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.
Study nilpotent Lie algebras with Ricci negative solvable extensions.
problem Understanding curvature behavior in Ricci negative solvmanifolds.
method Characterize derivations of nilpotent Lie algebras leading to Ricci negative solvable extensions.
result Found unexpected behaviors in nilpotent Lie algebras with Ricci negative solvable extensions.
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.
Study shows Kähler-Ricci flow on manifolds with negative curvature converges to a Kähler-Einstein metric.
problem Analyzing the behavior of Kähler-Ricci flow on manifolds with negative holomorphic curvature.
method Investigates the normalized Kähler-Ricci flow on complete Kähler manifolds of negative holomorphic sectional curvature.
result The flow exists for all time and converges to a Kähler-Einstein metric of negative scalar 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. 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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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. Local Ricci flow under negative curvature conditions, with applications to metric space smoothing.
problem Finding solutions to Ricci flow under local negative curvature conditions.
method Local solution to Ricci flow equation with uniform existence time and bounded curvature.
result Local Ricci flow exists for a uniform time with bounded curvature, generalizing previous results.
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.
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.
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.
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…
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.
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.
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…
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.
New results on tori restrict sectional curvature when Ricci curvature is negative and bounded.
problem Restricting sectional curvature on tori with mixed Ricci bounds.
method Using Lohkamp's theorem and explicit constants.
result Explicit constants show sectional curvature is positive in some directions.
Complete negative Kähler-Einstein metric found on Stein manifolds.
problem Existence of complete Kähler-Einstein metrics on Stein manifolds with negative curvature.
method Normalized Kähler-Ricci flow to deform metrics to complete negative Kähler-Einstein metric.
result Existence of complete negative Kähler-Einstein metric on Stein manifolds with negatively pinched holomorphic sectional curvature.
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 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.
New non-solvable Lie groups found with negative Ricci curvature.
problem Finding new Lie groups with negative Ricci curvature.
method Using a general construction from a previous article, the authors produce metric Lie algebras with negative Ricci curvature for compact semisimple Lie algebras.
result The constructed Lie algebras have negative Ricci curvature for all but finitely many finite-dimensional irreducible representations of the Lie algebra.
The paper provides gradient estimates for heat kernels on manifolds with negative Ricci curvature.
problem Estimating gradients of heat kernels on manifolds with negative Ricci curvature.
method Pointwise and Lp gradient estimates, uniform boundedness results for the heat operator of the Hodge Laplacian. result Uniform boundedness results and gradient estimates for heat kernels and Hodge Laplacian.
Directly proves Li-Yau estimates on manifolds with negative Ricci curvature.
problem Proving Li-Yau estimates on manifolds with negative Ricci curvature.
method Uses classical maximum principle argument and Hamilton's techniques.
result Directly proves sharp Li-Yau estimates simplifying previous methods.
Study on gradient steady Kähler Ricci solitons with specific curvature properties.
problem Characterize gradient steady Kähler Ricci solitons with non-negative Ricci curvature and integrable scalar curvature.
method Analyzing the structure of these solitons and their universal covering spaces.
result Gradient steady Kähler Ricci solitons are quotients of specific manifolds.
We give examples of pinched negatively curved manifolds for which the Ricci flow does not converge smoothly.
In this paper we announce the following result: ``Every manifold of dimension ≥3 admits a complete negatively Ricci curved metric.'' Furthermore we describe some sharper results and sketch proofs.
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.
Constructs Lie groups with negative Ricci curvature.
problem Finding Lie groups with negative Ricci curvature.
method Constructs Lie groups with specific algebraic structures and representations.
result Proves existence of Lie groups with negative Ricci curvature for various Levi factors.
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.
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.
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.
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 study proves triviality and rigidity results for Ricci solitons and estimates their conjugate radius.
problem Understanding the properties and behavior of Ricci solitons.
method Analytical proofs and estimates for various types of Ricci solitons.
result Upper bounds and estimates for conjugate radius of Ricci solitons.
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.
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 finds minimal hypersurface in convex manifolds with non-negative Ricci curvature.
problem Finding minimal hypersurfaces in manifolds with specific curvature properties.
method Min-max method applied to one-parameter families of hypersurfaces.
result The min-max minimal hypersurface is orientable, of index one and multiplicity one.
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…