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.
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.
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.
Complete classification of symmetric spaces actions.
problem Classifying symmetric spaces actions.
method Isometric cohomogeneity-one actions classification.
result Classification completed for all symmetric spaces of noncompact type.
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…
Uniformizes complete noncompact Kähler manifolds with maximal volume growth.
problem Uniformizing complete noncompact Kähler manifolds with maximal volume growth.
method Proved biholomorphic equivalence to Cn for specific manifolds. result Confirms Yau's uniformization conjecture for maximal volume growth.
In this paper we derive a precise estimate on the growth of potential functions of complete noncompact shrinking solitons. Based on this, we prove that a complete noncompact gradient shrinking Ricci soliton has at most Euclidean volume growth. The latter result can be viewed as an analog of the well-known theorem of Bi…
Study on anisotropic curvature flow for noncompact convex hypersurfaces.
problem Anisotropic curvature flow of noncompact convex hypersurfaces.
method Flow of complete noncompact convex hypersurfaces with anisotropy determined by a Wulff shape.
result The flow exists for all positive time for initial conditions.
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…
The noncompact Yamabe flow can lead to incomplete metrics over infinite time.
problem Incompleteness of noncompact Yamabe flow solutions over infinite time.
method Analysis of long-time behavior of the noncompact Yamabe flow.
result Existence of a long-time solution that is complete for each time but converges to an incomplete metric.
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.
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…
In this paper we survey the recent developments of the Ricci flows on complete noncompact Kähler manifolds and their applications in geometry.
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.
We discuss some geometric conditions under which a complete noncompact shrinking gradient Ricci soliton will split at infinity.
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. 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.
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.
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…
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.
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.
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.
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.
We obtain a Calabi-Yau type lower volume growth estimates for complete noncompact self-shrinkers of the mean curvature flow, more precisely, every complete noncompact properly immersed self-shrinker has at least linear volume growth.
Study equivalence of metrics on noncompact Kähler manifolds with Bergman kernel properties.
problem Equivalence of invariant metrics on noncompact Kähler manifolds with bounded curvature.
method Use boundedness of ratio between Bergman kernel and wedge product of metric in fundamental domain.
result Equivalence of Bergman metric and Kähler-Einstein metric under specific conditions.
New proof shows curvature integrability for noncompact surfaces.
problem Proving curvature integrability for noncompact surfaces.
method Short proof using complete Riemannian metrics and Gaussian curvature.
result Gaussian curvature must be integrable for noncompact surfaces.
Complete Calabi-Yau metrics on abelian fibrations over complex space.
problem Constructing complete Calabi-Yau metrics on noncompact abelian fibrations.
method Using abelian fibrations over C, we construct complete Calabi-Yau metrics and provide compactification. result We provide a compactification for the abelian fibration X such that the compactified variety has a negative canonical bundle. 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.
In this paper, we classify n-dimensional (n>2) complete noncompact locally conformally flat gradient steady solitons. In particular, we prove that a complete noncompact non-flat conformally flat gradient steady Ricci soliton is, up to scaling, the Bryant soliton.
Study local curvature estimates and existence of conformal metrics on noncompact manifolds.
problem Deriving local C0-estimates and existence of conformal metrics with prescribed curvature. method Utilizing Aviles-McOwen's result and its nonlinear extension, combined with asymptotic conditions.
result Proved existence of complete conformal metrics with prescribed curvature functions.
Survey on bottom of spectrum of Hodge Laplacian on complete noncompact Kähler manifolds
problem Bottom of the spectrum of Hodge Laplacian on complete noncompact Kähler manifolds
method Survey on Kähler hyperbolic manifolds and bounded symmetric domains
result Proposed several open problems
In this paper we give a partial affirmative answer to a conjecture of Greene-Wu and Yau. We prove that a complete noncompact Kähler surface with positive and bounded sectional curvature and with finite analytic Chern number c1(M)2 is biholomorphic to ${\C}^2$.
No expanding breathers found in noncompact Ricci flows with certain curvature conditions.
problem Finding expanding breathers in noncompact Ricci flows.
method Curvature positivity conditions (weak PIC-2 or nonnegative bisectional curvature).
result Complete noncompact expanding breathers are gradient solitons.
Study proves nonexistence of certain Kähler-Einstein metrics on specific noncompact manifolds.
problem Proving the nonexistence of complete Kähler-Einstein metrics on noncompact manifolds.
method Analyzing compact Kähler manifolds and subvarieties, applying blow-downs and canonical/terminal singularities.
result Nonexistence of complete Kähler-Einstein metrics on M−N and M−E under specified conditions. Solves CR Poincaré-Lelong equation on CR manifolds, revealing structures and solitons.
problem Solving CR Poincaré-Lelong equation on CR manifolds.
method Solves CR Poisson equation on CR (2n+1)-manifolds with specific curvature properties. result Discovers structures and CR Yamabe steady solitons on complete noncompact Sasakian manifolds.
This paper is to study the conformal scalar curvature equation on complete noncompact Riemannian manifold of nonpositive curvature. We derive some estimates and properties of supersolutions of the scalar curvature equation, and obtain some nonexistence results for complete solutions of scalar curvature equation.
We show that a totally geodesic submanifold of a symmetric space satisfying certain conditions admits an extension to a minimal submanifold of dimension one higher, and we apply this result to construct new examples of complete embedded minimal submanifolds in simply connected noncompact globally symmetric spaces.
The paper studies how the shape of surfaces changes over time using curvature.
problem Understanding how the shape of surfaces evolves over time using curvature.
method The authors use curvature flow with a power of a function of principal curvatures to study the evolution of surfaces.
result The complete smooth strictly convex solution exists and remains a graph until the maximal time of existence.
We study the spectrum of complete noncompact manifolds with bounded curvature and positive injectivity radius. We give general conditions which imply that their essential spectrum has an arbitrarily large finite number of gaps. In particular, for any noncompact covering of a compact manifold, there is a metric on the b…
We investigate complete noncompact Ricci-flat manifolds which are not of maximal volume growth. We show that the manifolds with a curvature decay condition and a holonomy decay condition are asymptotic to torus fibrations over ALE spaces. In particular, we classify complete noncompact 4-dimensional hyperkäler manifold…
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 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.
Let (M,g) be a noncompact complete n-manifold with harmonic curvature and positive Sobolev constant. Assume that L2 norms of Weyl curvature and traceless Ricci curvature are finite. We prove that (M,g) is Einstein if n≥5 and Ln/2 norms of Weyl curvature and traceless Ricci curvature are small enough…
Study finds conditions for minimal surfaces in noncompact spaces.
problem Existence of minimal surfaces in noncompact spaces with nonnegative scalar curvature.
method Conditions on a 2-sphere and its boundary.
result Existence of a closed minimal surface homologous to the boundary.
Study Liouville theorem for specific harmonic maps.
problem Proving Liouville theorem for harmonic maps.
method Establish gradient estimates under specific conditions.
result Proved Liouville theorem for harmonic maps.
Characterizes pseudo-collarable manifolds with noncompact boundaries.
problem Characterizing manifolds with noncompact boundaries.
method Extends earlier work by Guilbault and Tinsley to include noncompact boundaries, extending Siebenmann's work.
result Complete characterization of pseudo-collarable n-manifolds for n≥6. We construct complete noncompact Riemannian metrics with G2-holonomy on noncompact orbifolds that are R3-bundles with the twistor space Z as a spherical fiber.