigidity on manifolds.
method Sharp upper bounds for the bottom spectrum of the Beltrami Laplacian, scalar curvature rigidity theorem.
result Sharp upper bound for the bottom spectrum of the Beltrami Laplacian and scalar curvature rigidity theorem.
Study bounds on curvature for special Finsler metrics.
problem Curvature and topological properties of ∞-Einstein Finsler metrics. method Construct special metrics, analyze equivalence, impose curvature bounds.
result Establish bounds for curvature and distortion on ∞-Einstein Finsler manifolds. Extends Ricci flow theory to Kato-type curvature bounds, proving manifold properties.
problem Extending Ricci flow theory under Kato-type curvature bounds.
method Extends non-collapsed Ricci flow existence theory to Kato-type lower bounds.
result Compact three-dimensional non-collapsed strong Kato limit spaces are homeomorphic to smooth manifolds.
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 establishes distance estimates for manifolds with lower scalar curvature bounds.
problem Distance estimates on manifolds with lower scalar curvature bounds.
method Introduced a definition of relative index via a deformed Dirac operator trick and proved index coincidence with Callias operators.
result Proved short neck inequality and quantitative shielding result with positive scalar curvature.
Sharp comparison theorems for 3D manifolds with scalar curvature bound.
problem Understanding the geometry and topology of 3D manifolds with scalar curvature constraints.
method Sharp comparison results for Green's function and spectrum, derived from scalar curvature bounds.
result Sharp upper and lower bounds for the Green's function and spectrum of 3D manifolds.
3D metrics get scalar curvature bounds via IMCF.
problem Bounding scalar curvature for C0 metrics. method Inverse Mean Curvature Flow (IMCF) and Hawking mass monotonicity.
result Stability theorem for nonnegative scalar curvature.
Compactness theorem for manifolds with scalar curvature and entropy bounds.
problem Understanding the structure of manifolds with specific curvature and entropy bounds.
method Using volume upper bounds to prove Gromov-Hausdorff closeness to Euclidean balls.
result Unit balls in such manifolds are bi-Hölder and bi-W1,p homeomorphic to Euclidean balls. We overview main topics and ideas in spaces with their scalar curvatures bounded from below, and present a more detailed exposition of several known and some new geometric constraints on Riemannian spaces implied by the lower bounds on their scalar curvatures
In this paper we propose a class of local definitions of weak lower scalar curvature bounds that is well defined for C0 metrics. We show the following: that our definitions are stable under greater-than-second-order perturbation of the metric, that there exists a reasonable notion of a Ricci flow starting from C0…
In this note we reprove a theorem of Gromov using Ricci flow. The theorem states that a, possibly non-constant, lower bound on the scalar curvature is stable under C0-convergence of the metric.
The study finds a limit on the volume growth of certain 3-manifolds.
problem Volume growth of noncompact 3-manifolds with specific curvature properties.
method Analyzes 3-dimensional complete non-compact Riemannian manifolds with asymptotically nonnegative Ricci curvature and positive scalar curvature.
result Optimal asymptotic volume ratio for manifolds with finite first Betti number and linear volume growth.
On a pre-quantized symplectic manifold, we show that the symplectic Futaki invariant, which is an obstruction to the existence of constant Hermitian scalar curvature almost-Kähler metrics, is actually an asymptotic invariant. This allows us to deduce a lower bound for the L^2-norm of the Hermitian scalar curvature as o…
Upper bound for conjugate radius in open manifolds with scalar curvature and spectrum constraints.
problem Bounding the conjugate radius of open manifolds with specific curvature and spectrum conditions.
method Established an upper bound using scalar curvature and bottom-of-spectrum constraints.
result For certain conditions, the conjugate radius is no more than π.
In this paper we will give a rigorous proof of the lower bound for the scalar curvature of the standard solution of the Ricci flow conjectured by G. Perelman. We will prove that the scalar curvature R of the standard solution satisfies R(x,t)≥C0/(1−t)∀x∈R3,0≤t<1, for some constant $C_0>0…
Proves curvature comparison for Riemannian bands in low dimensions.
problem Curvature comparison in Riemannian bands with lower bounds.
method Uses warped products over scalar-flat manifolds with log-concave warping.
result Scalar and mean curvature comparison results proven.
Positive mass theorem for asymptotically flat manifolds with non-negative distributional scalar curvature
problem Positive mass theorem
method Ricci flow smoothing
result Asymptotically flat manifolds with non-negative ADM mass
New decay estimates for scalar curvature of steady gradient Ricci solitons.
problem Understanding scalar curvature behavior in steady gradient Ricci solitons.
method Using μ-bubbles introduced by Gromov.
result Provide new decay estimates for scalar curvatures.
We introduce a Z--coefficient version of Guth's macroscopic stability inequality for almost-minimizing hypersurfaces. In manifolds with a lower bound on macroscopic scalar curvature, we use the inequality to prove a lower bound on areas of hypersurfaces in terms of the Gromov simplicial norm of their homolog…
Smooth approximations bound dihedral angles of convex polytopes.
problem Bounding dihedral angles of convex polytopes.
method Approximating polytopes with smooth hypersurfaces and using geometric relations.
result Established lower bounds on dihedral angles.
Study connects manifold complexity to scalar curvature bounds.
problem Understanding the relationship between manifold complexity and scalar curvature.
method Combining quantitative operator K-theory, Lipschitz topological K-theory, and a vanishing theorem.
result Established a relationship between covering complexity and scalar curvature bounds.
Let M be a closed spin manifold which supports a positive scalar curvature metric. The set of concordance classes of positive scalar curvature metrics on M forms an abelian group P(M) after fixing a positive scalar curvature metric. The group P(M) measures the size of the space of positive scalar curvature metr…
In this note, we obtain a sharp volume estimate for complete gradient Ricci solitons with scalar curvature bounded below by a positive constant. Using Chen-Yokota's argument we obtain a local lower bound estimate of the scalar curvature for the Ricci flow on complete manifolds. Consequently, one has a sharp estimate of…
Study shows upper limit for torical band width with spectral curvature bounds.
problem Understanding the band width of torical bands with spectral curvature constraints.
method Used the warped \( μ\)-bubble method with spectral curvature bounds.
result Upper bound for the band width of torical bands is established.
Study shows no large mean curvature fill-ins for nonnegative scalar curvature.
problem Existence of fill-ins with nonnegative scalar curvature and large mean curvature.
method Analyzes closed Riemannian manifolds and scalar curvature properties.
result No closed Riemannian manifold admits a fill-in with nonnegative scalar curvature and large mean curvature.
Study shows rigidity of polyhedrons in hyperbolic spaces.
problem Rigidity of polyhedrons in hyperbolic spaces.
method Extending Gromov's comparison theory to metrics with negative scalar curvature lower bounds.
result Localization of the positive mass theorem for asymptotically hyperbolic manifolds.
The study examines how average scalar curvature influences geometric properties of Riemannian manifolds.
problem Investigating the geometric properties of Riemannian manifolds influenced by average scalar curvature.
method Analyzing the conjugate radius, average area of geodesic spheres, average volume of metric balls, and total volume of closed manifolds.
result Improves the Bishop-Gromov estimate on the average volume of metric balls and proves monotone decreasing properties of certain geometric integrals.
We show that the scalar curvature of a steady gradient Ricci soliton satisfying that the ratio between the square norm of the Ricci tensor and the square of the scalar curvature is bounded by one half, is boundend from below by the hyperbolic secant of one half the distance function from a fixed point.
Non-rigidity of hyperbolic manifold under scalar curvature constraints.
problem Non-rigidity of hyperbolic manifold under scalar curvature constraints.
method Compactly supported deformations, topological constraints.
result Non-rigidity under scalar curvature constraints, rigidity under topological constraints.
The paper bounds radii and curvatures in Riemannian manifolds.
problem Bounding radii and curvatures in Riemannian manifolds.
method Analyzing scalar curvature, injectivity radius, and mean curvature.
result Proves bounds on injectivity and focal radii under specific conditions.
We collect a few guesses on possible implications of a lower bound on the scalar curvature of a Riemannian manifold on the size and shape of this manifold.