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.
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.
Study of eta invariant for non-compact manifolds via Dirac-type operators.
problem Defining and studying the relative eta invariant for non-compact manifolds.
method Defined the relative eta function and studied its variation and gluing law.
result Shows the relative eta invariant coincides with a previously defined version.
New hierarchy for a special group type.
problem Classifying relatively hyperbolic virtually special groups.
method Constructing a new virtual quasiconvex hierarchy.
result Generalized Malnormal Special Quotient Theorem.
Develops theory of relatively geometric actions on CAT(0) cube complexes.
problem Understand actions of relatively hyperbolic groups on CAT(0) cube complexes.
method Introduces and studies relatively geometric actions, proving key results.
result Proves full relatively quasi-convex subgroups are convex compact.
Study eta invariant on non-compact manifolds with positive scalar curvature.
problem Proving geometric formulas and index theorems for uniformly positive scalar curvature metrics.
method Using Dirac-Schrödinger operators and relative eta invariant.
result New geometric formula for spectral flow and index formula for uniformly positive scalar curvature metrics.
Study on opers over complex manifolds of dimension one.
problem Investigating opers over complex manifolds of dimension one.
method Introducing relative opers and differential operators, analyzing their equivalence.
result Bijective correspondence between relative opers and differential operators.
Construct Kähler metrics with negative holomorphic bisectional curvature on compact relative Kähler fibrations.
problem Constructing Kähler metrics with negative holomorphic bisectional curvature on compact relative Kähler fibrations.
method Constructing Kähler metrics with negative holomorphic bisectional curvature on compact relative Kähler fibrations using specific curvature conditions.
result Explicit construction of Kähler metrics with negative holomorphic bisectional curvature on compact relative Kähler fibrations.
Non-uniform lattices in PU(n,1) cannot geometrically act on CAT(0) cube complexes.
problem Non-uniform lattices in PU(n,1) cannot geometrically act on CAT(0) cube complexes.
method Proving non-uniform lattices in PU(n,1) cannot geometrically act on CAT(0) cube complexes.
result Non-uniform lattices in PU(n,1) cannot geometrically act on CAT(0) cube complexes.
The paper characterizes convex co-compact groups with one-dimensional boundary faces.
problem Characterizing convex co-compact groups with specific boundary properties.
method Proving relative hyperbolicity and using coarse Hilbert dimension.
result Convex co-compact groups with one-dimensional boundary faces are relatively hyperbolic.
We introduce a notion of relative isospectrality for surfaces with boundary having possibly non-compact ends either conformally compact or asymptotic to cusps. We obtain a compactness result for such families via a conformal surgery that allows us to reduce to the case of surfaces hyperbolic near infinity recently stud…
Projective geometry aids in analyzing fields near compact manifolds.
problem Analyzing fields near compact manifolds.
method Developed a projective exterior differential tractor calculus.
result Analogous calculus for projectively compact manifolds.
Analytic torsion defined for non-compact Lie groups and discrete subgroups.
problem Defining and calculating analytic torsion for non-compact Lie groups and their discrete subgroups.
method Localised analytic torsion and relative analytic torsion defined for Lie groups of type I, using representations and discrete subgroups.
result Relative analytic torsion of (G,Γ) coincides with Lott L2 analytic torsion of a covering space. The study of topological groups with compact open subgroups and their geometric properties.
problem Characterizing and understanding topological groups with compact open subgroups.
method Geometric techniques, discrete actions on complexes, quasi-isometry invariance, and hyperbolic fine graphs.
result Generalizations of discrete group results to topological groups with compact open subgroups.
We show that the aspherical manifolds produced via the relative strict hyperbolization of polyhedra enjoy many group-theoretic and topological properties of open finite volume negatively pinched manifolds, including relative hyperbolicity, nonvanishing of simplicial volume, co-Hopf property, finiteness of outer automor…
Formula for Dolbeault cohomologies of complex manifolds via relative cohomology.
problem Calculating Dolbeault cohomologies of complex manifolds after blowing up.
method Introducing relative Dolbeault cohomology, proving blow-up formula, and using bimeromorphic invariance.
result Uniform proof of bimeromorphic invariance of Dolbeault cohomology numbers.
Given a compact Alexadrov n-space Z with curvature curv ≥κ, and let f:Z→X be a distance non-increasing onto map to another Alexandrov n-space with curv ≥κ. The relative volume rigidity conjecture says that if X achieves the relative maximal volume i.e. vol(Z)=vol(X), then X is isometric to $…
The paper defines conditions for groups acting on convex domains to be relatively hyperbolic.
problem Understanding conditions for groups acting on convex domains to be relatively hyperbolic.
method Analyzing the geometry of the convex domain to determine relative hyperbolicity.
result Established necessary and sufficient conditions for groups to be relatively hyperbolic.
Proves compact connected components in character varieties of punctured spheres.
problem Character varieties of punctured spheres into Hermitian Lie groups.
method Non-Abelian Hodge correspondence and parabolic Higgs bundles.
result Compact connected components with counter-intuitive properties.
Benjamini-Schramm convergence equals spectral convergence for lattices.
problem Equivalence of convergence notions for lattices.
method Extending conditions to locally compact groups and using relative L2-theory.
result Equivalence of Benjamini-Schramm and spectral convergence under mild conditions.
The paper establishes distance estimates for manifolds with lower scalar curvature bounds.
problem Distance estimates on manifolds with lower scalar curvature bounds.
method Introduced a definition of relative index via a deformed Dirac operator trick and proved index coincidence with Callias operators.
result Proved short neck inequality and quantitative shielding result with positive scalar curvature.
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.
Study on symmetric operators on non-compact manifolds, focusing on their index modulo 2.
problem Investigating elliptic operators with a specific symmetry and their index modulo 2.
method Analysis of Callias-type operators on non-compact manifolds, establishing mod 2 versions of index theorems.
result Established mod 2 versions of the Gromov-Lawson relative index theorem, Callias index theorem, and Boutet de Monvel's index theorem for Toeplitz operators.
We show how to construct absolutely exotic smooth structures on compact 4-manifolds with boundary, including contractible manifolds. In particular, we prove that any compact smooth 4-manifold W with boundary that admits a relatively exotic structure contains a pair of codimension-zero submanifolds homotopy equivalent t…
The main theorem shows that if M is an irreducible compact connected orientable 3-manifold with non-empty boundary, then the classifying space BDiff(M rel dM) of the space of diffeomorphisms of M which restrict to the identity map on boundary(M) has the homotopy type of a finite aspherical CW-complex. This answers, for…
Study the APS index theorem for even-dimensional manifolds with non-compact boundaries.
problem Define and compute the relative η-invariant for Callias-type operators.
method Use the index of the APS boundary value problem and spectral flow.
result The relative η-invariant behaves as the difference of η-invariants of two operators.
Study on complex spaces and their properties.
problem Characterizing relative properties of complex spaces.
method Analyzing indefinite Euclidean and non-flat complex spaces.
result Indefinite Euclidean complex space is not a relative of an indefinite non-flat complex space form.
Recently, Honda, Kazez and Matic described an adapted partial open book of a compact contact 3-manifold with convex boundary by generalizing the work of Giroux in the closed case. They also implicitly established a one-to-one correspondence between isomorphism classes of partial open book decompositions modulo positive…
Non-relatively hyperbolic separating curve graph for surfaces.
problem Classifying hyperbolicity of separating curve graphs.
method Proof of non-relatively hyperbolic property.
result Separating curve graph is not relatively hyperbolic for surfaces with genus ≥ 3 and one boundary component.
In this two papers we deal with the relative homotopy Dirichlet problem for p-harmonic maps from compact manifolds with boundary to manifolds of non-positive sectional curvature. Notably, we give a complete solution to the problem in case the target manifold is either compact and a new proof in case it is rotationally …
Absolute index theorem for warped product manifolds.
problem Equivariant index computation for manifolds with warped product structures.
method Warped product structure, Fredholm operator, Atiyah-Segal-Singer index theorem.
result Equivariant relative index theorem for manifolds with warped product structures.
We characterize the boundaries of positive holomorphic chains (with both compact and non-compact support) in an arbitrary complex manifold. We then consider a compact oriented real submanifold of dimension 2p-1 in a compact Kahler manifold X and address the question of which relative homology classes in H_{2p}(X,M;Z) a…
We define relative Gromov-Witten invariants of a symplectic manifold relative to a codimension two symplectic submanifold. These invariants are the key ingredients in the symplectic sum formula of [IP4]. The main step is the construction of a compact space of `V-stable' maps. Simple special cases include the Hurwitz nu…
Paper defines a new invariant for 4-manifolds with boundary.
problem Defining a new invariant for 4-manifolds with boundary.
method Defined the relative L-invariant by measuring path lengths in the cut complex of a trisection surface.
result Shows that if a rational homology ball has a zero relative L-invariant, it must be the 4-ball.
Compact character varieties of punctured spheres are proven.
problem Compact relative SO0(2,q)-character varieties of punctured spheres. method Non-abelian Hodge correspondence and Geometric Invariant Theory (GIT).
result Proves the existence of compact, totally non-hyperbolic character varieties.
Abstract: Heat kernel estimates lead to compactness results for perturbations by potentials.
problem Relative compactness of perturbations by potentials.
method Heat kernel estimates and domination principle.
result Abstract results on relative compactness of perturbations by potentials.
A relatively simple algebraic framework is given, in which all the compact symmetric spaces can be described and handled without distinguishing cases. We also give some applications and further results.
The paper decomposes metrics on manifolds with boundaries.
problem Characterizing Riemannian metrics on manifolds with boundaries.
method Koiso-type decomposition and Ebin-type slice theorems.
result Characterization of relative Einstein metrics.
We study a relative trace formula for a compact Riemann surface with respect to a closed geodesic C. This can be expressed as a relation between the period spectrum and the ortholength spectrum of C. This provides a new proof of asymptotic results for both the periods of Laplacian eigenforms along C as well estim…
The paper explores how the unit inclusion affects topological quantum field theories in non-semisimple categories.
problem Understanding the effects of unit inclusion in non-semisimple braided tensor categories on topological quantum field theories.
method Analyzes the dualizability of the unit inclusion morphism in Morita 4-category of braided tensor categories and applies the Cobordism Hypothesis.
result Shows that the unit inclusion in non-semisimple modular categories leads to non-compact relative 3D topological quantum field theories.
Characterizes relative hyperbolicity using Morse and contracting boundaries.
problem Understanding relative hyperbolicity in groups.
method Boundary-theoretic characterization using Morse and contracting boundaries.
result Non-elementary relatively hyperbolic groups have non-empty and compact Morse or contracting boundaries.
Dehn fillings for relatively hyperbolic groups generalize the topological Dehn surgery on a non-compact hyperbolic 3-manifold such as a hyperbolic knot complement. We prove a rigidity result saying that if two non-elementary relatively hyperbolic groups without suitable splittings have sufficiently many isomorphic De…
Local minimality proven for stable free-boundary minimal hypersurfaces.
problem Proving local minimality for stable free-boundary minimal hypersurfaces.
method Using relative current setting and strict stability, proving local minimality among relative cycles.
result Local minimality of stable free-boundary minimal hypersurfaces in a small tubular neighborhood.
Handles verify a manifold conjecture.
problem Verifying a conjecture about submanifolds in manifolds.
method Using handlebodies to construct a family of sets.
result A family of sets forms a neighbourhood basis of the manifold.
New condition controls quasiconvex subgroups in hyperbolic 3-manifolds.
problem Controlling behavior of quasiconvex subgroups in hyperbolic 3-manifolds.
method New condition on relatively hyperbolic Dehn filling.
result Provides new proof of virtual fibering of hyperbolic 3-manifolds.
Defines a spinorial quasilocal mass for compact manifolds.
problem Calculating mass for compact manifolds with boundaries.
method Spinorial construction using pullback of dual spinors.
result Positivity of quasilocal energy in both Riemannian and Lorentzian settings.
Study complex Monge-Ampère equations on compact Kähler manifolds.
problem Finite energy range of complex Monge-Ampère operator.
method Survey and general answer to Guedj-Zeriahi's question.
result General answer to Guedj-Zeriahi's question about finite energy range.
The study proves stabilizing of ascending chains in specific groups.
problem Stabilization of ascending chains in bounded rank subgroups of 3-manifold groups.
method Reduction to hyperbolic 3-manifolds and use of geometrization.
result Ascending chains in toral relatively hyperbolic groups stabilize.