Paper compares total quotient curvature and proves bounds for Einstein metric.
problem Comparing total quotient curvature and proving bounds for Einstein metric.
method Established comparison theorems and proved integral inequalities.
result Background Einstein metric achieves a sharp bound on total quotient curvature.
The study provides interior curvature estimates for convex graphs satisfying a specific quotient equation.
problem Interior curvature estimates for convex graphs.
method Analyzes convex graphs satisfying the quotient equation σn−2σn(λ)=f(X)>0. result Interior curvature estimates for convex graphs.
The study finds only spheres shrink self-similarly with quotient curvature speeds.
problem Characterizing self-similar shrinkers for quotient curvature speeds.
method Examined closed hypersurfaces shrinking with quotient curvature speeds.
result Only shrinking spheres are self-similar solutions.
Study shows infinitely many metrics with nonnegative sectional or positive Ricci curvature on specific 5D quotients.
problem Finding metrics with specific curvature properties on Brieskorn quotients.
method Analyzing moduli spaces of metrics with nonnegative sectional or positive Ricci curvature.
result Moduli spaces have infinitely many path components for both nonnegative sectional and positive Ricci curvature.
Paper solves curvature equations in Minkowski space for non-convex domains.
problem Solving curvature equations in non-convex domains of Minkowski space.
method Existence theorem proved via \emph{a priori} estimates and Serrin-type condition.
result Existence of solutions for curvature equations in non-convex domains.
In this paper, we study rigidity problems for hypersurfaces with constant curvature quotients H2kH2k+1 in the warped product manifolds. Here H2k is the k-th Gauss-Bonnet curvature and H2k+1 arises from the first variation of the total integration of $…
Characterizes orbifolds with upper curvature bounds as reflectofolds.
problem Understanding orbifolds with upper curvature bounds.
method Characterization through Alexandrov curvature and reflectofolds.
result Quotients of Riemannian manifolds by isometries have locally bounded curvature if and only if they are reflectofolds.
We study some cases when the sectional curvature remains positive under the taking of quotients by certain nonfree isometric actions of Lie groups. We consider the actions of the groups S1 and S3 such that the quotient space can be endowed with a smooth structure using the fibrations S3/S1≃S2 and $S^7…
The study examines the smoothness of submetries in Riemannian manifolds.
problem Regularity of submetries in Riemannian manifolds and their quotient spaces.
method Analysis of equidistant decompositions and quotient spaces.
result Strata of quotient spaces have curvature bounded from both sides.
We prove estimates for the sectional curvature of hyperkaehler quotients and give applications to moduli spaces of solutions to Nahm's equations and Hitchin's equations.
The paper examines gradient Ricci solitons on orbifolds and proves their rigidity properties.
problem The rigidity of positively curved gradient Ricci solitons on orbifolds.
method Analyzes scalar curvature, uses nonnegative curvature operator, κ-noncollapsed condition, and asymptotic quotient cylindrical properties.
result Steady gradient Ricci solitons on orbifolds with positive curvature are rigid and must be quotients of the Bryant soliton.
The study proves curvature bounds for quotient spaces of isometric actions.
problem Proving curvature bounds for quotient spaces of isometric actions.
method Disintegrate absolutely continuous measures and define a functional to prove curvature bounds.
result Necessary and sufficient conditions for Ricci curvature to be bounded below.
Paper investigates curvature problems and existence of solutions.
problem Existence of admissible solutions to curvature problems.
method Investigates curvature problems with prescribed Lp quotient type, proving existence under specific conditions. result Proves existence of admissible solutions without additional conditions.
Global convergence proved for Gursky-Malchiodi Q-curvature flow in dimensions n≥5.
problem Resolving the constant Q-curvature problem in dimensions n≥5. method Established a non-local version of the Łojasiewicz-Simon inequality for the Paneitz-Sobolev quotient, constructed test bubbles, and derived a stability inequality for the Paneitz-Sobolev quotient.
result Global convergence of the flow for arbitrary initial energy under the same positivity assumptions.
Study of a G2 cone quotient with U(1) action.
problem Understanding a G2 cone quotient with U(1) action. method Analyzing the quotient space R6 from R+imesCP3 with U(1) action, finding induced metrics and curvature/symplectic forms. result Closed expressions for the induced metric, curvature, and symplectic forms are found, invariant under SO(3) action. We show that a four-dimensional complete gradient shrinking Ricci soliton with positive isotropic curvature is either a quotient of S^4 or a quotient of S^3 cross R. This gives a clean classification result removing the earlier additional assumptions in [13] by Wallach and the second author.
The study classifies Riemannian manifolds with curvature nullity.
problem Classifying Riemannian manifolds with nontrivial curvature nullity.
method Classification theorems based on curvature nullity, scalar curvature, and quotient existence.
result New classification theorems and revisited previous results.
The study verifies a conjecture about homogeneous quotients of manifolds with positive curvature.
problem Verifying the Homogeneity Conjecture for manifolds with positive curvature.
method Examining globally homogeneous Riemannian quotients of homogeneous manifolds, focusing on those admitting a positive curvature metric.
result Further evidence supports the Homogeneity Conjecture for certain manifolds with positive curvature.
4D gradient solitons with constant curvature are rigid.
problem Characterizing 4D gradient Ricci solitons with constant scalar curvature.
method Proving rigidity using constant scalar curvature and quotient structures.
result 4D gradient solitons with constant curvature are rigid.
We prove that the quotient space of a variationally complete group action is a good Riemannian orbifold. The result is generalized to singular Riemannian foliations without horizontal conjugate points.
The study examines moduli spaces of metrics with positive Ricci or non-negative sectional curvature on sphere bundles.
problem Classifying and understanding moduli spaces of metrics with specific curvature properties on sphere bundles.
method Analyzing total spaces of S7-bundles over S8 and quotients of Milnor and Shimada spheres. result The moduli space of metrics has infinitely many path components.
Let (M,g) be a smooth Riemannian manifold and G a compact Lie group acting on M effectively and by isometries. It is well known that a lower bound of the sectional curvature of (M,g) is again a bound for the curvature of the quotient space, which is an Alexandrov space of curvature bounded below. Moreo…
The study constructs metrics with positive 2nd Ricci curvature on various manifolds.
problem Constructing metrics with positive 2nd Ricci curvature on closed manifolds.
method Generalization of the concept of fatness to ensure the existence of metrics with positive 2nd Ricci curvature on certain homogeneous bundles.
result Infinitely many examples of manifolds with positive 2nd Ricci curvature, including non-simply connected spaces.
Study on curvature properties and Shafarevich conjecture for complex hyperbolic manifolds.
problem Existence and nonexistence of Kähler metrics with nonpositive curvature on toroidal compactifications.
method Analysis of toroidal compactifications of finite volume complex hyperbolic manifolds, verification of Shafarevich conjecture.
result Verification of Shafarevich conjecture for compactifications of quotients of complex hyperbolic space by non-uniform arithmetic lattices.
Develops arithmetic PDE geometry using Fermat quotients.
problem Creating an arithmetic PDE analogue of Riemannian geometry.
method Using Fermat quotients and Frobenius elements in the absolute Galois group of a p-adic field. result Existence and uniqueness of geodesics and connections proved.
New proof for curved 3-cohom manifold rational ellipticity.
problem Rational ellipticity of curved manifolds with specific cohomogeneity.
method Proved rationally elliptic for cohomogeneity-three manifolds with positive curvature and no boundary quotient.
result Closed, simply connected, positively curved, cohomogeneity-three manifolds without boundary are rationally elliptic.
The Siu-Yang conjecture is reviewed for complex 2-manifolds.
problem Determining biholomorphic quotients of complex 2-balls.
method Review and analysis of existing results.
result Negative sectional curvature implies biholomorphic quotients of complex 2-balls.
We find complete hypersurfaces of constant curvature in hyperbolic space with a prescribed asymptotic boundary at infinity for a general class of (elliptic) curvature functions which includes the higher order mean curvatures and their curvature quotients.
We prove that a four-dimensional gradient shrinking Ricci soliton with δW±=0 is either Einstein, or a finite quotient of S3×R, S2×R2 or R4. We also prove that a four-dimensional cscK gradient Ricci soliton is either Kähler-Einstein, or a finite quotient of $M\times\…
New findings on shrinking solitons with positive isotropic curvature.
problem Characterizing shrinking solitons with positive isotropic curvature.
method Analyzing properties of gradient shrinking solitons in dimensions 5 and above.
result Non-flat complete shrinking solitons with positive isotropic curvature are quotients of the round sphere or the cylinder.
In this paper we prove new classification results for nonnegatively curved gradient expanding and steady Ricci solitons in dimension three and above, under suitable integral assumptions on the scalar curvature of the underlying Riemannian manifold. In particular we show that the only complete expanding solitons with no…
In this paper we study minimal and constant mean curvature (cmc) periodic surfaces in H^2 x R. More precisely, we consider quotients of H^2 x R by discrete groups of isometries generated by horizontal hyperbolic translations f and/or a vertical translation T. In the quotient by the Z^2 subgroup of the isometry group ge…
The study proves CR structures on specific three-manifolds are equivalent to standard structures.
problem Proving CR structures on three-manifolds are equivalent to standard structures.
method Analyzing Yamabe constant and total Q′-curvature to deduce CR equivalence. result Closed CR three-manifolds with certain curvature properties are equivalent to standard structures.
Paper solves Hessian quotient equations in Lorentz-Minkowski space with Dirichlet boundary conditions.
problem Existence and uniqueness of solutions to Hessian quotient equations in Lorentz-Minkowski space.
method Suitable settings to prove existence and uniqueness of solutions.
result Existence and uniqueness of solutions to the class of Hessian quotient equations.
The study proves unique static manifolds with positive scalar curvature and boundary.
problem Characterizing static three-manifolds with boundary and positive scalar curvature.
method Analyzing Ricci curvature bounds and quotient spaces.
result The only orientable quotient of the Nariai static manifold with boundary Nar−1,1(S2) is the only such manifold with connected boundary under certain conditions. The study examines 4D steady gradient Ricci solitons with nonnegative curvature away from a compact set.
problem Analyzing noncompact steady gradient Ricci solitons with nonnegative curvature operator.
method Examining the asymptotic behavior of noncompact κ-noncollapsed steady gradient Ricci solitons with nonnegative curvature operator away from a compact set.
result 4D noncompact κ-noncollapsed steady gradient Ricci solitons with nonnegative sectional curvature must be a Bryant Ricci soliton up to scaling.
We define and study the signature, A-hat genus and higher signatures of the quotient space of an S1-action on a closed oriented manifold. We give applications to questions of positive scalar curvature and to an Equivariant Novikov Conjecture.
We show that any closed biquotient with finite fundamental group admits metrics of positive Ricci curvature. Also, let M be a closed manifold on which a compact Lie group G acts with cohomogeneity one, and let L be a closed subgroup of G which acts freely on M. We show that the quotient N := M/L carries metrics of nonn…
The paper compares isoperimetric quotients and capacities in weighted manifolds.
problem Comparing isoperimetric quotients and capacities in weighted manifolds.
method Analysis of weighted Laplacian of the distance function and techniques for non-compact submanifolds.
result Parabolicity and hyperbolicity criteria for weighted manifolds.
Study convex capillary hypersurfaces with prescribed curvature in a spherical cap.
problem Prescribed curvature problem for convex capillary hypersurfaces.
method Reformulated as Hessian quotient equation with Robin boundary condition.
result Existence of strictly convex capillary hypersurface with prescribed curvature.
Study on convex capillary hypersurfaces with Lp curvature in half-space.
problem Prescribed Lp curvature for convex capillary hypersurfaces.
method Reduction to Hessian quotient equation with Robin boundary condition.
result Existence and uniqueness of smooth admissible solutions.
The Gromoll-Meyer sphere is constructed and studied in a homogeneous space.
problem Constructing and studying the Gromoll-Meyer sphere in a specific homogeneous space.
method Constructing a homogeneous isoparametric foliation and transnormal system on the quotient space, analyzing the induced metrics.
result The Gromoll-Meyer sphere has positive Ricci curvature and quasi-positive sectional curvature.
Indecomposable symmetric Lorentzian manifolds of non-constant curvature are called Cahen-Wallach spaces. Their isometry classes are described by continuous families of real parameters. We derive necessary and sufficient conditions for the existence of compact quotients of Cahen-Wallach spaces in terms of these paramete…
Study shows Sasaki solitons with harmonic Weyl tensor are spheres.
problem Characterizing gradient shrinking Sasaki-Ricci solitons.
method Integral curvature estimates and quotient analysis.
result Gradient shrinking Sasaki-Ricci solitons with harmonic Weyl tensor are finite quotients of spheres.
The paper studies quotient spaces of Spin(7) manifolds by S^1 actions and their G2 structures.
problem Understanding quotient spaces of Spin(7) manifolds by S^1 actions and their geometric properties.
method Derives equations relating intrinsic torsion of Spin(7)-structures to G2-structures, focusing on three torsion classes.
result Shows that the quotient space cannot have G2 holonomy unless the original manifold is a Calabi-Yau 4-fold.
We use the quaternion Kahler reduction technique to study old and new self-dual Einstein metrics of negative scalar curvature with at least a two-dimensional isometry group, and relate the quotient construction to the hyperbolic eigenfunction Ansatz. We focus in particular on the (semi-)quaternion Kahler quotients of (…
We study the classification of smooth toroidal compactifications of nonuniform ball quotients in the sense of Kodaira and Enriques. Moreover, several results concerning the Riemannian and complex algebraic geometry of these spaces are given. In particular we show that there are compact complex surfaces which admit Riem…
Researchers develop geodesics for a new metric on correlation matrices.
problem Lack of intrinsic tools for statistical analyses of correlation matrices.
method Developed geodesics for the quotient-affine metric on full-rank correlation matrices.
result Provided fundamental Riemannian operations for the quotient-affine metric.