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.
Second Ricci flow proves existence of Kaehler-Einstein metrics on noncompact manifolds.
problem Existence of Kaehler-Einstein metrics on noncompact Hermitian manifolds.
method Established short time existence and Shi's type estimate of second Ricci flow.
result Second Ricci flow proves existence of Kaehler-Einstein metrics on complete noncompact Hermitian manifolds.
Study Poisson metrics on noncompact Kähler manifolds and their Higgs bundle applications.
problem Existence of Poisson metrics on flat vector bundles over noncompact Riemannian manifolds.
method Generalization of Corlette-Donaldson-Hitchin-Simpson's nonabelian Hodge correspondence to noncompact Kähler manifolds.
result Existence of Poisson metrics on Higgs bundles over noncompact Kähler manifolds.
The paper solves the Yamabe problem on certain noncompact manifolds.
problem Solving the Yamabe problem on noncompact manifolds.
method Examined geometric conditions on complete noncompact Riemannian manifolds.
result Found conditions for the existence of bounded positive solutions.
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…
Paper solves complex equations on noncompact manifolds.
problem Establishing estimates and existence for fully non-linear equations.
method General class of fully non-linear equations on Kähler and Hermitian manifolds.
result Constructs complete Kähler metrics with prescribed volume forms.
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.
The paper proves properties of geometric flows on noncompact manifolds.
problem Existence criteria for geometric flows on noncompact affine Riemannian manifolds.
method Obtained existence criteria through a geometric flow on noncompact affine Riemannian manifolds.
result Complete noncompact Hessian manifolds with nonnegative Hessian sectional curvature and bounded geometry are diffeomorphic to \(\mathbb{R}^n\) if their tangent bundle has maximal volume growth.
The paper proves uniqueness of Ricci flow on noncompact manifolds.
problem Proving uniqueness of Ricci flow on noncompact manifolds.
method Pseudolocality theorem for L-complete Ricci flow. result Strong uniqueness theorem for L-complete Ricci flow on Euclidean space. New examples show manifolds with noncompact boundaries that can't be pseudo-collarable.
problem Identifying manifolds with noncompact boundaries that cannot be pseudo-collarable.
method Provided examples of Z-compactifiable manifolds with noncompact boundaries that fail to be pseudo-collarable. result Found manifolds with noncompact boundaries that cannot be pseudo-collarable.
We prove a Schwarz-type lemma for noncompact manifolds with possibly noncompact boundary. The result is a consequence of a suitable form of the weak maximum principle of independent interest. The paper is enriched with applications to conformal deformations of noncompact manifolds with boundary, among them a generaliza…
Noncompact Ricci-flat solutions have infinite unstable dimensions.
problem Understanding unstable dimensions of noncompact Ricci-flat solutions.
method Derived sufficient conditions for infinite-dimensional unstable manifolds.
result Noncompact Ricci-flat solutions have uncountably many unstable perturbations.
Study geodesic orbits on noncompact curved spaces, proving their distribution and counting.
problem Counting and equidistribution of periodic orbits on noncompact manifolds.
method Proved equidistribution in narrow topology, deduced exact asymptotic counting.
result Exact asymptotic counting of periodic orbits on noncompact manifolds.
Existence criteria for Chern-Ricci flows on noncompact manifolds established.
problem Existence and properties of Chern-Ricci flows on noncompact complex manifolds.
method Generalization of results for Kahler-Ricci flows to Chern-Ricci flows, existence criteria established.
result Existence of complete Kahler metrics with nonnegative and bounded bisectional curvature on noncompact complex manifolds.
Morse inequalities for noncompact manifolds with group action.
problem Establishing inequalities for noncompact manifolds with group action.
method Using L2-Betti numbers and functions describing critical points. result Morse inequalities given in terms of L2-Betti numbers and group functions. The study of potential functions on noncompact quasi-Einstein manifolds, focusing on dimensions and flatness.
problem Characterizing potential functions on noncompact quasi-Einstein manifolds.
method Analyzing the dimensionality and structure of potential functions on specific manifolds.
result The dimension of positive potential functions on a three-dimensional noncompact quasi-Einstein manifold is at most two, with equality if and only if the manifold is a product of a λ-Einstein surface and R. 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…
The study restricts scalar curvature on certain noncompact manifolds.
problem Ruling out complete metrics with nonnegative scalar curvature.
method Conditions and examples of positive scalar curvature metrics.
result Examples of manifolds with positive scalar curvature.
Paper explores prescribing Chern scalar curvatures on noncompact Hermitian manifolds.
problem Prescribing Chern scalar curvatures on complete noncompact Hermitian manifolds.
method Generalizes Aviles-McOwen's existence results to higher-dimensional Hermitian manifolds.
result Existence results for Chern scalar curvatures on Hermitian manifolds.
The study proves that certain noncompact Hessian manifolds are diffeomorphic to R^n.
problem Characterizing complete noncompact Hessian manifolds with nonnegative Hessian sectional curvature.
method Using a geometric flow on noncompact affine Riemannian manifolds, constructing Hessian metrics, and proving diffeomorphism.
result Complete noncompact Hessian manifolds with nonnegative Hessian sectional curvature are diffeomorphic to R^n if their tangent bundle has maximal volume growth.
New nontrivial breathers found for Ricci flow on noncompact manifolds.
problem Perelman's theorem that nontrivial breathers don't exist for closed manifolds.
method Constructing nontrivial expanding breathers in all dimensions for noncompact manifolds.
result Nontrivial breathers found for Ricci flow on noncompact manifolds.
The paper explores Ricci forms on noncompact complex manifolds, including Kähler-Einstein and canonical metrics.
problem Existence of specific metrics on noncompact complex manifolds with prescribed Ricci curvature.
method Analyzes geometric problems on noncompact complex manifolds using Ricci curvature.
result Improves the main theorem in Cheng-Yau [4] and constructs Hesse-Einstein metrics.
Optimizes heat equation estimates on noncompact manifolds.
problem Improving gradient estimates for heat equations on noncompact manifolds.
method Localized and global noncompact versions of Hamilton's gradient estimate for positive solutions to the heat equation.
result Essentially optimal estimates significantly improve previous results.
In this paper, we first obtain the sub-Laplacian comparison theorem in a complete noncompact pseudohermitian manifold of vanishing torsion (i.e. Sasakian manifold). Secondly, we derive the sub-gradient estimate for positive pseudoharmonic functions in a complete noncompact pseudohermitian manifold which satisfies the C…
The paper disproves a conjecture about Ricci flow solutions on noncompact 3-manifolds.
problem Existence of κ-solutions of Ricci flow on noncompact 3-manifolds with positive curvature. method Analyzing the blow-up behavior of solutions and integrating curvature over time.
result Proves the nonexistence of κ-solutions under certain conditions, partially confirming a Perelman conjecture. We prove the uniqueness of solutions of the Ricci flow on complete noncompact manifolds with bounded curvatures using the De Turck approach. As a consequence we obtain a correct proof of the existence of solution of the Ricci harmonic flow on complete noncompact manifolds with bounded curvatures.
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.
New uniformity definition on noncompact manifolds without metrics.
problem No Riemannian metric for uniformity on noncompact manifolds.
method Definition of uniformity without metrics, equivalent to bounded geometry.
result Equivalent definition of uniformity on noncompact manifolds.
The Ricci flow is an evolution system on metrics. For a given metric as initial data, its local existence and uniqueness on compact manifolds was first established by Hamilton \cite{Ha1}. Later on, De Turck \cite{De} gave a simplified proof. In the later of 80's, Shi \cite{Sh1} generalized the local existence result to…
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.
Though Trudinger-Moser inequalities on compact Riemannian manifolds or Euclidean space are well understood, we know little about them on complete noncompact Riemannian manifolds. In this paper, we established respectively necessary condition and sufficient condition under which Trudinger-Moser inequalities hold on comp…
We prove a Hitchin-Thorpe inequality for noncompact Einstein 4-manifolds with asymptotic geometry at infinity. The asymptotic geometry at infinity is either a cusp bundle over a compact space (the fibered cusps) or a fiber bundle over a cone with a compact fiber (the fibered boundary). Many noncompact Einstein manifold…
Paper extends noncompact Llarull's theorem to manifolds with boundary.
problem Finding upper bounds for scalar curvature infimum in noncompact manifolds.
method Using deformed Dirac operators to relax boundary conditions.
result Upper bound for scalar curvature infimum in terms of Laplacian spectrum.
The paper confirms a conjecture about the fundamental groups of ends of certain noncompact manifolds.
problem Understanding the fundamental groups of ends of noncompact manifolds with specific curvature properties.
method Analyzing the universal cover and using the concept of visibility manifolds.
result The fundamental group of each end of a manifold is almost nilpotent if its universal cover is a visibility manifold.
The study provides volume growth estimates for specific types of manifolds.
problem Estimating volume growth for Ricci solitons and quasi-Einstein manifolds.
method Similar to classical results, the study proves volume growth estimates for gradient Ricci solitons and quasi-Einstein manifolds.
result Sharp volume growth estimates for gradient shrinking Ricci solitons and upper bound volume growth estimates for quasi-Einstein manifolds.
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.
Paper investigates prescribing Chern scalar curvatures on specific manifolds.
problem Prescribing Chern scalar curvatures on noncompact Hermitian manifolds with nonpositive curvatures.
method Establishes existence results and sufficient conditions for negative curvature metrics.
result Obtains sufficient conditions for the existence of a constant negative Chern scalar curvature metric.
The paper proves a new inequality for 3-manifolds with noncompact boundaries.
problem Proving positivity of a convex combination of ADM masses on 3-manifolds with noncompact boundaries.
method Obtained an integral inequality for asymptotically linear harmonic functions.
result Positivity of a convex combination of ADM masses under a positivity condition on scalar curvatures and boundary mean curvatures.
The goal of this article is to study the geometry of Bach-flat noncompact steady quasi-Einstein manifolds. We show that a Bach-flat noncompact steady quasi-Einstein manifold (Mn,g) with positive Ricci curvature such that its potential function has at least one critical point must be a warped product with Einstei…
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…
In this paper we survey the recent developments of the Ricci flows on complete noncompact Kähler manifolds and their applications in geometry.
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.
Classifies 3D Lorentz manifolds with noncompact isometry groups.
problem Classifying Lorentz manifolds with specific isometry properties.
method Complete topological and geometrical classification.
result Complete classification of 3D Lorentz manifolds.
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. 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.
In this paper we obtain three results concerning the geometry of complete noncompact positively curved Kähler manifolds at infinity. The first one states that the order of volume growth of a complete noncompact Kähler manifold with positive bisectional curvature is at least half of the real dimension (i.e., the complex…