Counterexamples show Marchal's lemma fails for certain N-body systems.
problem Understanding when Marchal's lemma for N-body collisions holds or fails.
method Using metric geometry and the Jacobi-Maupertuis reformulation of mechanics, the team created counterexamples.
result Counterexamples demonstrate Marchal's lemma does not always apply to N-body systems.
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…
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.
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.
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.
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.
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.
For a symplectic manifold (M,ω), not necessarily hard Lefschetz, we prove a version of the Merkulov dδ--lemma. We also study the dδ--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 Θ-positive representations. result Closed subsets of representation varieties are characterized by Θ-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.
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.
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.
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.
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.
Two new boundary Schwarz lemmas for holomorphic maps established.
problem Establishing boundary Schwarz lemmas for holomorphic maps.
method Two boundary versions of the Schwarz lemma for holomorphic maps.
result First boundary Schwarz lemmas for general holomorphic self maps and biholomorphisms with invariant Kähler metrics.
Abstract: Generalizes Milnor-Schwarz lemma to inverse monoids.
problem Applying Milnor-Schwarz lemma to inverse monoids.
method Two proofs provided: elementary and using Vietoris-Rips complex.
result Generalization of Milnor-Schwarz lemma to inverse monoids.
The paper extends the Discrete Schwarz-Pick Lemma to circle packings with obtuse intersections and disjoint packings.
problem Proving the Discrete Schwarz-Pick Lemma for circle packings with various inversive distances.
method Using a variational principle for circle packings with inversive distances, the paper extends the lemma to a broader range of packings.
result The Discrete Schwarz-Pick Lemma holds for circle packings with inversive distances in (−1,1], provided an additional condition on triangle weights. New proof and insights on Elliptical Potential Lemma for online learning.
problem Limitations in the original proof of the Elliptical Potential Lemma.
method Proposes a new proof and new perspectives on the lemma.
result New flexibility in the type of potentials considered.
For the convenience of readers of the article {\em No-arbitrage pricing under systemic risk: accounting for cross-ownership} (Fischer, 2012, arXiv:1005.0768), a full proof of Lemma A.5 and a shorter proof of Lemma A.6 of that paper are provided.
This paper proves an C∞ closing lemma for Hamiltonian flows on symplectic 4-manifolds.
problem Proving the C∞ closing lemma for Hamiltonian flows on symplectic 4-manifolds. method Combining results from geodesic flows on Finsler surfaces with the dual lens map technique, extending to Hamiltonian flows with certain restrictions.
result Established the C∞ closing lemma for a large family of Hamiltonian flows on 4-dimensional symplectic manifolds. Develops a generalized version of Chung's Lemma for stochastic optimization methods.
problem Establishing asymptotic convergence rates for stochastic optimization methods under various step size rules.
method Generalized version of Chung's Lemma for a broader family of step size rules.
result Demonstrates tight non-asymptotic convergence rates for various stochastic methods.
In this article, we prove an analog of the classical collar lemma in the setting of Hitchin representations.
The L2-∂∂-Lemma is extended to complete Kähler manifolds with a gap in the spectrum.
problem Extending the L2-∂∂-Lemma to non-compact Kähler manifolds. method Proving the L2-∂∂-Lemma on complete Kähler manifolds with a gap in the spectrum. result The L2-∂∂-Lemma is generalized to complete Kähler manifolds. The famous Švarc-Milnor Lemma says that a group G acting properly and cocompactly via isometries on a length space X is finitely generated and induces a quasi-isometry equivalence g→g⋅x0 for any x0∈X. We redefine the concept of coarseness so that the proof of the Lemma is automatic.
New CR almost Schur Lemma estimates curvature on compact manifolds.
problem Estimating curvature on compact pseudohermitian manifolds.
method Established a new CR almost Schur Lemma with specific positivity conditions.
result Estimates pseudohermitian scalar curvature as a constant.
Simplified proof of Morrey's lemma for metric spaces.
problem Existence of area-minimizing surfaces in metric spaces.
method Simpler proof using Korn and Lichtenstein's smooth isothermal coordinates.
result Strengthened application to Sobolev maps in metric spaces.
Paper extends Weyl's lemma to RCD(K,N) spaces.
problem Applying Weyl's lemma to RCD(K,N) metric measure spaces.
method Extending Weyl's lemma to RCD(K,N) spaces and proving applications.
result Local regularity of solutions for Poisson equations and Liouville-type results for harmonic functions.
In this note, we prove the sharp Davies-Gaffney-Grigor'yan lemma for minimal heat kernels on graphs.