We give a lower estimate of the framing function of knots, and prove a strengthened version of Dehn's lemma conjectured by Greene-Wiest.
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48 results for “Dehn's Lemma”
Dehn's Lemma for loops in 3-manifolds.
problem Dehn's Lemma for loops in 3-manifolds.
method Null-homotopic curve in a 3-manifold M can be displaced by a height function in a collar of the boundary.
result An immersed closed curve in the boundary of a 3-manifold can be displaced to a simple closed curve bounding a disk in the 3-manifold.
The study examines subgroups of torus mapping class group generated by Dehn twists powers.
problem Characterizing subgroups generated by powers of Dehn twists.
method Using the ping pong lemma and geometric intersection numbers.
result Subgroups can be free groups, direct products, or have specific ranks.
A lantern lemmamath.GT
We show that in the mapping class group of a surface any relation between Dehn twists of the form T_x^j T_y^k = M (M a multitwist) is the lantern relation, and any relation of the form (T_x T_y)^k = M (where T_x commutes with M) is the 2-chain relation.
We investigate certain -dimensional analogues of the classical -dimensional Dehn's lemma, giving examples where such analogues do or do not hold, in the smooth and topological categories. In particular, we show that an essential -sphere in the boundary of a simply connected -manifold such that i…
Explains the Schwarz lemma in lecture notes.
problem None explicitly stated; focuses on explanation.
method Expository notes on the Schwarz lemma.
result Explains the Schwarz lemma.
Whitehead's Lemma is a simple result of Stallings folds.
problem None explicitly stated; Whitehead's Lemma is observed as a consequence.
method Observation of Whitehead's Lemma as a consequence of Stallings folds.
result Whitehead's Lemma is a simple result of Stallings folds.
Author provides an alternate proof of the free ribbon lemma.
problem Proving that every free sphere-link in the 4-sphere is a ribbon sphere-link.
method An alternate proof of the free ribbon lemma.
result Provides an alternate proof of the free ribbon lemma.
The paper extends Schwarz's lemma to RC-positivity and complex manifolds.
problem Comparing metrics with RC-positivity in complex manifolds.
method Establishing Schwarz lemmas for RC-positivity and applying them to complex manifolds.
result New diameter and volume comparison theorems.
New CR-structures lemma simplifies CR-manifold deformation proof.
problem Deformation unobstructedness of CR-manifolds.
method New Tian-Todorov lemma applied to CR-manifolds.
result Reproved deformation unobstructedness of CR-manifolds.
Survey on strong closing lemmas in Hamiltonian dynamics.
problem Understanding dynamics in Hamiltonian systems.
method Use spectral invariants in symplectic geometry.
result Proofs of strong closing lemmas in various dimensions.
Study several weak forms of a lemma on compact complex manifolds.
problem Understanding weak forms of a lemma on compact complex manifolds.
method Complete unified study of weak forms of the $\ddb-$Lemma.
result Unified understanding of various weak forms of the $\ddb-$Lemma.
Unified Schwarz lemma in Kähler and Hermitian geometry.
problem Various forms of the Schwarz lemma in Kähler and Hermitian geometry.
method Introducing new curvatures to refine and elucidate the real bisectional curvature.
result Unified Chern-Lu, Aubin-Yau, and Chen-Cheng-Look Schwarz lemmas.
Paper generalizes Schwarz Lemma for VT harmonic maps with conditions.
problem Generalizing Schwarz Lemma for a specific type of harmonic maps.
method Conditions on eigenvalues and Ricci curvature are used to prove the lemma.
result Schwarz Lemma for VT harmonic maps proved with distance and volume decreasing properties.
Formulates Index III lemma and Rauch III theorem with applications.
problem Develops new mathematical theorems based on existing ones.
method Formulation of Index III lemma and Rauch III theorem based on Index I, II lemmas and Rauch I, II theorems.
result Presented Rauch's type theorem and volume comparison result as applications.
New Schwarz Lemma for Bergman metrics in bounded domains.
problem Finding bounds for Bergman metrics in bounded domains.
method Using Cauchy-Schwarz inequality from probability theory.
result Established a new Schwarz Lemma for Bergman metrics.
Selberg's Lemma fails for certain curved manifolds.
problem Applying Selberg's Lemma to negatively curved Hadamard manifolds.
method Proving the failure of Selberg's Lemma for specific groups of manifolds.
result Selberg's Lemma does not hold for discrete isometry groups of negatively curved Hadamard manifolds.
Tucker and Ky Fan's lemma are combinatorial analogs of the Borsuk-Ulam theorem (BUT). In 1996, Yu. A. Shashkin proved a version of Fan's lemma, which is a combinatorial analog of the odd mapping theorem (OMT). We consider generalizations of these lemmas for BUT-manifolds, i.e. for manifolds that satisfy BUT. Proofs rel…
Boundary Value Problem and Discrete Schwarz-Pick Lemma for Generalized Hyperbolic Circle Packingsmath.DG
Paper proves a discrete Schwarz-Pick lemma for generalized circle packings.
problem Comparing geometric quantities of circle packings with different boundary values.
method Combinatorial Calabi flows and maximum principle.
result Discrete Schwarz-Pick lemma proven for generalized circle packings.
The paper improves Zakalyukin's lemma for frontals and applies it to surface singularities.
problem Improving the conditions under which wave front germs imply map germs.
method Generalization of Zakalyukin's lemma for frontals and applications to surface singularities.
result The paper provides a more general version of Zakalyukin's lemma for map germs.
Paper proves almost Schur Lemma on smooth metric measure spaces.
problem Proving a specific lemma on metric measure spaces.
method Proves almost Schur Lemma using closed smooth metric measure spaces.
result Implications of the lemma for X. Cheng's and De Lellis-Topping's results.
Meridian lemma extended to fully alternating links in thickened surfaces.
problem Extending Menasco's meridian lemma to fully alternating links in thickened surfaces.
method Developed a new meridian lemma for fully alternating links in thickened orientable surfaces of positive genus.
result The meridian lemma holds for fully alternating links in thickened surfaces.
A quantitative closing Lemma and Partner orbits on Riemannian manifolds with negative curvaturemath.DG
Proves a quantitative closing lemma for negatively curved manifolds.
problem Closing lemma for negatively curved manifolds.
method Quantitative closing lemma proof.
result Study of partner and pseudo-partner orbits for self-crossing closed geodesics.
Paper generalizes Schwarz lemma for harmonic maps between Riemannian manifolds.
problem Generalizing Schwarz lemma for harmonic maps.
method Using Bochner techniques and sub-Laplacian comparison theorem.
result Established a generalization of Schwarz lemma for transversally harmonic maps.
Margulis Lemma on spacesmath.DG
Extends Margulis Lemma to RCD(K,N) spaces.
problem Applying Margulis Lemma to new geometric structures.
method Improved Regularity Estimates for Regular Langrangian Flows.
result Margulis Lemma extended to RCD(K,N) spaces.
Cyclic surgery, degrees of maps of character curves, and volume rigidity for hyperbolic manifoldsmath.GT
This paper proves a theorem about Dehn surgery using a new theorem about PSL(2, C) character varieties. Confirming a conjecture of Boyer and Zhang, this paper shows that a small hyperbolic knot in a homotopy sphere having a non-trivial cyclic slope r has an incompressible surface with non-integer boundary slope strictl…
Proves Poincaré lemma on specific sub-Riemannian structures.
problem Solving an open problem on Poincaré lemma for corank 1 sub-Riemannian structures.
method Combines Poincaré lemma on Riemannian manifolds with a path integral formula.
result Necessary and sufficient 'curl-vanishing' conditions for Poincaré lemma.
Study on homological Dehn functions of groups of type .
problem Understanding the homological Dehn functions of groups of type .
method Proved foundational results, studied homological Dehn functions of Leary's groups, and provided methods to obtain groups with specific homological Dehn functions.
result Found groups of type with quartic homological Dehn function and unsolvable word problem.
Corrects a lemma in a 2009 paper about contact homology and Floer homology.
problem An error in a lemma about contact homology and Floer homology.
method None, as it is a correction of an existing lemma.
result Corrects an error in a previously published lemma.
The paper characterizes when the -lemma holds for twistor spaces.
problem Characterizing the -lemma for twistor spaces.
method Study Bott-Chern and Aeppli cohomologies of twistor spaces.
result Explicit computation of Dolbeault cohomology for flat torus twistor space.
New Morse lemma handles functions going to negative infinity.
problem Deforming functions with negative infinity behavior.
method Generalized Morse Theory lemma for functions going to negative infinity.
result Valid for functions with suitable growth condition.
For a symplectic manifold , not necessarily hard Lefschetz, we prove a version of the Merkulov --lemma. We also study the --lemma and related cohomologies for compact symplectic solvmanifolds.
Positive representations on surfaces have positive cross-ratios and satisfy a collar lemma.
problem Characterizing representations of surface groups with positive properties.
method Proving a collar lemma and showing positivity of cross-ratios for -positive representations.
result Closed subsets of representation varieties are characterized by -positive representations.
Proves a generalized Whitehead cut vertex lemma for tree groups.
problem Extending Whitehead's cut vertex lemma to tree group conjugacy classes.
method Proves a version of Whitehead's lemma for tree groups.
result Establishes a cut vertex in star graphs for tree group conjugacy classes.
Precise computations of Dehn functions for subgroups of free group products.
problem Computing precise Dehn functions for subgroups of direct products of free groups.
method Analyzing specific subgroups and using algebraic methods to compute Dehn functions.
result Quartic and quadratic Dehn functions for specific subgroups of free group products.
This paper contains a purely topological theorem and a geometric application. The topological theorem states that if M is a simple closed orientable 3-manifold such that π_1(M) contains a genus g surface group and H_1(M;Z/2Z) has rank at least 4g-1 then M contains a closed incompressible surface of genus at most g. Thi…
Proves a general ∂∂̄-lemma and applies it to a Fujino conjecture.
problem Establishing a general ∂∂̄-lemma and its applications.
method Develops a general ∂∂̄-lemma and applies it to Fujino's conjecture.
result Establishes a Kähler version of Fujino's injectivity theorem.
Enhanced Schwarz lemma for Hermitian manifolds with new curvature constraints.
problem Improving Schwarz lemma for holomorphic maps between Hermitian manifolds.
method Introducing new curvature constraints on source and target manifolds, controlling by holomorphic sectional curvature.
result Significant improvements on the Wu--Yau theorem and Schwarz lemma for Gauduchon connections.
Corrects a proof gap in the Morse lemma for hyperbolic spaces.
problem Gap in the proof of optimal bounds for the Morse lemma in Gromov-hyperbolic spaces.
method Describes a computer certification of the corrected result.
result Corrects the main theorem of [ShCh13a] in optimal bounds for the Morse lemma.
Schwarz lemma extended to equality cases and curvature on manifolds.
problem Extending Schwarz lemma to equality cases and studying curvature.
method Analyzing Schwarz lemma inequalities and equalities, studying holomorphic sectional curvature.
result Holomorphic maps are totally geodesic and have constant rank when Schwarz lemma equality holds.
Dehn quandles of groups and surfaces unify various quandle constructions.
problem Understanding and unifying various quandle constructions.
method Introducing Dehn quandles of groups and subsets, proving properties and embeddings.
result Dehn quandles embed naturally into their enveloping groups, and enveloping groups of certain quandles are the quandles themselves.
Generalized Stacey-Roberts lemma for Banach manifolds.
problem Constructing Lie groupoids of smooth mappings in infinite-dimensional geometry.
method Generalization of the Stacey-Roberts lemma to Banach manifolds with smooth partitions of unity.
result Remedied an error in the original proof for finite-dimensional setting.
The paper presents methods to write presentations for Dehn quandles.
problem Writing explicit presentations for Dehn quandles.
method Two approaches: for Garside and Gaussian groups, and for general Dehn quandles with known centralisers.
result Examples of Dehn quandles including spherical Artin groups, surface groups, and mapping class groups.
Thurston's jigglingmath.GT
Thurston's jiggling lemma simplifies triangulations.
problem Simplifying triangulations into a general position.
method Alternative, conceptual proof and generalization to manifolds.
result A more straightforward proof of Thurston's jiggling lemma.
On the order of Dehn twistsmath.GT
Study finds the order of Dehn twists in various groups.
problem Determining the order of Dehn twists in different topological groups.
method Analyzing the specific groups mentioned in the abstract.
result The order of Dehn twists was calculated in the specified groups.
Extends Stein's lemma to exponential-family mixtures for gradient computation.
problem Computing gradients for complex distributions with weak assumptions.
method Generalizes Stein's lemma to exponential-family mixtures and applies it to reparameterization trick.
result Derives new gradient identities for various distributions.
Study shows connectedness of Bowditch boundary persists in long Dehn fillings.
problem Persistence of Bowditch boundary connectedness in Dehn fillings.
method Analysis of relatively hyperbolic group pairs and peripheral subgroups.
result Connectedness of Bowditch boundary persists in sufficiently long Dehn fillings without needing restrictions.
The study explores the Dehn functions of Kähler groups and their properties.
problem Which functions can arise as Dehn functions of Kähler groups?
method Analyzes examples of Kähler groups with various Dehn functions and proves the existence of a Kähler group with a cubic bounded Dehn function.
result There exists a Kähler group with a cubic bounded Dehn function and an exponential upper bound.