Study on equivariant Heegaard genus of reducible 3-manifolds with group actions.
problem Understanding the equivariant Heegaard genus of reducible 3-manifolds with group actions.
method Thin position theory for 3-dimensional orbifolds to establish bounds on equivariant Heegaard genus.
result Sharp bounds on equivariant Heegaard genus of reducible manifolds, similar to tunnel number results.
Estimates on Dirac operator eigenvalues for reducible manifolds.
problem Estimating eigenvalues of the Dirac operator on specific manifolds.
method Using Laplace-Beltrami operator and scalar curvature for estimates.
result Sharp estimates on the first eigenvalue, reducing to Alexandrov's result.
Killing tensors on reducible spaces are reducible, except for special cases.
problem Characterizing Killing tensors on reducible spaces.
method Analyzing Killing tensors on product manifolds and their lifts.
result Killing tensors on product manifolds are reducible, except for specific cases.
Study mapping class groups on invariant incompressible surfaces in reducible 3-manifolds.
problem Understanding the effect of mapping class groups on incompressible surfaces in reducible 3-manifolds.
method Analyzing the mapping class group of reducible 3-manifolds and their incompressible surfaces.
result Characterize reducible 3-manifolds that admit Anosov tori.
We give three infinite families of examples of nonhyperbolic Dehn fillings on hyperbolic manifolds. A manifold in the first family admits two Dehn fillings of distance two apart, one of which is toroidal and annular, and the other is reducible and ∂-reducible. A manifold in the second family has boundary consi…
We construct infinitely many manifolds admitting both strongly irreducible and weakly reducible minimal genus Heegaard splittings. Both closed manifolds and manifolds with boundary tori are constructed.
Study 4D steady gradient Ricci solitons reducing to 3D manifolds.
problem Understanding 4D steady gradient Ricci solitons that reduce to 3D.
method Analyzing asymptotic geometry and curvature properties.
result 4D solitons either reduce to spherical space forms or the 3D Bryant soliton.
Study conformal product structures on reducible Riemannian manifolds.
problem Characterize conformal product structures on compact reducible Riemannian manifolds.
method Analyzing technical assumptions and properties of conformal product structures.
result Under suitable conditions, underlying manifolds are either conformally flat or triple products.
Study homogeneous Lorentzian manifolds under reductive Lie groups, reducing descriptions to semisimple groups.
problem Characterize homogeneous Lorentzian manifolds under reductive Lie groups.
method Analyze manifolds M=G/L for connected reductive Lie groups G and reductive subgroup L; focus on totally reducible isotropy representations. result Homogeneous Lorentzian manifolds reduce to semisimple Lie groups, and are reductive.
The study finds knots with large bridge numbers in handlebodies that become boundary-reducible after surgery.
problem Understanding the relationship between knot bridge numbers and boundary-reducibility after Dehn surgery.
method Examined hyperbolic knots in 3-dimensional handlebodies of genus g and performed Dehn surgeries.
result Knots with arbitrarily large bridge numbers can lead to boundary-reducible manifolds after surgery.
This paper constructs exotic diffeomorphisms on reducible 4-manifolds with odd b+.
problem Detecting exotic diffeomorphisms on reducible 4-manifolds with odd b+.
method Using a families Bauer-Furuta invariant and pin-cobordism.
result Constructs exotic diffeomorphisms on reducible 4-manifolds with odd b+.
If a simple 3-manifold M admits a reducible and a toroidal Dehn filling, the distance between the filling slopes is known to be bounded by three. In this paper, we classify all manifolds which admit a reducible Dehn filling and a toroidal Dehn filling with distance 3.
We consider cones over manifolds admitting real Killing spinors and instanton equations on connections on vector bundles over these manifolds. Such cones are manifolds with special (reduced) holonomy. We generalize the scalar ansatz for a connection proposed by Harland and Nolle in such a way that instantons are parame…
This work proposes a model for geodesic distances and flows on manifolds.
problem Geodesic distances and flows on differentiable manifolds.
method Manifold-augmented Eikonal equation solutions.
result Geodesic flow provides globally length-minimizing curves.
A study on 3-manifolds and handle additions with constraints on minimal intersection.
problem Understanding ∂-reducible handle additions in 3-manifolds. method Analyzing the geometric intersection number of separating slopes on a manifold boundary.
result The minimal intersection number of separating slopes is at most 8 for ∂-reducible handle additions. If a hyperbolic 3-manifold M admits a reducible and a finite Dehn filling, the distance between the filling slopes is known to be 1. This has been proved recently by Boyer, Gordon and Zhang. The first example of a manifold with two such fillings was given by Boyer and Zhang. In this paper, we give examples of hyperboli…
New proof confirms 4-manifolds with weakly reducible genus-three trisections are standard.
problem Proving 4-manifolds with weakly reducible genus-three trisections are standard.
method Using weak reducibility from Heegaard theory, the tools and techniques borrowed from 3-manifold topology.
result Proves Meier's conjecture for weakly reducible genus-three trisections.
We prove a lower estimate for the first eigenvalue of the Dirac operator on a compact locally reducible Riemannian spin manifold with positive scalar curvature. We determine also the universal covers of the manifolds on which the smallest possible eigenvalue is attained.
The study of universal links in 3-manifolds and their properties.
problem Existence and characterization of universal links in 3-manifolds.
method Analyzing branched coverings and distinguishing between universal and complement universal links.
result Closed spherical 3-manifolds are the only ones admitting universal links.
We prove a homological version of a conjecture about the homotopy type of diffeomorphism spaces of reducible 3-manifolds.
problem Proving a conjecture about the homotopy type of diffeomorphism spaces of reducible 3-manifolds.
method Homological approach to show finitely many nonzero homology groups, each finitely generated.
result BDiff(M, rel ∂) has finitely many nonzero homology groups, each finitely generated, for connected sums of irreducible 3-manifolds with nontrivial and non-spherical boundaries.
The study identifies GOF-knots in 3-manifolds with reducible genus two Heegaard splittings.
problem Determining GOF-knots in 3-manifolds with reducible genus two Heegaard splittings.
method Using a necessary and sufficient condition for decomposing a genus two handlebody, the study identifies GOF-knots by analyzing simple closed curves on the Heegaard surface.
result The number and positions of GOF-knots in 3-manifolds with reducible genus two Heegaard splittings are determined.
The study of flat manifolds and their reducible holonomy groups.
problem Holonomy groups of compact flat manifolds and their reducibility.
method Algebraic and geometric analysis of holonomy-invariant subspaces and foliations.
result Compact flat manifolds admit nonzero proper parallel distributions with compact leaves.
Let M be a simple 3-manifold such that one component of ∂M, say F, has genus at least two. For a slope α on F, we denote by M(α) the manifold obtained by attaching a 2-handle to M along a regular neighborhood of α on F. If M(α) is reducible, then α is called a reducing slope. In this paper…
We give new bounds for the distance between two exceptional filling slopes for a 1-cusped hyperbolic 3-manifold in several different situations. The distance between a reducible slope and a slope that produces a manifold with finite fundamental group is at most 2. The distance between a reducible slope and one that pro…
New proof of Haken's Lemma and Scharlemann's theorem using thin position.
problem Proving Haken's Lemma and Scharlemann's theorem about Heegaard surfaces in 3-manifolds.
method Using thin position of Heegaard splittings.
result A new proof of Haken's Lemma and Scharlemann's theorem.
Constructs knots with maximal Thurston-Bennequin number and creates reducible 3-manifolds.
problem Creating knots with maximal Thurston-Bennequin number and reducible 3-manifolds.
method Uses Stein handle decompositions of D4 to construct Legendrian representatives of knots. result Constructs infinitely many knots yielding reducible 3-manifolds by Legendrian surgery.
In this paper, we study a class of Finsler metrics which contains the class of P-reducible metrics. Finsler metrics in this class are called generalized P-reducible metrics. We consider generalized P-reducible metrics with scalar flag curvature and find a condition under which these metrics reduce to C-reducible metric…
Character scheme of small Seifert 3-manifolds is reduced if and only if no exceptional abelian character exists.
problem Character scheme of small Seifert 3-manifolds and its reduction condition.
method Complete description of the SL2(C)-character scheme of small Seifert 3-manifolds.
result Character scheme is reduced if and only if no exceptional abelian character exists, and exceptional abelian character has multiplicity 2.
The holonomy group of flat manifolds acts reducibly, leading to compact leaf foliations.
problem Understanding the geometric and algebraic properties of flat manifolds and their foliations.
method Analyzing the holonomy group and sublattice-invariant subspaces of compact flat manifolds.
result Compact leaf foliations of flat manifolds have intrinsic geometric properties related to the original manifold.
Finite type for 3-manifolds with boundary.
problem Finite type for 3-manifolds with boundary.
method Contractible space of reducing spheres, homotopy type of finite CW complex.
result B Diff(M rel boundary) has the homotopy type of a finite CW complex.
Proof shows smooth minimal hypersurfaces for perimeter-minimizing sets in low-dimensional Riemannian manifolds.
problem Finding sets of least perimeter in Riemannian manifolds.
method Short proof using de Giorgi and Miranda's tradition in flat space.
result Reduced boundary of least perimeter sets is a smooth minimal hypersurface in low dimensions.
Study deformations of reducible SL(n,C) representations in fibered 3-manifolds.
problem Deformations of reducible representations in fibered 3-manifold groups.
method Analyzing the eigenvalues of monodromy and using representation theory.
result Reducible representations are contained in specific dimensional components of the representation variety.
A new proof shows a reducing sphere can be obtained from a given sphere without surgeries.
problem Finding a reducing sphere in a 3-manifold.
method Using a sequence of 1-surgeries, show a reducing sphere can be obtained without surgeries.
result A reducing sphere can be obtained from a given sphere without surgeries.
This paper establishes transversality for perturbed Vafa-Witten moduli spaces on 4-manifolds.
problem Transversality of Vafa-Witten moduli spaces on 4-manifolds with C≡0. method Constructing perturbation terms to ensure the moduli space is a smooth manifold of dimension zero.
result For generic perturbation terms, the moduli space of solutions is a smooth manifold of dimension zero.
Unified framework for gravity on manifolds with corners derived.
problem Unified description of gravity on manifolds with corners.
method Derivation of corner Poisson structure and reduction procedure.
result Unified framework for bulk, boundary, and corner structures of Palatini-Cartan gravity.
New findings on Weyl-reducible conformal manifolds and their universal covers.
problem Characterizing the structure of universal covers of Weyl-reducible conformal manifolds.
method Analyzing the properties of Weyl connections and their universal covers.
result For the case where the dimension of the universal cover's irreducible part is 2, the manifold must have dimension 1 or 2.
A compact 4-dimensional manifold is a non-singular graph-manifold if it can be obtained by the glueing T^2-bundles over compact surfaces (with boundary) of negative Euler characteristics. If none of glueing diffeomorphisms respect the bundle structures, the graph-structure is called reduced. We prove that any homotopy …
Sparse reduced-rank regression selects variables and ranks via manifold optimization.
problem Traditional rank selection fails when true rank is high.
method Sparse regularization and manifold optimization for rank and variable selection.
result Accurate estimation of coefficient parameter with high true rank.
Continues work on derived manifolds and symplectic schemes, constructing virtual classes.
problem Constructing virtual fundamental classes for derived manifolds and schemes.
method Cosection localization, reduced virtual fundamental classes, and applications to Donaldson-Thomas theory.
result Virtual fundamental classes for (−2)-shifted symplectic derived schemes are consistent with algebraic and differential geometric constructions. We determine the Dehn surgeries on 2-bridge links, which yield reducible 3-manifolds. Further, we show the conditions that we obtain a torus or cable knot from one component of a 2-bridge link by a surgery on another component.
Let M be a simple 3-manifold with a toral boundary component partial_0 M. If Dehn filling M along partial_0 M one way produces a toroidal manifold and Dehn filling M along partial_0 M another way produces a boundary-reducible manifold, then we show that the absolute value of the intersection number on partial_0 M of th…
Probabilistic descent on manifolds using triangular sets and embedding theorems.
problem Descent over manifolds defined by polynomials.
method Triangularization, embedding theorem, numerical continuation.
result Effective numerical method for probabilistic descent.
In this thesis, we use normal surface theory to understand certain properties of minimal triangulations of compact orientable 3-manifolds. We describe the collapsing process of normal 2-spheres and disks. Using some geometrical constructions to take connected sums of triangulated 3-manifolds, we obtain the following re…
In this paper, we shall prove that any Heegaard splitting of a ∂-reducible 3-manifold M, say M=W∪V, can be obtained by doing connected sums, boundary connected sums and self-boundary connected sums from Heegaard splittings of n manifolds M1,...,Mn where Mi is either a solid torus or a $…
In this paper we investigate the relationship between the existence of parallel semi-Riemannian metrics of a connection and the reducibility of the associated holonomy group. The question as to whether the holonomy group necessarily reduces in the presence of a specified number of independent parallel semi-Riemannian m…
Solves modified conjecture for Fano manifolds using Ding stability.
problem Finding Kähler-Einstein metrics on Fano manifolds.
method Interprets Ding semistability and solves modified conjecture.
result Solves modified conjecture for coupled Kähler-Einstein metrics on Fano manifolds.
Study on metrics on closed manifolds with specific holonomy groups.
problem Characterizing metrics on closed manifolds with reducible holonomy groups.
method Analyzing locally conformally Berwald metrics and quotient spaces by homotheties.
result A de Rham type splitting theorem for analytic manifolds.
The study provides a structure theorem for a new class of noncompact 3-manifolds.
problem Understanding a new class of noncompact 3-manifolds.
method Proved a structure theorem for irreducible open graph manifolds.
result A canonical 'reduced' decomposition of irreducible open graph manifolds along embedded, incompressible 2-tori.