Proves almost flat manifolds with mixed curvature bounds.
problem Finding structures with mixed curvature bounds.
method Mixed curvature analogue of Gromov's almost flat manifolds theorem.
result Proves upper and lower curvature bounds for almost flat manifolds.
The paper studies geometric PDEs for flatness on Riemannian manifolds.
problem Understanding flatness in geometric PDEs.
method Study geometric PDEs of connection-flatness, curvature-flatness, Ricci-flatness, scalar curvature-flatness.
result Introduce new Theorems about flatness in Differential Geometry.
New manifolds with negative curvature limit to one with negative curvature.
problem Preserving nonnegative scalar curvature under intrinsic flat convergence.
method Constructing sequences of manifolds with positive scalar curvature.
result Intrinsic flat limit of manifolds with negative scalar curvature.
Statistical manifolds with constant curvature are projectively flat and symmetric.
problem Characterizing statistical manifolds with constant curvature.
method Analyzing the curvature and projective flatness properties of statistical manifolds.
result Statistical manifolds with constant curvature are projectively flat and symmetric.
The paper classifies special types of contact metric manifolds with curvature conditions.
problem Classifying N(κ)-contact metric manifolds with specific curvature tensors. method Examining flatness conditions on T-curvature tensor and analyzing specific curvature tensors. result A classification of N(κ)-contact metric manifolds under various curvature conditions. Piecewise flat approximations for curvature in Euclidean and non-Euclidean spaces.
problem Approximating local extrinsic curvature on discrete manifolds.
method Constructing discrete curvature forms on piecewise flat manifolds, using weighted sums of hinge angles.
result Converges to smooth curvature values as mesh refinement occurs, favorably comparing with other discrete approaches.
Study gap phenomenon in flat manifolds with Ricci curvature.
problem Understanding curvature decay in flat manifolds.
method Construct solutions to Yamabe flow and analyze curvature decay.
result If curvature decays quickly, manifold must be flat.
Ricci flow on flat manifolds converges to Euclidean space under curvature pinching.
problem Curvature pinching on asymptotically flat manifolds.
method Ricci flow on asymptotically flat manifolds with integral curvature pinching.
result Ricci flow converges to flat Euclidean space for sufficiently pinched curvature.
Flat space for manifolds with tiny curvature.
problem Understanding manifolds with curvature concentration.
method Analyzing non-compact manifolds with non-negative Ricci curvature and small curvature concentration.
result Manifolds with curvature concentration are flat.
The paper examines flatness conditions on normal metric contact pairs and proves properties of Einstein manifolds.
problem The study of flatness conditions on normal metric contact pairs.
method Analysis of conformal, concircular, and quasi-conformal curvature tensors.
result Normal metric contact pair manifolds with flat conformal, concircular, and quasi-conformal curvature tensors are Einstein manifolds with specific scalar and sectional curvatures.
Study on the regularity of p-Gauss curvature flow near flat interfaces.
problem Regularity of p-Gauss curvature flow near flat interfaces. method Analysis of convex hypersurface near the interface.
result Regularity of the convex hypersurface near the interface.
Flat Ricci-flat manifolds with bounded gradient of Green function are flat.
problem Understanding the rigidity of Ricci-flat manifolds with specific curvature decay.
method Analyzing the gradient of the Green function and using curvature decay conditions.
result Flat Ricci-flat manifolds with bounded gradient of Green function are flat.
The paper defines curvature at infinity for flat manifolds.
problem Defining curvature at the boundary of flat manifolds.
method Constructing coordinates at infinity for asymptotically flat ends.
result A Weyl tensor and renormalized volume defined at infinity.
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 $(…
Flat surfaces in Lie groups with constant curvature are flat.
problem Characterizing surfaces in Lie groups with constant Gaussian curvature.
method Analyzing surfaces as products of curves and using bi-invariant metrics.
result All surfaces of constant curvature in 3D Lie groups are flat.
Classifies special submanifolds with specific curvature properties.
problem Classifying submanifolds with constant Moebius curvature and flat normal bundle.
method Analyzes isometric immersions with constant Moebius curvature and flat normal bundle.
result Classifies submanifolds with these curvature properties.
The paper explores curvatures on graphs and their implications for Ricci flatness.
problem Comparing and understanding different curvature notions on graphs and their implications for Ricci flatness.
method Analyzing Ollivier Ricci curvature and Bakry-Émery curvature on combinatorial graphs, investigating graph products, and proving curvature properties.
result Non-negativity of Ollivier Ricci curvature implies non-negativity of Bakry-Émery curvature under specific conditions.
In Finsler geometry, there are infinitely many models of constant curvature. The Funk metrics, the Hilbert-Klein metrics and the Bryant metrics are projectively flat with non-zero constant curvature. A recent example constructed by the author is projectively flat with zero curvature. In this paper, we introduce a techn…
Warped tori with almost non-negative scalar curvature converge to a flat torus.
problem Understanding the behavior of warped product metrics on a 3-torus.
method Analyzing sequences of warped product metrics with specific curvature and volume bounds.
result A subsequence of warped product metrics converges to a flat torus.
Study flat flow solutions to Mullins-Sekerka and area-preserving curvature flows on planar flat torus.
problem Behavior of flat flow solutions on planar flat torus.
method Sharp quantitative Alexandrov inequality derivation for periodic smooth sets.
result Flat flows converge to specific configurations exponentially fast.
Classifies Einstein submanifolds with flat normal bundle and parallel mean curvature.
problem Classifying Einstein submanifolds in space forms.
method Extending previous results for isometric immersions of Riemannian manifolds with constant sectional curvature.
result Classification of Einstein submanifolds in space forms with flat normal bundle and parallel mean curvature.
Constructs metrics with negative curvature on specific manifold types.
problem Creating negatively curved metrics on locally conformally flat manifolds.
method Using Morse functions to construct conformal metrics.
result Successfully constructs conformal metrics with negative sectional curvature.
Yamabe flow proves compactness of certain locally conformally flat manifolds with positive Ricci curvature.
problem Proving compactness of locally conformally flat manifolds with positive Ricci curvature.
method Using the Yamabe flow to prove compactness.
result Locally conformally flat manifolds with positive pinched Ricci curvature are compact.
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. New graphs with maximum degree 4 found to be Ricci-flat.
problem Characterizing Ricci-flat graphs with maximum degree 4.
method Defined Ricci curvature on graphs and used previous results to find all such graphs.
result All Ricci-flat graphs with maximum degree at most 4 were determined.
The paper investigates properties of the σ₂-curvature and its implications on metric rigidity.
problem Understanding the properties and rigidity of metrics related to the σ₂-curvature.
method Introducing a symmetric 2-tensor, defining σ₂-singular spaces, and proving rigidity results.
result Proves that a metric has to be flat if it is close to a flat metric under certain conditions.
In this note we study constant mean curvature surfaces in asymptotically flat 3-manifolds. We prove that, in an asymptotically flat 3-manifold with positive mass, stable spheres of given constant mean curvature outside a fixed compact subset are unique. Therefore we are able to conclude that there is a unique foliation…
Small mass implies a bilipschitz diffeomorphism to flat space
problem Given a 3-dimensional asymptotically flat manifold with non-negative scalar curvature and L2-norm of the curvature tensor at most 1, if the mass is small, is there a bilipschitz diffeomorphism from the manifold to the flat Euclidean space? method Using previous work
result A strong positive answer to the problem
Study shows tori metrics converging to flat under specific conditions.
problem Understanding convergence of metrics on tori with non-negative scalar curvature.
method Uniformly conformal metrics and controlled geometry sequences.
result Sequence of metrics converges to flat metric in multiple senses.
Flat open manifolds with full first Betti number have zero curvature.
problem Maximal first Betti number rigidity for open manifolds with nonnegative Ricci curvature.
method Proving rigidity for open manifolds with specific curvature conditions and Betti numbers.
result Open manifolds with maximal first Betti number are flat.
In this article, we investigate deformation problems of Q-curvature on closed Riemannian manifolds. One of the most crucial notions we use is the Q-singular space, which was introduced by Chang-Gursky-Yang during 1990's. Inspired by the early work of Fischer-Marsden, we derived several results about geometry relate…
Detect spacetime curvature with event causality measurements.
problem Detecting spacetime curvature without rulers and clocks.
method Prove spacetime non-flatness through causal relations.
result Sixteen measurements verify non-flatness of non-conformally flat spacetimes.
The paper examines Einstein warped product manifolds and finds a flat Base-manifold condition.
problem Investigating conditions for (2+2)-Einstein warped product manifolds. method Analyzing the (2+2)-Einstein warped product manifolds with specific curvature conditions. result The f-curvature-Base condition is equivalent to a flat Base-manifold. The paper classifies PMCV hypersurfaces in non-flat pseudo-Riemannian space forms.
problem Characterizing PMCV hypersurfaces in non-flat pseudo-Riemannian space forms.
method Analyzing the properties of hypersurfaces with at most two distinct principal curvatures.
result PMCV hypersurfaces are either minimal or locally isoparametric.
The paper proves volume comparison theorems for metrics near stable Einstein and Ricci flat manifolds.
problem Investigating volume comparison theorems for metrics near stable Einstein and Ricci flat manifolds.
method Applying similar techniques to derive local rigidity theorems for strictly stable Ricci flat manifolds.
result Derives local rigidity theorems for strictly stable Ricci flat manifolds with respect to σ2-curvature.
Proves a special type of submanifolds in a curved space.
problem Characterizing submanifolds with specific properties in a curved space.
method Uses the properties of flat normal bundle and parallel mean curvature to prove the submanifolds are warped products.
result Einstein submanifolds with flat normal bundle and parallel mean curvature are warped product of isometric immersions.
The paper proves constant mean curvature surfaces in specific manifold types.
problem Existence of surfaces with constant mean curvature in asymptotically flat and hyperbolic manifolds.
method Combines min-max theory with inverse mean curvature flow.
result Existence of compact surfaces with constant mean curvature in asymptotically flat and hyperbolic manifolds.
New metrics with non-negative scalar curvature are always Ricci-flat on certain surgeries.
problem Understanding metrics with non-negative scalar curvature on surgeries of manifolds.
method Analyzing spin surgeries and their impact on metrics with non-negative scalar curvature.
result Complete metrics with non-negative scalar curvature are Ricci-flat on certain surgeries.
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.
We study curvature restrictions of Levi-flat real hypersurfaces in complex projective planes, whose existence is in question. We focus on its totally real Ricci curvature, the Ricci curvature of the real hypersurface in the direction of the Reeb vector field, and show that it cannot be greater than -4 along a Levi-flat…
Surface diffusion and mean curvature flows converge to stable critical sets in flat tori.
problem Stability of surface diffusion and mean curvature flows in flat tori.
method Existence and convergence of flows starting close to stable critical sets, proven for all times.
result Flows converge exponentially fast to stable critical sets in flat tori.
Flow of spacelike hypersurfaces converges to flat slice in asymptotically flat spacetimes.
problem Long-time behavior of mean curvature flow in asymptotically flat spacetimes.
method Analysis of mean curvature flow in Lorentzian product manifolds.
result Mean curvature flow converges uniformly to a flat slice as time goes to infinity.
In this note, we consider the isoperimetric inequality on an asymptotically flat manifold with nonnegative scalar curvature, and improve it by using Hawking mass. We also obtain a rigidity result when equality holds for the classical isoperimetric inequality on an asymptotically flat manifold with nonnegative scalar cu…
The study examines complete conformally flat submanifolds with nullity in Euclidean space.
problem Investigating properties of conformally flat submanifolds with nullity.
method Analyzing the index of relative nullity and scalar curvature to deduce manifold properties.
result Conditions for the manifold to be flat and the immersion to be a cylinder over a submanifold.
A piecewise flat Finsler metric on a triangulated surface M is a metric whose restriction to any triangle is a flat triangle in some Minkowski space with straight edges. One of the main purposes of this work is to study the properties of geodesics on a piecewise flat Finsler surface, especially when it meets a vertex…
Study on immersions with flat normal bundle in curved spaces.
problem Behavior of isometric immersions with negative curvature.
method Investigation of second fundamental form growth in space forms.
result Second fundamental form grows exponentially if normal bundle is flat.
New result on Levi-flat hypersurfaces' normal bundles without positive curvature.
problem Understanding Levi-flat hypersurfaces' normal bundles and their curvature properties.
method Analyzing the normal bundle of Levi-flat real hypersurfaces in complex manifolds.
result The normal bundle to the Levi foliation does not admit a Hermitian metric with positive curvature.
The paper proves convergence of graphs of functions to flat tori under certain curvature conditions.
problem Stability of graphical tori with scalar curvature approaching zero.
method Adapting results from previous works, the paper proves convergence of sequences of graphs of functions to flat tori under specific curvature and diameter bounds.
result Graphical tori with scalar curvature approaching zero converge to flat tori.