Study examines noncompact cases of Gauss Curvature Flow on revolution surfaces.
problem Noncompact cases of Gauss Curvature Flow on revolution surfaces.
method Examines two noncompact cases of Gauss Curvature Flow on revolution surfaces.
result Examines noncompact cases of Gauss Curvature Flow on revolution surfaces.
Classification extended for reducible symmetric spaces.
problem Classifying homogeneous foliations on symmetric spaces.
method Extended classification from irreducible to reducible symmetric spaces.
result Classification completed for all noncompact symmetric spaces.
Study gives conditions for noncompact manifolds to have positive scalar curvature.
problem Conditions for noncompact manifolds to have positive scalar curvature.
method Analyzes finite and infinite volume cases to find obstructions.
result Provides conditions for noncompact manifolds to admit metrics with positive scalar curvature.
Study extends holomorphic forms on noncompact Kahler manifolds.
problem Extension of holomorphic canonical forms on noncompact Kahler manifolds.
method L2 analytic methods and L2 Hodge theory.
result Generalizes classical results to noncompact cases.
Study shows scalar curvature bounds for noncompact foliations.
problem Bounding scalar curvature on noncompact foliations.
method Analyzes scalar curvature on enlargeable Riemannian manifolds and foliations.
result Infimum of leafwise scalar curvature is non-positive.
Study mapping class groups of infinite type surfaces with noncompact boundaries.
problem Classify pure mapping class groups of infinite type surfaces.
method Developed a method to cut surfaces into simpler ones and combined recent results.
result Complete classification of perfect and uniformly perfect pure mapping class groups.
The Lichnerowicz conjecture asserts that all harmonic manifolds are either flat or locally symmetric spaces of rank 1. This conjecture has been proved by Z.I. Szabo for harmonic manifolds with compact universal cover. E. Damek and F. Ricci provided examples showing that in the noncompact case the conjecture is wrong. H…
Study extends convergence results to noncompact Ricci flows.
problem Convergence of noncompact nonsingular solutions of the Ricci flow.
method Extends convergence results from compact to noncompact cases.
result Blow up limit of finite time singularity is a gradient shrinking soliton.
In this paper the Gromov-Witten invariants on a class of noncompact symplectic manifolds are defined by combining Ruan-Tian's method with that of McDuff-Salamon. The main point of the arguments is to introduce a method dealing with the transversality problems in the case of noncompact manifolds. Moreover, the technique…
We give a definition of the fractional Laplacian on some noncompact manifolds, through an extension problem introduced by Caffarelli-Silvestre. While this definition in the compact case is straightforward, in the noncompact setting one needs to have a precise control of the behavior of the metric at infinity and geomet…
Let $\M$ be a complete, connected noncompact manifold with bounded geometry. Under a condition near infinity, we prove that the Log Sobolev functional (\ref{logfanhan}) has an extremal function decaying exponentially near infinity. We also prove that an extremal function may not exist if the condition is violated. This…
Study on solitary waves on noncompact manifolds, proving existence and stability, with instability in most cases.
problem Existence and stability of solitary waves on noncompact manifolds.
method Existence via perturbation, stability via spectral analysis of linearized operator, Vakhitov-Kolokolov criterion for instability.
result Existence and instability of solitary waves on most noncompact manifolds, with numerical evidence.
Study calculates mass for certain hyperbolic spaces with noncompact boundaries.
problem Calculating mass for specific types of hyperbolic spaces.
method Defined a mass-type invariant for asymptotically hyperbolic manifolds with noncompact boundaries.
result Proved a sharp positive mass inequality under suitable conditions.
Proves symplectomorphism of noncompact manifolds with embedded rays.
problem Symplectic manifolds with embedded rays.
method Constructs explicit symplectomorphisms using excision trick.
result Symplectomorphic to complement of the ray in standard Euclidean space.
We find upper and lower bounds for the first eigenvalue and the volume entropy of a noncompact real analytic Kähler manifold, in terms of Calabi's diastasis function and diastatic entropy, which are sharp in the case of the complex hyperbolic space. As a corollary we obtain explicit lower bounds for the first eigenvalu…
A theorem on odd dimensional noncompact manifolds shows curvature bounds.
problem Understanding curvature bounds on noncompact manifolds.
method Analyzing scalar curvature and maps with specific properties.
result Curvature lower bound is negative under given conditions.
Extends fixed point theorem to noncompact manifolds using KK-theory.
problem Generalizing fixed point theorem to noncompact manifolds.
method Using KK-theory, extends equivariant index to noncompact setting.
result Obtains fixed point formula for noncompact manifolds.
Derives heat equation estimates linked to Ricci flow on compact and noncompact manifolds.
problem Estimating heat equation coupled to Ricci flow on noncompact manifolds.
method Local derivative estimates for the heat equation coupled to the Ricci flow.
result Extends results on distance distortion and backward pseudolocality to noncompact manifolds.
Paper constructs Witten and elliptic genera for noncompact manifolds with proper actions.
problem Constructing genera for noncompact manifolds with proper actions.
method Using proper cocompact actions by almost connected Lie groups, the paper constructs and proves properties of Witten and elliptic genera.
result Vanishing and rigidity results generalizing known compact group actions.
Formula for fixed points on noncompact spaces.
problem Calculating fixed points on noncompact manifolds.
method Equivariant index theorem, localised functional, asymptotically local operators.
result Obtained a new Lefschetz fixed-point formula.
RG-2 flow reveals horizons in noncompact black hole metrics.
problem Understanding the evolution of black hole metrics under RG-2 flow.
method Adapted proofs for asymptotically flat case, analyzed horizons and monotonicity of entanglement entropy.
result Entanglement entropy is monotonic under RG-2 flow, related to horizon appearance.
Researchers prove Weyl laws for Schrödinger operators on noncompact manifolds.
problem Proving Weyl laws for Schrödinger operators on noncompact manifolds.
method Heat kernel asymptotics, Karamata-Hardy-Littlewood Tauberian theorem, and semiclassical analysis.
result Established both classical and semiclassical Weyl laws for Schrödinger operators on noncompact manifolds.
Mass calculation extended to manifolds with noncompact boundaries.
problem Computing mass in asymptotically flat and hyperbolic manifolds with boundaries.
method Coordinate-free approach using Einstein and Newton tensors.
result Mass can be computed as surface integrals involving these tensors.
We study boundary value problems for the Dirac operator on Riemannian Spinc manifolds of bounded geometry and with noncompact boundary. This generalizes a part of the theory of boundary value problems by C. Bär and W. Ballmann for complete manifolds with closed boundary. As an application, we derive the lower bound …
Note on Einstein-type manifolds with constant curvature.
problem Characterizing Einstein-type manifolds with constant scalar curvature.
method Analyzing properties of gradient Einstein-type manifolds, including compact and noncompact cases.
result Nontrivial gradient Einstein-type manifolds are isometric to the standard sphere.
We obtain some nonexistence results for complete noncompact stable hyppersurfaces with nonnegative constant scalar curvature in Euclidean spaces. As a special case we prove that there is no complete noncompact strongly stable hypersurface M in R4 with zero scalar curvature S2, nonzero Gauss-Kronecker…
Investigates noncompact warped product Ricci solitons, proving nonexistence results.
problem Proving nonexistence results for noncompact warped product Ricci solitons.
method Proves nonexistence results using pde estimates and properties of the first eigenvalue of a weighted Laplacian.
result Nonexistence theorem for shrinking Ricci solitons.
Generalizes Roe's theorem to noncompact hypersurfaces.
problem Obtaining index theorems for noncompact manifolds.
method Extending Roe's partitioned manifold index theorem to noncompact hypersurfaces.
result Equality between two classes in the K-theory of the Roe algebra of N.
Symplectic stability holds for noncompact manifolds with specific end conditions.
problem Symplectic stability on noncompact manifolds with cylindrical ends.
method Study of symplectic forms on noncompact manifolds with cylindrical ends, showing stability under a growth condition.
result Symplectic stability holds for noncompact manifolds with cylindrical ends under a natural growth condition.
Paper proves new inequalities for Einstein-Maxwell data sets.
problem Establishing area-charge inequalities for Einstein-Maxwell initial data sets.
method Applying Gromov's μ-bubble technique in a new geometric context.
result Novel rigidity theorems for noncompact Einstein-Maxwell data sets.
In this note, we consider the Dirac operator D on a Riemannian symmetric space M of noncompact type. Using representation theory we show that D has point spectrum iff the A^-genus of its compact dual does not vanish. In this case, if M is irreducible then M=U(p,q)/U(p)×U(q) with p+q odd, and …
New classification of gradient steady Ricci solitons with vanishing D-tensor.
problem Classifying gradient steady Ricci solitons with specific properties.
method Extending Cao-Chen's work on Bach-flat gradient Ricci solitons, proving properties for D-flat solitons. result Any n-dimensional complete noncompact gradient steady Ricci soliton with vanishing D-tensor is either Ricci-flat or isometric to the Bryant soliton. Study on ancient Ricci flows with positive curvature, proving noncollapsedness.
problem Characterizing ancient Ricci flows with positive sectional curvature.
method Analyzing complete and noncompact Type I ancient Ricci flows with positive sectional curvature.
result Ancient solutions are noncollapsed on all scales in complete and noncompact cases, and in even-dimensional closed cases.
Researchers compute the full spectrum of Laplace operator on distance spheres in symmetric spaces.
problem Computing the full Laplace spectrum on distance spheres in symmetric spaces.
method Lie-theoretic methods to explicitly compute the spectrum.
result Unified formula for the full spectrum of Laplace operator on distance spheres in symmetric spaces of rank one.
Study of irreversible metric-measure spaces, proving convergence and stability results.
problem Understanding Gromov-Hausdorff convergence and stability in noncompact irreversible metric-measure spaces.
method Introducing a nondecreasing function to bound reversibility of larger balls, proving convergence/stability results in Gromov-Hausdorff topology.
result Satisfactory convergence/stability results in Gromov-Hausdorff topology for various irreversible spaces, including Finsler manifolds.
Under suitable conditions near infinity and assuming boundedness of curvature tensor, we prove a no breathers theorem in the spirit of Ivey-Perelman for some noncompact Ricci flows. These include Ricci flows on asymptotically flat (AF) manifolds with positive scalar curvature. Since the method for the compact case face…
Study on noncompact steady quasi-Einstein manifolds with specific tensor conditions.
problem Classifying noncompact steady quasi-Einstein manifolds with vanishing Weyl tensor condition.
method Analyzing manifolds with nonnegative Ricci curvature and zero radial Weyl curvature under fourth-order divergence-free Weyl tensor condition.
result Proves that such manifolds must be a warped product with (n−1)−dimensional Einstein fiber. We formulate a quantization commutes with reduction principle in the setting where the Lie group G, the symplectic manifold it acts on, and the orbit space of the action may all be noncompact. It is assumed that the action is proper, and the zero set of a deformation vector field, associated to the momentum map and a…
Paper answers Gromov's compactness question on noncompact manifolds.
problem Gromov's compactness question on positive scalar curvature metrics.
method Construct examples and prove vanishing of certain index theoretic invariants.
result Positive answer for a class of 1-tame manifolds.
Perelman proves compact and complete noncompact shrinking breather properties.
problem Proving properties of shrinking breathers in both compact and noncompact settings.
method Using Perelman's entropy formula and L-geometry, constructing ancient solutions.
result Complete noncompact case proven without technical assumptions.
Suppose M is a noncompact connected smooth 2-manifold without boundary and let D(M)_0 denote the identity component of the diffeomorphism group of M with the compact-open C^infty-topology. In this paper we investigate the topological type of D(M)_0 and show that D(M)_0 is a topological ell_2-manifold and it has the hom…
The paper proves positive mass theorems for initial data sets with noncompact boundaries.
problem Proving positive mass theorems for initial data sets with noncompact boundaries.
method Defining an energy-momentum vector at spatial infinity and proving positive mass inequalities under DECs.
result Proves positive mass inequalities for initial data sets with noncompact boundaries under suitable DECs.
Every noncompact surface has a 3-rigid triangulation.
problem Classifying noncompact surfaces and proving their rigidity.
method Triangulation and minimally rigid structures.
result Every noncompact surface has a (3,6)-tight triangulation that is minimally 3-rigid.
Suppose M is a noncompact connected PL 2-manifold and let H(M)_0 denote the identity component of the homeomorphism group of M with the compact-open topology. In this paper we classify the homotopy type of H(M)_0 by showing that {\cal H}(M)_0 has the homotopy type of the circle if M is the plane, an open or half open a…
Extends Llarull's theorem to noncompact manifolds with boundary.
problem Generalizing Llarull's theorem to noncompact manifolds with boundary.
method Extends previous results to include compact boundaries.
result Generalized theorem to noncompact manifolds with compact boundaries.
Study noncompact manifolds' Chern scalar curvatures, proving existence and multiplicity.
problem Prescribing Chern scalar curvatures on noncompact manifolds.
method Solving a Kazdan-Warner type equation on noncompact non-Kähler manifolds with an analytic condition.
result Established existence results and provided a new proof of multiplicity theorem.
Proves a Schwarz-type lemma for noncompact manifolds with boundary.
problem Noncompact manifolds with boundary and their geometric applications.
method Suitable form of the weak maximum principle.
result Generalization of a classical result by Escobar.
Sharp gradient bound found for compact manifolds.
problem Finding a sharp gradient bound for compact manifolds.
method Using a specific function α=1 in the Li-Yau gradient bound.
result A sharp gradient bound is found for compact manifolds.