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.
Geodesic completeness proven for all compact locally symmetric Lorentz manifolds.
problem Geodesic completeness of compact locally symmetric Lorentz manifolds.
method Proof in all remaining cases using completeness result.
result All compact, locally symmetric Lorentz manifolds are geodesically complete.
Paper proves stability of positive mass theorem for specific types of manifolds.
problem Stability of positive mass theorem for compact graphical manifolds.
method Used Federer--Fleming flat distance and static quasi-local Brown-York energy.
result Proved stability of positive mass theorem for compact (locally) hyperbolic graphical manifolds.
Study on compact toric locally conformally Kähler manifolds, finding specific properties.
problem Characterizing properties of compact toric locally conformally Kähler manifolds.
method Analyzing Kodaira dimension, using specific examples and mappings.
result Kodaira dimension is -∞ for underlying complex manifolds, and specific properties for surfaces and Vaisman manifolds.
Abstract: Necessary conditions found for compact manifolds locally modeled on nonreductive spaces.
problem Conditions for existence of compact manifolds locally modeled on nonreductive homogeneous spaces.
method Relative Lie algebra cohomology to generalize earlier results.
result No compact manifold locally modeled on positive dimensional coadjoint orbits of real linear solvable algebraic groups.
The paper shows a 3D shape can't fit into certain compact spaces.
problem Whether a specific 3D shape can fit into compact spaces.
method Using techniques from Sternfeld's thesis, the paper proves the impossibility.
result The contractible open 3-manifold cannot embed in compact, locally connected and locally 1-connected metric spaces.
Study on Einstein manifolds with specific properties.
problem Identifying all locally homogeneous compact pseudo-Riemannian Einstein manifolds.
method Analyzing standard compact Clifford-Klein forms of simple non-compact Lie groups and conjecturing based on T. Kobayashi's work.
result Found at least one Einstein metric in standard compact Clifford-Klein forms and conjecturing these are the only possible ones.
New compact examples of pseudo-Kähler manifolds with essential conformal transformations found.
problem Existence of essential conformal transformations in pseudo-Riemannian manifolds.
method Construction of compact locally conformally pseudo-Kähler manifolds with essential conformal transformations.
result Found compact examples of pseudo-Kähler manifolds with essential conformal transformations that are not conformally flat.
Study locally conformal symplectic manifolds on compact nilmanifolds.
problem Characterize locally conformal symplectic structures on compact nilmanifolds.
method Left-invariant structures, foliation analysis, Lie group theory.
result Structure theorem for locally conformal symplectic manifolds of the first kind.
Short proof shows parallel Lee form for certain compact manifolds.
problem Understanding parallel Lee form in compact pluricanonical locally conformally Kähler manifolds.
method Short proof using known properties of pluricanonical and locally conformally Kähler manifolds.
result Compact pluricanonical locally conformally Kähler manifolds have parallel Lee form.
Study compactifies Einstein metrics on 4D manifolds.
problem Compactification of conformally compact Einstein metrics.
method Assumptions on local and non-local conformal invariants, boundary metrics, and manifold topology.
result Established compactness results for 4D manifolds.
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.
The Laplace spectrum uniquely identifies five out of eight metrically maximal three-dimensional geometries.
problem Characterizing compact locally homogeneous three-manifolds using their Laplace spectra.
method Analyzing geometric structures and their spectral properties.
result For five out of eight metrically maximal three-dimensional geometries, compact locally homogeneous three-manifolds are uniquely determined by their spectra.
Proves regularity of extremal function on compact Kähler manifolds.
problem Regularity of extremal function on compact Kähler manifolds.
method Local property analysis and equivalence of continuity and Hölder continuity.
result Equivalence of classical notions of local L-regularity and locally Hölder continuous property. Investigate local Lie group structure of bisections over compact manifolds
problem Study the local Lie group structure associated with the space of admissible bisections of a local Lie groupoid over a compact manifold.
method Investigate the relation of this local Lie group to the Lie algebra of sections of the associated Lie algebroid.
result Prove that the globalizability of a local Lie groupoid implies the globalizability of its associated local Lie group of bisections.
New compact ECS manifolds with rank 2 discovered, differing from previous rank 1 examples.
problem Finding new compact ECS manifolds with rank 2.
method Constructing new examples of compact pseudo-Riemannian manifolds with parallel Weyl tensor, rank 1 or 2.
result New compact ECS manifolds of rank 2, locally homogeneous, and geodesically incomplete.
We characterize compact locally conformal parallel G2 (respectively, Spin(7)) manifolds as fiber bundles over S1 with compact nearly Kähler (respectively, compact nearly parallel G2) fiber. A more specific characterization is provided when the local parallel structures are flat.
The paper classifies holonomy groups of lcK manifolds and their conformal structures.
problem Classifying the holonomy groups of locally conformally Kähler manifolds.
method Analyzing the conformal classes, Einstein properties, and holonomy groups of lcK manifolds.
result Holonomy groups of compact lcK manifolds are restricted to specific values, with exceptions for Vaisman manifolds.
We prove a structure theorem for compact aspherical Lorentz manifolds with abundant local symmetry. If M is a compact, aspherical, real-analytic, complete Lorentz manifold such that the isometry group of the universal cover has semisimple identity component, then the local isometry orbits in M are roughly fibers of a f…
Study on isometric immersions in complex surfaces.
problem Understanding isometric immersions of locally conformally Kaehler manifolds.
method Investigation of isometric immersions into Hopf manifolds, focusing on compact complex surfaces.
result Characterization of Hopf-induced metrics on compact complex surfaces.
New definition for a special type of complex manifolds.
problem Characterizing compact locally conformally hyperkähler manifolds.
method Equivalent definition using a nondegenerate complex two-form.
result Established a conformal analogue of Beauville's theorem.
Study non-Kähler metrics on complex nilmanifolds, proving torus structure under certain conditions.
problem Understanding special non-Kähler metrics on complex nilmanifolds.
method Analyzing locally conformally Kähler, k-Gauduchon, balanced, and locally conformally balanced metrics on compact complex manifolds. result Compact complex nilmanifolds with balanced or k-Gauduchon metrics are tori, extending previous results. Localized deformation of scalar curvature and mean curvature on manifolds.
problem Deforming scalar curvature and mean curvature on compact manifolds with boundary.
method Proving localized surjection of scalar curvature and mean curvature map, handling non-variational linearized problem.
result Localized deformations of scalar curvature and mean curvature on compact manifolds are possible.
The Green function helps in creating evenly spaced points on compact manifolds.
problem Creating evenly spaced points on compact manifolds.
method Using the Green function for the Laplacian to minimize energy and achieve uniform distribution.
result A sequence of minimizers for the Green energy is asymptotically uniformly distributed.
We verify a conjecture for compact balanced threefolds and LCK manifolds with constant holomorphic sectional curvature.
problem Verify a conjecture about constant curvature Hermitian manifolds.
method Inspired by Huang and Wan's approach, investigate canonical metric connections.
result Establish the conjecture for connected compact locally conformally Kähler manifolds.
The study characterizes compact homogeneous manifolds with Bismut parallel torsion.
problem Characterizing compact homogeneous manifolds with specific geometric properties.
method Investigating Hermitian manifolds with Bismut parallel torsion, focusing on locally homogeneous manifolds.
result Characterization of compact Chern flat BTP manifolds and properties of BTP compact Hermitian locally homogeneous manifolds.
Yamabe flow proves compactness of certain locally conformally flat manifolds with positive Ricci curvature.
problem Proving compactness of locally conformally flat manifolds with positive Ricci curvature.
method Using the Yamabe flow to prove compactness.
result Locally conformally flat manifolds with positive pinched Ricci curvature are compact.
Compact ECS manifolds are proven to be bundles over the circle.
problem Compact ECS manifolds and their local structure.
method Proof of a specific result for compact rank-one ECS manifolds.
result Compact rank-one ECS manifolds, not locally homogeneous, are bundles over the circle.
New insights into compact rank-one ECS manifolds, proving they are bundles over circles.
problem Understanding the structure of compact rank-one ECS manifolds.
method Analyzing the properties of pseudo-Riemannian manifolds with parallel Weyl tensor.
result Compact rank-one ECS manifolds are bundles over the circle with specific leaf structures.
Compact lcK manifolds with holomorphic Lee field are Vaisman under certain conditions.
problem Characterizing compact locally conformally Kähler manifolds with holomorphic Lee fields.
method Analyzing conditions for a compact lcK manifold to be Vaisman when it has a holomorphic Lee vector field.
result Compact lcK manifolds with holomorphic Lee field are Vaisman if the Lee field has constant norm or the metric is Gauduchon.
We study two kinds of transformation groups of a compact locally conformally Kahler (l.c.K.) manifold. First we study compact l.c.K. manifolds with parallel Lee form by means of the existence of a holomorphic l.c.K. flow. Next, we introduce the Lee-Cauchy-Riemann (LCR) transformations as a class of diffeomorphisms pres…
Study on special metrics on complex manifolds, focusing on degeneracy.
problem Understanding degenerate metrics on complex manifolds.
method Investigation of special-Hermitian metrics, including Kähler and locally conformally Kähler metrics.
result Characterization of degenerate metrics on non-Kähler manifolds.
We characterize compact locally conformally Kähler (l.c.K.) manifolds under the assumption of a purely conformal, holomorphic circle action. As an application, we determine the structure of the compact l.c.K. manifolds with parallel Lee form. We introduce the Lee-Cauchy-Riemann (LCR) transformations as a class of diffe…
In this paper we show as main results two structure theorems of a compact homogeneous locally conformally Kaehler (or shortly l.c.K.) manifold, a holomorphic structure theorem asserting that it has a structure of holomorphic principal fiber bundle over a flag manifold with fiber a 1-dimensional complex torus, and a met…
Compact locally conformal Kähler manifolds with constant Chern holomorphic sectional curvature are necessarily Kähler.
problem Chern version of the constant holomorphic sectional curvature conjecture for compact locally conformal Kähler manifolds
method Prove the conjecture using curvature identities and properties of Kähler metrics
result Compact locally conformal Kähler manifolds with constant Chern holomorphic sectional curvature are necessarily Kähler
Study characterizes toric LCK manifolds, proving conjecture and showing differences from symplectic case.
problem Characterizing compact toric locally conformally Kähler manifolds.
method Proves conjecture about toric LCK manifolds, constructs examples to show differences from symplectic case.
result Proves a conjecture about toric LCK manifolds and shows differences from symplectic case.
The study examines special properties of compact Riemannian manifolds with harmonic Weyl curvature.
problem Characterizing compact Riemannian manifolds with specific curvature properties.
method Rigidity theorems and geometric analysis on manifolds with positive scalar curvature and constant σ2. result Theorems proving the isometry of certain manifolds under specific curvature conditions.
We formulate a conjecture that arithmetic locally symmetric manifolds have simple homotopy type, and prove it for the non-compact case. More precisely, we show that, for any symmetric space S of non-compact type without Euclidean de Rham factors, there are constants a=a(S) and d=d(S) such that any non-compact arithmeti…
Compact LCK manifolds of algebraic codimension one are bimeromorphically equivalent to elliptic fibrations.
problem Characterizing compact locally conformally Kähler manifolds of algebraic codimension one.
method Proving bimeromorphic equivalence to elliptic fibrations.
result Compact LCK manifolds of algebraic codimension one are bimeromorphically equivalent to elliptic fibrations.
Classifies Lie groups acting on compact Lorentz manifolds.
problem Classifying Lie groups acting on compact Lorentz manifolds.
method Classification up to local isomorphisms.
result Classification of Lie groups without compact factors.
Generalizes tools for studying collapsed manifolds to new geometry.
problem Studying collapsed manifolds with bounded sectional curvature.
method Generalizes fibration and stability theorems for compact group actions on manifolds with local bounded Ricci covering geometry.
result Two generalized results used in Xiaochun Rong's work on almost flat manifolds.
We prove that any compact homogeneous locally conformally Kähler manifold has parallel Lee form.
Local smoothing of metrics with small curvature, removing Ricci curvature condition.
problem Establishing local smoothing of metrics with curvature concentration.
method Local mollification, removing Ricci curvature condition, Sobolev constants and volume growth.
result Compactness of manifolds with small curvature concentration under Ahlfors regularity and Sobolev constant.
Compact ECS manifolds have a simple topological structure.
problem Understanding the structure of compact ECS manifolds.
method Review of basic facts, construction of examples, and proof of a topological structure result.
result Compact rank-one ECS manifolds are bundles over S^1 with fibers being leaves of D⊥.
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.
Positive simplicial volume implies locally symmetric space structure.
problem Understanding simplicial volume in locally homogeneous spaces.
method Analyzing properties of locally homogeneous Riemannian manifolds.
result Closed locally homogeneous manifolds with positive simplicial volume are locally symmetric.
Local invertibility of higher order tensor transforms on compact manifolds.
problem Invertibility of higher order tensor transforms on compact manifolds.
method Local invertibility of transverse and mixed ray transforms of tensors on compact Riemannian manifolds.
result Local invertibility of transverse and mixed ray transforms of tensors for specific dimensions.
New results on localization of exotic diffeomorphisms in 4-manifolds.
problem Understanding when groups of exotic diffeomorphisms can be localized to smaller embedded submanifolds.
method Analyzing families Seiberg-Witten theory, Dehn twists, and properties of Seifert fibered homology spheres.
result Exotic diffeomorphisms cannot be localized to topologically embedded rational homology balls or homology spheres.