Complete Finsler spaces with negative Ricci curvature are reversible.
problem Characterizing Finsler spaces with constant negative Ricci curvature.
method Utilizing projectively invariant pseudo-distance and Schwarzian derivative.
result Every connected complete Finsler space with constant negative Ricci scalar is reversible.
Sharp inequality in spaces with non-negative Ricci curvature.
problem Proving a sharp isoperimetric inequality in metric measure spaces.
method Using volume entropy in non-compact metric measure spaces with non-negative synthetic Ricci curvature.
result Proved a sharp dimension-free isoperimetric inequality.
Study the structure of Kähler foliations with negative Ricci curvature.
problem Characterize the structure of Kähler foliations with negative Ricci curvature.
method Prove a de Rham type theorem decomposition on the leaf space.
result Characterize each factor in the decomposition of the leaf space.
Optimal diameter estimates for 3D spaces with non-negative Ricci curvature.
problem Estimating the diameter of 3D spaces with non-negative Ricci curvature.
method Proving positive scalar curvature passes to Ricci limit spaces of non-negative curvature.
result Optimal Bonnet-Myers upper bound for 3D spaces.
Three-manifolds with non-negative pinched Ricci curvature have complete Ricci flows.
problem Proving Hamilton's pinching conjecture for three-manifolds.
method Ricci flow with scale-invariant curvature decay and pinching preservation.
result Hamilton's pinching conjecture is proven without additional hypotheses.
The paper bends Riemannian manifolds to achieve metrics with negative Ricci curvature.
problem Achieving metrics with negative Ricci curvature on closed Riemannian manifolds.
method Solving a fully nonlinear equation to conformally bend the manifold.
result Metrics of quasi-negative Ricci curvature are conformal to metrics with negative Ricci curvature.
Study on G2-structures with negative Ricci curvature on closed and noncompact manifolds.
problem Existence and properties of closed G2-structures with negative Ricci curvature. method Analyzing existence and nonexistence of closed G2-structures with negative Ricci curvature on closed and noncompact manifolds. result No closed manifold admits a closed G2-structure with negative Ricci curvature. For noncompact manifolds, restrictions on lengths of geodesics are found. Compact Kähler orbifolds with non-negative Ricci curvature are simply connected.
problem Understanding the topology of Kähler orbifolds with non-negative Ricci curvature.
method Proved orbifold versions of Kobayashi's theorem.
result Compact Kähler orbifolds with non-negative Ricci curvature are simply connected under certain conditions.
Smooth 3D flows from non-smooth starting points.
problem Creating smooth Ricci flows from non-smooth initial conditions.
method Generalized singular Ricci flow applied to 3D complete manifolds.
result Existence of smooth Ricci flows starting from non-smooth initial conditions.
Ricci flow deforms metrics with positive curvature to include negative curvature.
problem Preserving positive sectional curvature under Ricci flow in dimension four.
method Evolved cohomogeneity one metrics on S4 and CP2 via Ricci flow. result Metrics with positive sectional curvature lose this property under Ricci flow.
Sharp inequality for submanifolds in manifolds with non-negative Ricci curvature.
problem Establishing a Fenchel-Willmore inequality for submanifolds in manifolds with non-negative Ricci curvature.
method Analyzing submanifolds in manifolds with non-negative intermediate Ricci curvature and Euclidean volume growth.
result Sharp Fenchel-Willmore inequality for submanifolds in manifolds with non-negative intermediate Ricci curvature.
Survey on rigidity and almost rigidity of Green functions in non-negative Ricci curvature spaces.
problem Rigidity and almost rigidity of Green functions in non-negative Ricci curvature spaces.
method Survey and observation on Cheeger-Yau inequality on RCD spaces.
result Observations on the Cheeger-Yau inequality and its applications.
In this paper, we compare Ollivier Ricci curvature and Bakry-Émery curvature notions on combinatorial graphs and discuss connections to various types of Ricci flatness. We show that non-negativity of Ollivier Ricci curvature implies non-negativity of Bakry-Émery curvature under triangle-freeness and an additional in-de…
Extends Bochner's theorem to include small positive Ricci curvature.
problem Classical Bochner theorem limitations.
method Extensions with small positive Ricci curvature.
result Isometry group is finite for small positive curvature.
New proof shows certain 3D spaces are essentially like infinite space.
problem Characterizing 3D spaces with non-negative Ricci curvature.
method Integrable Ricci curvature, Sobolev inequality, spectral non-negativity.
result Proves complete Riemannian 3-manifolds are diffeomorphic to R3. Formal manifolds with non-negative Ricci curvature have formal covers.
problem Formality of manifolds with non-negative Ricci curvature.
method Study of universal covers and formal properties.
result Closed non-orientable manifolds with non-negative Ricci curvature are formal.
New Bochner technique for foliations with non-negative Ricci curvature.
problem Analyzing foliations with non-negative transverse Ricci curvature.
method Generalizing Bochner technique to foliations with non-negative transverse Ricci curvature.
result Obtained a new vanishing theorem for basic cohomology.
Study on Lie groups with negative Ricci curvature, including open questions and a new cone.
problem Understanding Lie groups with negative Ricci curvature metrics.
method Overview and introduction of a new cone C(n) for solvable Lie algebras.
result Introduction of a new open cone C(n) that parametrizes solvable Lie algebras with negative Ricci curvature metrics.
A version of the singular Yamabe problem in bounded domains yields complete conformal metrics with negative constant scalar curvatures. In this paper, we study whether these metrics have negative Ricci curvatures. Affirmatively, we prove that these metrics indeed have negative Ricci curvatures in bounded convex domains…
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 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.
Classifies Einstein metrics on R^4 with Heisenberg symmetry, finding incomplete Ricci-flat metrics and two complete negative-curvature examples.
problem Classifying Einstein metrics on R4 with Heisenberg symmetry. method Invariant under a four-dimensional group of isometries including the Heisenberg group, analyzing Ricci-flat and negative-curvature metrics.
result Found two complete negative-curvature examples: complex hyperbolic metric and one-loop deformed universal hypermultiplet.
Sharp isoperimetric inequality for Finsler manifolds with non-negative Ricci curvature.
problem Proving an isoperimetric inequality for Finsler manifolds with specific curvature properties.
method Analyzing measured Finsler manifolds with non-negative Ricci curvature and Euclidean volume growth.
result Sharp isoperimetric inequality and rigidity results for the inequality.
Graphs with bounded degrees and non-negative Ollivier-Ricci curvature have subexponential growth and diffusive random walk.
problem Understanding geometric properties of graphs with non-negative Ollivier-Ricci curvature.
method Analyzing the geometric properties of graphs with non-negative Ollivier-Ricci curvature, proving subexponential growth and diffusive random walk.
result For graphs with bounded degrees and non-negative Ollivier-Ricci curvature, the average log-volume growth and random walk displacement are subexponential.
Study improves understanding of Ricci curvature in manifolds.
problem Understanding Ricci curvature in manifolds with specific assumptions.
method Exploring m-intermediate Ricci curvature and proving comparison theorems.
result Stable weighted slicing in manifolds with non-negative m-intermediate Ricci curvature has almost non-negative Ricci curvature.
Study on stable minimal hypersurfaces under Ricci curvature constraints.
problem Stability of weighted minimal hypersurfaces under Ricci curvature bounds.
method Derive geometric consequences and prove a Schoen-Yau type criterion.
result Structure theorem for three-dimensional weighted manifolds of non-negative Ricci curvature.
For all complex dimensions n>=2, we construct complete Kaehler manifolds of bounded curvature and non-negative Ricci curvature whose Kaehler--Ricci evolutions immediately acquire Ricci curvature of mixed sign.
Sharp isoperimetric inequality on Finsler manifolds with non-negative Ricci curvature.
problem Proving an isoperimetric inequality on Finsler metric measure manifolds.
method Defining volume entropy and second Cheeger constant, proving sharp inequality.
result Sharp isoperimetric inequality involving volume entropy and weighted Ricci curvature.
The study shows finite measure-preserving isometry groups for certain metric measure spaces.
problem Understanding the structure of isometry groups in metric measure spaces.
method Analyzing synthetic negative Ricci curvature and Bakry-Émery Ricci curvature.
result The measure-preserving isometry group is finite for compact metric measure spaces with specific curvature conditions.
We study the behaviour of the normalized Kähler-Ricci flow on complete Kähler manifolds of negative holomorphic sectional curvature. We show that the flow exists for all time and converges to a Kähler-Einstein metric of negative scalar curvature, recovering a result of Wu and Yau.
The study proves a new inequality and formula for manifolds with non-negative Ricci curvature.
problem Proving a sharp mean value inequality for non-negative superharmonic functions.
method Develops a new sharp mean value inequality and an explicit formula for weighted scalar curvature.
result The new inequality removes the radius restriction of Schoen-Yau's result and provides an explicit formula for integral of weighted scalar curvature.
The paper proves finite topological type theorems for open manifolds with non-negative Ricci curvature and almost maximal local rewinding volume.
problem Proving finite topological type theorems for open manifolds with non-negative Ricci curvature and almost maximal local rewinding volume.
method The approach removes constraints of sectional curvature or conjugate radius and extends to previous related studies.
result Theorems are proven for open manifolds with non-negative Ricci curvature and almost maximal local rewinding volume, without the need for triangle comparison of Toponogov type.
The paper proves isoperimetric inequalities in manifolds with small negative Ricci curvature.
problem Proving isoperimetric inequalities in manifolds with small negative Ricci curvature.
method Expanding on the ABP method, the paper uses the elliptic Kato constant to control the non-negativity of the Ricci-tensor and applies techniques from Li-Tam and Kasue.
result Sharp isoperimetric inequalities in the limit are proven in the presence of small negative curvature.
Study derivations for nilpotent Lie algebras with negative Ricci curvature.
problem Characterize derivations leading to solvable extensions with negative Ricci curvature.
method Investigate the space of diagonalizable derivations for specific Lie algebras.
result Prove conjecture about derivations in dimension 5 and for Heisenberg and standard filiform Lie algebras.
We study minimal hypersurfaces in manifolds of non-negative Ricci curvature, Euclidean volume growth and quadratic curvature decay at infinity. By comparison with capped spherical cones, we identify a precise borderline for the Ricci curvature decay. Above this value, no complete area-minimizing hypersurfaces exist. Be…
The study uses Ricci flow to prove flatness of certain Riemannian manifolds.
problem Proving the flatness of Riemannian manifolds with specific curvature properties.
method Ricci flow approach, quantitative existence theory, curvature estimates, and regularization.
result Manifolds with non-negative curvature and specific decay rates are necessarily flat.
Two remarks on curvature properties of Kähler manifolds.
problem Curvature properties of Kähler manifolds.
method Analyzing semi-positive holomorphic sectional curvature and quasi-negative k-Ricci curvature. result For semi-positive holomorphic sectional curvature, the rational dimension of the MRC fibration equals the number of non-truly-flat directions. For quasi-negative k-Ricci curvature, the canonical bundle is ample. Lower bounds for eigenvalues on manifolds with negative Ricci curvature.
problem Estimating eigenvalues on non-compact manifolds with negative Ricci curvature.
method Using a one-dimensional differential equation model to bound the principal p−frequency. result The lower bound for the principal p−frequency is sharp and depends on the diameter and curvature. We show that under Ricci curvature integral assumptions the dimension of the first cohomology group can be estimated in terms of the Kato constant of the negative part of the Ricci curvature. Moreover, this provides quantitative statements about the cohomology group, contrary to results by Elworthy and Rosenberg.
Proves uniqueness of Ricci flow with scaling invariant estimates.
problem Proving uniqueness of Ricci flow with scaling invariant curvature bound.
method Solving Ricci-harmonic map heat flow in unbounded curvature background.
result Complete Ricci flow starting from uniformly non-collapsed, non-negatively curved manifold is unique in dimension three.
Lower bounds for eigenvalues on manifolds with negative Ricci curvature.
problem Estimating eigenvalues on non-compact manifolds with negative Ricci curvature.
method Using a one-dimensional differential equation model, the paper establishes a lower bound for the principal p−frequency. result The lower bound for the principal p−frequency is sharp and depends on the diameter and curvature. The Wu-Yau theorem is proven for Sasakian manifolds with specific curvature conditions.
problem Proving properties of Sasakian manifolds with negative transverse holomorphic sectional curvature.
method Analyzing the curvature properties and applying the Wu-Yau theorem.
result Compact Sasakian manifolds with negative transverse holomorphic sectional curvature have negative transverse Ricci curvature.
Along the line of the Yang Conjecture, we give a new estimate on the lower bound of the first non-zero eigenvalue of a closed Riemannian manifold with negative lower bound of Ricci curvature in terms of the in-diameter and the lower bound of Ricci curvature.
The paper generalizes a Steklov eigenvalue inequality for substatic triples under non-negative Ricci curvature.
problem Estimating Steklov eigenvalues for substatic triples under non-negative Ricci curvature.
method Generalization of Fraser-Li type inequality for substatic triples under non-negative Ricci curvature associated with an affine connection.
result The paper provides a new inequality for Steklov eigenvalues of substatic triples.
Log-Sobolev inequality proven for submanifolds in specific types of manifolds.
problem Proving Log-Sobolev inequality for submanifolds in asymptotic non-negative intermediate Ricci curvature manifolds.
method Extending previous results, proving inequality for submanifolds in specific types of manifolds.
result Sharp Log-Sobolev inequality proven for submanifolds in complete non-compact Riemannian manifolds with asymptotic non-negative intermediate Ricci curvature and Euclidean volume growth.
The paper proves a quantitative rigidity result for spaces with specific curvature bounds.
problem Understanding the rigidity of spaces with almost maximal volume entropy.
method Analyzing Riemannian manifolds and RCD-spaces with specific curvature conditions. result Spaces with almost maximal volume entropy are closely related to hyperbolic space forms.
We prove that the space of smooth Riemannian metrics on the three-ball with non-negative Ricci curvature and strictly convex boundary is path connected; and, moreover, that the associated moduli space (i.e., modulo orientation-preserving diffeomorphisms of the three-ball) is contractible. As an application, using resul…
In this paper we study the evolution of almost non-negatively curved (possibly singular) three dimensional metric spaces by Ricci flow. The non-negatively curved metric spaces which we consider arise as limits of smooth Riemannian manifolds (M_i,g_i), i \in N, whose Ricci curvature is not less than -c^2(i), where c^2(i…