The study classifies all compact hyperbolic polytopes with eight facets.
problem Classifying compact hyperbolic Coxeter four-polytopes with specific numbers of facets.
method Complete classification through mathematical analysis.
result The complete classification of compact hyperbolic Coxeter four-polytopes with eight facets.
The paper establishes a duality between non-compact and compact symmetric pairs.
problem Understanding the relationship between non-compact and compact symmetric pairs.
method Developed a duality theorem between non-compact pseudo-Riemannian semisimple symmetric pairs and commutative compact semisimple symmetric triads.
result Explicit description of a one-to-one correspondence between non-compact and compact symmetric pairs.
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.
Compact curve solution emerges from non-compact curve.
problem Constructing solutions from non-compact curves.
method Slingshot solution to curve shortening flow.
result Compact embedded solution exists for a finite time.
Compact metrics on Heisenberg manifolds have a specific condition for being relatively compact.
problem Conditions for relatively compact sets of left invariant metrics on Heisenberg manifolds.
method Necessary and sufficient condition for relatively compact sets of left invariant metrics.
result A condition for a set of left invariant metrics to be relatively compact in the moduli space.
Compact Lie groups have compact isometry groups with pseudo-Riemannian metrics.
problem Characterizing isometry groups of compact Lie groups with pseudo-Riemannian metrics.
method Analyzing left-invariant pseudo-Riemannian metrics on compact Lie groups.
result Isometry groups of compact Lie groups are compact.
Study on GL(2) geometries on complex manifolds, focusing on Kähler-Einstein and Fano manifolds.
problem Characterizing compact complex manifolds with holomorphic GL(2)-geometry.
method Analyzing Kähler-Einstein and Fano manifolds, using GL(2) and SL(2) geometries.
result Only compact Kähler-Einstein manifolds with holomorphic GL(2)-geometry are covered by compact complex tori, three dimensional quadric, or three dimensional Lie ball.
Paper proves structure for compact Kähler manifolds with pseudo-effective tangent bundles.
problem Compact Kähler manifolds with pseudo-effective tangent bundles.
method Smooth or locally constant rationally connected fibration onto a quotient of a compact complex torus.
result Compact Kähler manifolds with pseudo-effective tangent bundles admit a fibration structure.
Compact embeddings for invariant functions in metric-measure spaces.
problem Embedding functions with symmetry in metric-measure spaces.
method Analyzing H-invariant functions in compact metric-measure spaces, extending to Riemannian manifolds. result Obtained compact Sobolev embeddings for critical exponents.
Compact theorem on Hamiltonian stationary submanifolds in symplectic manifolds.
problem Compactness of Hamiltonian stationary Lagrangian submanifolds in symplectic manifolds.
method Proving a compactness theorem with area and extrinsic curvature bounds.
result Uniform bounds on area and total extrinsic curvature lead to compactness of Hamiltonian stationary Lagrangian submanifolds.
Compact groups with polynomial growth have specific embeddings.
problem Understanding groups with polynomial growth.
method Embedding into semidirect products of Lie and compact groups.
result Groups can be embedded as co-compact subgroups.
Let F be a foliation of codimension 2 on a compact manifold with at least one non-compact leaf. We show that then F must contain uncountably many non-compact leaves. We prove the same statement for oriented p-dimensional foliations of arbitrary codimension if there exists a closed p form which evaluates positively on e…
Study on compact strong HKT manifolds and their properties.
problem Characterizing the structure of compact strong HKT manifolds.
method Geometric analysis, rigidity theorems, classification, and properties of Ricci foliations.
result Compact strong HKT manifolds are Hopf fibrations over compact 4-dimensional orbifolds.
Study compact symplectic solvmanifolds' hard Lefschetz property.
problem Compact symplectic solvmanifolds' hard Lefschetz property.
method Analysis of compact symplectic solvmanifolds as quotients of solvable Lie groups by lattices.
result Characterization of conditions for the hard Lefschetz property.
Paper classifies compact symmetric triads using double Satake diagrams and canonical forms.
problem Classifying compact symmetric triads.
method Introducing double Satake diagrams and canonical forms, proving their existence and properties.
result Existence and properties of canonical forms for compact simple symmetric triads.
In this paper, we establish some compactness results of conformally compact Einstein metrics on 4-dimensional manifolds. Our results were proved under assumptions on the behavior of some local and non-local conformal invariants, on the compactness of the boundary metrics at the conformal infinity, and on the topology…
Compact shrinkers with curvature pinching conditions proven.
problem Ensuring shrinkers are compact under curvature pinching conditions.
method Various curvature pinching conditions applied to shrinkers with positive Ricci curvature and asymptotically nonnegative sectional curvature.
result Shrinkers with curvature pinching conditions are proven to be compact.
No conformal product structures on compact manifolds with constant curvature.
problem Existence of conformal product structures on compact manifolds with constant curvature.
method Analyzing non-flat manifolds and irreducible, compact locally symmetric spaces of non-positive curvature.
result Compact non-flat manifolds with constant sectional curvature admit no conformal product structure.
We study compact Riemannian manifolds for which the light between any pair of points is blocked by finitely many point shades. Compact flat Riemannian manifolds are known to have this finite blocking property. We conjecture that amongst compact Riemannian manifolds this finite blocking property characterizes the flat m…
We study Bott-Chern cohomology on compact complex non-Kähler surfaces. In particular, we compute such a cohomology for compact complex surfaces in class VII and for compact complex surfaces diffeomorphic to solvmanifolds.
We classify compact homogeneous geometries of irreducible spherical type and rank at least 2 which admit a transitive action of a compact connected group, up to equivariant 2-coverings. We apply our classification to polar actions on compact symmetric spaces.
A transversely holomorphic foliation on a compact complex manifold, exhibits a compact stable leaf if and only if the set of compact leaves is not a zero measure subset of the manifold.
Extended Vaisman theorem to compact spaces with singularities.
problem Generalizing Vaisman's theorem to spaces with singularities.
method Extended Vaisman's theorem to compact complex spaces with singularities.
result Vaisman's theorem extended to compact spaces with singularities.
In this paper, we establish compactness results of some class of conformally compact Einstein 4-manifolds. In the first part of the paper, we improve the earlier results obtained by Chang-Ge. In the second part of the paper, as applications, we derive some compactness results under perturbation conditions when the L^2-…
When studying the causal propagation of a field in a globally hyperbolic spacetime M, one often wants to express the physical intuition that it has compact support in spacelike directions, or that its support is a spacelike compact set. We compare a number of logically distinct formulations of this idea, and of the com…
The study classifies all compact 5D polytopes with 9 facets.
problem Classifying compact hyperbolic Coxeter polytopes.
method Complete classification through mathematical analysis.
result A complete list of compact hyperbolic Coxeter 5D polytopes with 9 facets.
Classifies π1-injective maps between non-compact surfaces.
problem Characterizing maps with injective fundamental groups.
method Proper homotopy classification of maps.
result All π1-injective proper maps are classified. Circle packings on compact surfaces simplified.
problem Simplifying circle packings on complex surfaces.
method Uniformisation of weighted maps.
result Unified approach to circle packings.
Thirty years after the birth of foliations in the 1950's, André Haefliger has introduced a special property satisfied by holonomy pseudogroups of foliations on compact manifolds, called compact generation. Up to now, this is the only general property known about holonomy on compact manifolds. In this article, we give a…
Paper studies heat flow for VT harmonic maps on compact manifolds.
problem Existence of VT harmonic maps and geodesics on compact manifolds.
method Heat flow method to solve Dirichlet problem and existence of geodesics.
result Existence of VT harmonic maps and geodesics under certain conditions.
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.
Develops Lie algebraic approach for compact complex homogeneous manifolds.
problem Proves important results on compact complex homogeneous manifolds.
method Uses standard results in Lie theory to associate a canonical abelian Lie algebra with a given integrable complex structure.
result Provides a new method of associating a canonical abelian Lie algebra with a given integrable complex structure.
We construct a compact metric space that has any other compact metric space as a tangent, with respect to the Gromov-Hausdorff distance, at all points. Furthermore, we give examples of compact sets in the Euclidean unit cube, that have almost any other compact set of the cube as a tangent at all points or just in a den…
Defined and proved monotonicity of a product on compact Hermitian manifolds.
problem Defining and proving properties of a product on compact Hermitian manifolds.
method Proved the well-definedness and monotonicity of the relative non-pluripolar product.
result Monotonicity of the relative non-pluripolar product in terms of masses on compact Hermitian manifolds.
Extends weak continuity of Yang-Mills connections to a broader class.
problem Weak compactness of Ω-Yang-Mills connections. method Compensation compactness argument applied to Yang-Mills fields.
result Weak continuity result extended to Ω-Yang-Mills connections. The reduction of biharmonic maps equation in terms of the Maurer-Cartan form for all smooth map of any compact Riemannian manifolds into a compact Lie group with bi-invariant Riemannian metric is obtained. By this formula, all the biharmonic curves into a compact Lie group and all biharmonic maps from a 2-dimensional o…
We study locally compact contractive local groups, that is, locally compact local groups with a contractive pseudo-automorphism. We prove that if such an object is locally connected, then it is locally isomorphic to a Lie group. We also prove a related structure theorem for locally compact contractive local groups whic…
Compact, non-convex curve flows are created.
problem Creating compact, non-convex ancient solutions for curve shortening flow.
method Constructed an ancient solution asymptotic to Yin-Yang curve.
result Compact, non-convex ancient solutions for curve shortening flow are demonstrated.
We discuss our recent results on the existence and classification problem of complex and Kaehler structures on compact solvmanifolds. In particular, we determine in this paper all the complex surfaces which are diffeomorphic to compact solvmanifolds (and compact homogeneous manifolds in general).
We develop the basic topological properties of compact polygons, i.e. of compact topological Tits buildings of rank two. It is proved that the Coxeter diagram of such a building is always crystallographic, that is, compact connected n-gons exist only for n=3,4,6. We classify compact polygons which admit a transitive gr…
Paper proves uniqueness of Einstein metrics on balls.
problem Uniqueness of conformally compact Einstein metrics on balls.
method Compactness result for conformally compact Einstein metrics.
result Global uniqueness of Graham-Lee metrics on balls.
Smooth compactness theorem for elasticae, except straight segments.
problem Compactness of elasticae space.
method Smooth compactness theorem proof.
result Smooth stability results for minimizers.
Characterizes isolated compact subgroups in Lie groups.
problem Identifying isolated compact subgroups in Lie groups.
method Characterization based on intrinsic structure, irrelevant ambient group details.
result Characterization of isolated compact subgroups depends only on intrinsic structure.
In 1995, S. Adams and G. Stuck as well as A. Zeghib independently provided a classification of non-compact Lie groups which can act isometrically and locally effectively on compact Lorentzian manifolds. In the case that the corresponding Lie algebra contains a direct summand isomorphic to the two-dimensional special li…
Extends optimal regularity and Uhlenbeck compactness to non-Riemannian manifolds.
problem Establishing optimal regularity and compactness for connections on vector bundles over non-Riemannian manifolds.
method Proofs based on RT-equations for connections with Lp curvature, extending to non-compact gauge groups. result Removes singularities at GR shock waves, ensuring existence of geodesics and coordinates.
No exact G₂-structures on compact Lie group quotients.
problem Existence of exact G₂-structures on compact quotients of Lie groups.
method Analyzing compact quotients of seven-dimensional Lie groups by co-compact discrete subgroups.
result Compact quotients of seven-dimensional Lie groups by co-compact discrete subgroups do not admit exact G₂-structures induced by left-invariant ones.
Maps between non-compact surfaces can have geometric kernels under certain conditions.
problem Understanding when maps between non-compact surfaces have geometric kernels.
method Using Brown's proper fundamental group to establish sufficient conditions for geometric kernels.
result Characterization of conjugacy classes in the proper fundamental group and sufficient conditions for geometric kernels.
Compact hyperbolic complex manifolds are rigid under deformation.
problem Studying the deformation behavior of compact hyperbolic complex manifolds.
method Analyzing smooth families of compact complex manifolds over the unit disk and compact Riemann surfaces.
result The H-locus is either at most a discrete subset or the whole domain, depending on the family structure.