Study Szegő kernel on non-compact CR manifolds with specific conditions.
problem Analyzing Szegő kernel on non-compact CR manifolds.
method Establish Szegő kernel asymptotic expansions on non-compact strictly pseudoconvex CR manifolds with transversal CR R-action under natural geometric conditions. result Szegő kernel asymptotic expansions established on non-compact CR manifolds.
New method constructs G2-manifolds from Calabi-Yau 3-folds.
problem Creating complete non-compact G2-manifolds.
method Starting with a Calabi-Yau 3-fold, constructs a 1-parameter family of circle-invariant complete G2-metrics.
result Produces infinitely many diffeomorphism types of complete non-compact simply connected G2-manifolds.
In this article we study the limiting behavior of the Kähler Ricci flow on complete non-compact Kähler manifolds. We provide sufficient conditions under which a complete non-compact gradient Kähler-Ricci soliton is biholomorphic to $\ce^n$. We also discuss the uniformization conjecture by Yau \cite{Y} for complete non-…
Complete solutions found for Toda equations on non-compact surfaces.
problem Solving Toda equations on non-compact Riemann surfaces.
method Introduced complete solutions and proved existence and uniqueness using Toda equations and harmonic bundle techniques.
result Existence and uniqueness of complete solutions to Toda equations on non-compact Riemann surfaces.
We consider the Qk flow on complete non-compact graphs. We prove that a complete graph evolves by the Qk curvature up to some time T depending on the radius of a sphere enclosed by the initial graph.
In this paper, we give a delay estimate of scalar curvature for a complete non-compact expanding (or steady) gradient Ricci soliton with nonnegative Ricci curvature. As an application, we prove that any complete non-compact expanding (or steady) gradient Kähler-Ricci solitons with positively pinched Ricci curvature sho…
In this paper, we prove that the Lp essential spectra of the Laplacian on functions are [0,+∞) on a non-compact complete Riemannian manifold with non-negative Ricci curvature at infinity. The similar method applies to gradient shrinking Ricci soliton, which is similar to non-compact manifold with non-negative…
The paper explores geometry and positive scalar curvature on non-compact manifolds.
problem Understanding the relationship between geometry and positive scalar curvature on non-compact manifolds.
method Analysis of volume growth, scalar curvature integral, and width in different dimensions.
result Proves minimal volume growth and integral of scalar curvature in three dimensions, and volume growth with stronger conditions in higher dimensions.
The paper studies rigidity results for harmonic forms on Kähler manifolds.
problem Understanding harmonic forms on Kähler manifolds.
method Analyzes rigidity results for harmonic (p,q)-forms in complete Kähler manifolds. result Shows several rigidity results and applications to non-compact Kähler manifolds.
We study the evolution of convex complete non-compact graphs by positive powers of Gauss curvature. We show that if the initial complete graph has a local uniform convexity, then the graph evolves by any positive power of Gauss curvature for all time. In particular, the initial graph is not necessarily differentiable.
Proves a Liouville-type theorem for p-Laplacian on manifolds.
problem Proving Liouville-type theorems for p-Laplacian on manifolds.
method Proved a Liouville-type result for the p-Laplacian on complete Riemannian manifolds.
result Proved a Liouville-type theorem for the p-Laplacian on complete non-compact Riemannian manifolds.
We derive lower bounds on the scalar curvature of complete non-compact gradient Yamabe solitons under some integral curvature conditions. Based on this, we prove that the corresponding potential functions have at most quadratic growth in distance. We also obtain a finite topological type property on complete shrinking …
Condition for Anosov flows on non-compact manifolds without conjugate points.
problem Conditions for Anosov flows on non-compact manifolds.
method Formulated a condition for complete, connected and non-compact Riemannian manifolds.
result No conjugate points in geodesic flow if Anosov.
New local method solves Yamabe problems on compact and non-compact manifolds.
problem Yamabe problems on compact and non-compact manifolds.
method Local method for compact and non-compact manifolds.
result Generalizes Brezis and Nirenberg's nonlinear eigenvalue problem to subsets of manifolds.
Sharp lower bound for p-Laplacian eigenvalue on non-compact manifolds.
problem Estimating eigenvalues of p-Laplacian on non-compact manifolds. method Sharp lower bound established through domain properties and curvature conditions.
result Sharp lower bound for the first Dirichlet eigenvalue of p-Laplacian. In this paper we use a dynamical approach to prove some new divergence theorems on complete non-compact Riemannian manifolds.
Constructs complete Calabi-Yau metrics from smoothed Calabi-Yau intersections.
problem Creating complete Calabi-Yau metrics on non-compact manifolds.
method Extending Székelyhidi's work, constructing metrics with varying complex structures and possible singularities.
result Produces Calabi-Yau metrics with fibers having varying complex structures and possibly isolated singularities.
Study calculates index for non-compact manifolds with boundary.
problem Index of strongly Callias-type operators on non-compact manifolds.
method Computes index of local boundary value problem on non-compact manifold.
result Extends Freed's index theorem to non-compact manifolds.
Proves well-posedness of the Cauchy problem for the Dirac operator on non-compact spacetimes.
problem Proving well-posedness of the Cauchy problem for the Dirac operator on non-compact spacetimes.
method Analyzes globally hyperbolic manifolds with complete spacelike Cauchy hypersurfaces.
result Proves well-posedness of the Cauchy problem for the Dirac operator.
Logarithmic Sobolev inequality proven for non-compact self-shrinkers.
problem Establishing a logarithmic Sobolev inequality for non-compact self-shrinkers.
method Using Alexandrov-Bakelman-Pucci (ABP) method to prove the inequality for Euclidean space, then applying this method to non-compact self-shrinkers.
result Optimal logarithmic Sobolev inequality for complete, non-compact, properly embedded self-shrinkers.
In this paper we prove that Dirac operators on non-compact complete orbifolds which are sufficiently regular at infinity, admit a unique extension. Additonally, we prove a generalized orbifold Stokes'/Divergence theorem.
In this paper, we study the global Kähler-Ricci flow on a complete non-compact Kähler manifold. We prove the following result. Assume that (M,g0) is a complete non-compact Kähler manifold such that there is a potential function f of the Ricci tensor, i.e., Rijˉ(g0)=fijˉ. Assume that the quan…
New criterion found for Hermitian-Yang-Mills metrics on non-compact Kähler manifolds.
problem Existence of Hermitian-Yang-Mills metrics on non-compact Kähler manifolds.
method Algebraic criterion and stability condition introduced.
result New stability condition is both sufficient and necessary for the existence of Hermitian-Yang-Mills metrics.
New theorem on geodesics on non-compact manifolds.
problem Existence of geodesics with specific properties.
method Analyzing geodesics with local homology in maximal degree.
result Infinitely many closed geodesics on non-compact manifolds.
New complete minimal submanifolds found in specific Riemannian spaces.
problem Finding complete minimal submanifolds in non-compact Riemannian symmetric spaces.
method Constructing multidimensional families of submanifolds via complex-valued eigenfunctions.
result New families of complete minimal submanifolds in SL_n(R)/SO(n), Sp(n,R)/U(n), SO*(2n)/U(n), and SU*(2n)/Sp(n).
Study of Sasakian immersions in Heisenberg group and BNimesR, focusing on η-Einstein case.
problem Classification of Sasakian immersions in specific spaces.
method Analysis of complete regular Sasakian manifolds in Heisenberg group and BNimesR with standard Sasakian structures. result Complete classification of η-Einstein Sasakian immersions. We prove that for a solution (Mn,g(t)), t∈[0,T), where T<∞, to the Ricci flow with bounded curvature on a complete non-compact Riemannian manifold with the Ricci curvature tensor uniformly bounded by some constant C on Mn×[0,T), the curvature tensor stays uniformly bounded on Mn×[0,T).…
Unique shrinking gradient Kähler-Ricci solitons found on non-compact toric manifolds.
problem Existence and uniqueness of shrinking gradient Kähler-Ricci solitons on non-compact toric manifolds.
method Analyzing properties of Ricci curvature and Lie algebra constraints.
result At most one complete Tn-invariant shrinking gradient Kähler-Ricci soliton on a non-compact toric manifold. Extends soliton theory to non-compact cases.
problem Generalized solitons in non-compact settings.
method Place conditions on vector field and curvature, use tensor properties.
result Non-compact q-solitons are stationary and q-flat. New framework for Seiberg-Witten map on non-compact 4-manifolds.
problem L^2 self-dual forms on non-compact 4-manifolds cannot exhaust d^+ images.
method Functional analytic framework for Seiberg-Witten map.
result Construction of new functional analytic framework.
We prove that every complete non-compact manifold of finite volume contains a (possibly non-compact) minimal hypersurface of finite volume. The main tool is the following result of independent interest: if a region U can be swept out by a family of hypersurfaces of volume at most V, then it can be swept out by a fa…
In non-compact manifolds, geodesic flowers exist.
problem Existence of geodesic flowers in non-compact manifolds.
method Proving the existence of non-trivial geodesic flowers in complete non-compact manifolds with locally convex ends.
result Non-trivial geodesic flowers exist in every complete non-compact manifold with locally convex ends.
Study shows uniqueness of solutions on complex manifolds without requiring solution decay.
problem Uniqueness of solutions to Monge-Ampere equation on complex manifolds.
method Caccioppoli inequality techniques applied to Kähler manifolds with sub-quadratic volume growth.
result Uniqueness of bounded C1,1 solutions to Monge-Ampere equation without decay requirement. Study rigidifies non-compact manifolds with specific curvature conditions.
problem Analyzing non-compact generalized m-quasi-Einstein manifolds with constant scalar curvature and soliton function.
method Introduced a weighted function and proved its subharmonicity to derive rigidity results.
result Proves manifolds are Euclidean under specific conditions, with constant μ essential.
Anosov geodesic flow proven in non-compact manifolds with negative curvature.
problem Proving Anosov geodesic flow in non-compact manifolds with negative curvature.
method Proving the geodesic flow is Anosov by showing average sectional curvature is negative and uniformly away from zero.
result Constructed a non-compact manifold with Anosov geodesic flow.
Geometrically proves Lie algebras are identified by their Iwasawa subalgebras.
problem Determining semi-simple Lie algebras from their subalgebras.
method Geometry of Einstein solvmanifolds and algebraic procedure.
result Semi-simple Lie algebras are uniquely determined by their Iwasawa subalgebras.
In this paper, we study the volume growth property of a non-compact complete Riemannian manifold X. We improve the volume growth theorem of Calabi (1975) and Yau (1976), Cheeger, Gromov and Taylor (1982). Then we use our new result to study gradient Ricci solitons. We also show that on X, for any q∈(0,∞),…
Review of gravitational instantons in physics.
problem Understanding solutions to Einstein equations in four dimensions.
method Lecture review of examples.
result Examples of gravitational instantons.
Develops quantization for non-compact complex manifolds with spectral gap.
problem Quantization of non-compact complex manifolds with spectral gap.
method Berezin-Toeplitz quantization, spectral gap analysis, asymptotic expansion.
result Toeplitz operators form a closed algebra and satisfy a complete composition expansion.
The study counts ends on shrinkers using geometric covering methods.
problem Counting the number of ends on shrinkers.
method Geometric covering method to study the number of ends.
result Proves that the number of ends on any complete non-compact shrinker is at most polynomial growth with fixed degree.
The paper proves conditions for Anosov geodesic flows on non-compact manifolds.
problem Conditions for Anosov geodesic flows on non-compact manifolds.
method Analyzing sectional curvature and focal points to determine Anosov geodesic flows.
result A sufficient condition for Anosov geodesic flows on non-compact manifolds.
Symphonic maps on non-compact Riemannian manifolds are non-existent.
problem Existence of symphonic maps on compact or non-compact Riemannian manifolds
method Study of conformal vector fields and Ricci solitons
result No symphonic maps exist
Constructs Spin(7) manifolds from self-dual Einstein 4-orbifolds.
problem Creating complete non-compact Spin(7) manifolds.
method Analytic construction using adiabatic limits of Spin(7) metrics on Seifert bundles over G2 orbifolds.
result Produces complete non-compact Spin(7) manifolds with arbitrary second Betti numbers and infinitely many families of ALC metrics.
Sharp volume growth ratio for 3D manifolds with positive scalar curvature.
problem Volume growth and scalar curvature in non-compact Riemannian manifolds.
method Analyzing 3D complete, non-compact manifolds with non-negative Ricci and positive scalar curvature.
result Obtained sharp linear volume growth ratio and rigidity.
Global existence of Yamabe flow on non-compact manifolds with unbounded initial curvature.
problem Global existence of Yamabe flow on non-compact manifolds with unbounded initial curvature.
method Assumption of conformally equivalent initial metric to a complete background metric with bounded scalar curvature and positive Yamabe invariant.
result Global existence of Yamabe flow without requiring initial curvature bounds.
Study on scalar curvature decay on non-compact manifolds linked at infinity.
problem Understanding scalar curvature decay on non-compact manifolds with topological linking at infinity.
method Analyzing polynomial decay, developing obstruction theory, using μ--bubble exhaustions, and index theory. result Topological linking at infinity forces polynomial decay of scalar curvature on manifolds of weakly bounded geometry.
The paper proves conditions for Kähler-Einstein metrics to remain Kähler-Einstein under cscK perturbations.
problem Conditions for Kähler-Einstein metrics to remain Kähler-Einstein under cscK perturbations.
method Study of constant scalar curvature Kähler (cscK) metrics on complete non-compact Kähler--Einstein manifolds.
result Sufficient conditions for a cscK perturbation of a Kähler--Einstein metric to remain Kähler--Einstein.
We investigate the homogeneity of certain kind of slices of the complete complexification of a proper complex equifocal submanifold in a symmetric space of non-compact type.