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.
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
New method constructs potential functions for Kähler-Einstein metrics.
problem Constructing potential functions for Kähler-Einstein metrics on pseudoconvex domains.
method Method of potential scaling.
result Existence of 1-parameter family of automorphisms for certain pseudoconvex domains.
We establish a new symmetrization procedure for the isoperimetric problem in symmetric spaces of noncompact type. This symmetrization generalizes the well known Steiner symmetrization in euclidean space. In contrast to the classical construction the symmetrized domain is obtained by solving a nonlinear elliptic equatio…
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.
Introduces Kähler duality between domains in complex space.
problem None explicitly stated; focuses on concept introduction.
method None explicitly stated; focuses on concept introduction.
result Introduces Kähler duality between domains in complex space.
We prove that a complete noncompact Kähler manifold Mnof positive bisectional curvature satisfying suitable growth conditions is biholomorphic to a pseudoconvex domain of {\bf C}n and we show that the manifold is topologically {\bf R}2n. In particular, when Mn is a Kähler surface of positive bisecti…
In this paper we prove the existence of complete, noncompact convex hypersurfaces whose p-curvature function is prescribed on a domain in the unit sphere. This problem is related to the solvability of Monge-Ampère type equations subject to certain boundary conditions depending on the value of p. The special case of…
New bounds for Dirac eigenvalue involving boundary capacity.
problem Eigenvalue bounds for Dirac operator on hypersurfaces.
method Estimates for Dirac operator on boundaries of compact manifolds.
result Lower bounds for first eigenvalue involving boundary capacity.
Study of invariant solutions for certain PDEs on Riemannian manifolds.
problem Solving PDEs with group invariance on unbounded domains.
method Reduction of unbounded domains to bounded ones using group actions.
result Presentation of a method for studying invariant solutions.
We show the existence of isometric (or Ford) fundamental regions for a large class of subgroups of the isometry group of any rank one Riemannian symmetric space of noncompact type. The proof does not use the classification of symmetric spaces. All hitherto known existence results of isometric fundamental regions and do…
In this paper we prove that given a volume, among all domains with smooth boundary in rank-1 symmetric spaces of noncompact type, geodesic balls maximizes the first nonzero Steklov eigenvalue. We also prove a comparison result for the first nonzero Steklov eigenvalue for domains in simply connected Riemannian manifolds…
The paper explores symplectic geometry of Cartan-Hartogs domains.
problem Understanding the symplectic geometry of Cartan-Hartogs domains.
method Constructing a dual counterpart and computing symplectic capacity.
result A Cartan-Hartogs domain admits symplectic duality if and only if it reduces to a complex hyperbolic space.
Sharp isoperimetric inequalities for Neumann eigenvalues in symmetric spaces.
problem Finding bounds for eigenvalues of Neumann Laplacian on domains in symmetric spaces.
method Proving sharp inequalities for eigenvalues in compact and noncompact rank-1 symmetric spaces.
result Generalization of previous results for hyperbolic space and symmetric spaces.
Study shows different behaviors of noncompact hypersurfaces under mean curvature flow.
problem Analyzing stability of noncompact hypersurfaces with curvature blowup.
method Numerical overlap method to construct global solutions.
result Existence of near and far classes of initial data leading to distinct behaviors.
Study proves inequality for eigenvalues in symmetric spaces.
problem Proving an inequality for Steklov eigenvalues in symmetric spaces.
method Analyzes eigenvalues on bounded domains in noncompact rank-1 symmetric spaces.
result Extends previous results to non-Euclidean spaces.
Paper proves eigenvalue inequality for Hopf-symmetric domains.
problem Eigenvalue inequality for Hopf-symmetric domains in non-compact symmetric spaces.
method Used geometric and spectral analysis on non-compact rank one symmetric spaces.
result Eigenvalue inequality for bounded Hopf-symmetric domains in non-compact symmetric spaces.
Inspired by the work of Z. Lu and G. Tian [21] in the compact setting, in this paper we address the problem of studying the Szegö kernel of the disk bundle over a noncompact Kähler manifold. In particular we compute the Szegö kernel of the disk bundle over a Cartan-Hartogs domain based on a bounded symmetric domain. Th…
The main results of this paper consists of two parts. Firstly, we obtain an almost rigidity theorem which says that on a RCD(0, N) space, when a domain between two level sets of a distance function has almost maximal volume compared to that of a cylinder, then this portion is close to a cylinder as a metric space. Seco…
Estimates the index of the Laplace operator on planar domains with Robin boundary condition.
problem Index estimates for planar domains with Robin boundary condition
method Combines conformal and spectral techniques with topology of the domain.
result Lower bounds for the index in terms of the number of boundary components.
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.
Let Ω be a bounded domain with convex boundary in a complete noncompact Riemannian manifold with Bakry-Émery Ricci curvature bounded below by a positive constant. We prove a lower bound of the first eigenvalue of the weighted Laplacian for closed embedded f-minimal hypersurfaces contained in Ω. Using this estimat…
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.
New noncompact Coxeter polytopes found in various dimensions.
problem Classifying and constructing noncompact hyperbolic Coxeter polytopes.
method Maximal-cusp density and noncompact analog of Bogachev-Douba-Raimbault's argument.
result Infinitely many pairwise incommensurable noncompact Coxeter polytopes in dimensions 4-9.
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.
We are concerned about the coarse and precise aspects of a priori estimates for Green's function of a regular domain for the Laplacian-Betrami operator on any 3≤n-dimensional complete non-compact boundary-free Riemannian manifold through the square Sobolev/Nash/logarithmic-Sobolev inequalities plus the rough and s…
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.
A Margulis spacetime is a complete flat affine Lorentzian 3-manifold with free fundamental group. Associated to M is a noncompact complete hyperbolic surface Σ. We study double extensions of π1(M)≅π1(Σ) when Σ is homeomorphic to a projective plane minus two discs. We classify proper actions of this do…
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.
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…
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.
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.
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…
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. 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.
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.
Since non-compact RCD(0, N) spaces have at least linear volume growth, we study noncompact RCD(0, N) spaces with linear volume growth in this paper. One of the main results is that the diameter of level sets of a Busemann function grow at most linearly on a noncompact RCD(0, N) space satisfying the linear volume growth…
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.
Paper proves a noncompact version of Gromov's band-width estimate.
problem Proving a precise upper bound for noncompact Riemannian bands.
method Developed a quantitative partitioned manifold index theory.
result Proved a version of Gromov's band-width estimate for noncompact Riemannian bands.
Study solves Ricci curvature problem for specific noncompact spaces.
problem Solving the Prescribed Ricci Curvature problem for noncompact spaces with two isotropy summands.
method Classified and solved for all simply connected, noncompact G/H with semi-simple G and connected H having two irreducible summands. result Provided solutions to the Prescribed Ricci Curvature problem for all such spaces.
Study of elliptic boundary value problems on non-compact manifolds.
problem Analyzing elliptic differential operators on manifolds with non-compact boundaries.
method Regularity theory and trace theorems for sections in the maximal domain under various assumptions.
result Systematic study of local and nonlocal boundary conditions, including the Atiyah-Patodi-Singer condition.
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.
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. In this work, we first establish short time existence and Shi's type estimate of second Ricci flow on complete noncompact Hermitian manifolds. As an application, we use the second Ricci flow to discuss the existence of Kaehler-Einstein metric on complete noncompact Hermitian manifolds.
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…