Generalizes Laudenbach-Poénaru theorem for group actions on 4-manifolds.
problem Extending diffeomorphisms with group actions on specific 4-manifolds.
method Proves a generalization of the Laudenbach-Poénaru theorem for finite group actions on #n(S1imesS2), showing linearly parted actions and equivariant diffeomorphisms. result Finite group actions on #n(S1imesS2) extend to aturaln(S1imesB3), and extensions are equivariantly diffeomorphic. Study trisections of non-orientable 4-manifolds with boundary.
problem Understanding trisections in non-orientable 4-manifolds.
method Introduced trisections of non-orientable 4-manifolds with boundary, proved a non-orientable analogue of a theorem, and discussed adaptation of trisection theory.
result Existence of trisection diagrams and Kirby diagrams for closed non-orientable 4-manifolds.
We simplify and prove a splitting theorem for mapping class groups of connect sums of S2imesS1.
problem Understanding the structure of mapping class groups of specific 3-manifolds.
method We prove a splitting theorem for the mapping class group of connect sums of S2imesS1, simplifying Laudenbach's original proof. result The mapping class group of connect sums of S2imesS1 is the semidirect product of Out(Fn) by (Z/2)n. New proof of Laudenbach and Poénaru's theorem on 4D 1-handlebodies.
problem Proving diffeomorphisms extend to 4D 1-handlebodies and compression bodies.
method Classification of Heegaard splittings and Cerf's theorem on 3-ball diffeomorphisms.
result Every diffeomorphism of the positive boundary extends to the whole compression body.
Given a smooth compact manifold with boundary, we show that the subcomplex of the deformed de Rham complex consisting of eigenspaces of small eigenvalues of the Witten Laplacian is canonically isomorphic to the Thom-Smale complex constructed by Laudenbach. Our proof is based on Bismut-Lebeau's analytic localization tec…
We provide an explicit section for a mapping class group sequence.
problem Mapping class group of connect sums of S2imesS1. method Provided an explicit section for the split exact sequence.
result Explicit section s for the exact sequence. We consider a Morse function f and a Morse-Smale gradient-like vector field X on a compact connected oriented 3-manifold M such that f has only one critical point of index 3. Based on Laudenbach's ideas, we will show that the flow of X can be isotoped into one so that the trajectory spaces of the new flow pro…
Inspired by Katok's examples of Finsler metrics with a small number of closed geodesics, we present two results on Reeb flows with finitely many periodic orbits. The first result is concerned with a contact-geometric description of magnetic flows on the 2-sphere found recently by Benedetti. We give a simple interpretat…
Study the moduli space of reducible 3-manifolds using prime decomposition.
problem Understanding the homotopy type of moduli spaces of reducible 3-manifolds.
method Construct a splitting map from BextrmDiff+(M) to BextrmDiff+(P1⊔⋯⊔Pn), yielding a prime decomposition fibre sequence. result The fibre Hg(P1,…,Pn) is a finite, connected cell complex, and the prime decomposition fibre sequence is effective for computations. The paper studies mapping class groups of locally finite graphs and their associated sphere complexes.
problem Understanding mapping class groups of locally finite graphs and their geometric representations.
method Generalizes results from 3-manifolds to locally finite graphs, proving surjections and splitting properties.
result A sphere complex S(MΓ) associated with a graph Γ, with a faithful action of the mapping class group. The paper studies 3-manifolds with specific Morse-Smale diffeomorphisms and finds they are homeomorphic to lens spaces.
problem Understanding the topology of 3-manifolds with certain Morse-Smale diffeomorphisms.
method Analyzing the structure of fixed points and separatrices of diffeomorphisms in 3-manifolds.
result All supporting manifolds of these diffeomorphisms are homeomorphic to lens spaces.
AI generates theorems and proofs for training theorem provers.
problem Limited human-written theorems and proofs for supervised learning.
method Proposes a neural generator to automatically synthesize theorems and proofs.
result Synthetic data improves automated theorem proving in Metamath.
Global inverse function theorem proved easily using Riemannian geometry.
problem Global inverse function theorem in Riemannian geometry.
method Hopf--Rinow theorem in Riemannian geometry.
result Hadamard's global inverse function theorem is proven easily.
A new comparison theorem for geometric spaces.
problem Geometric space comparison theorems.
method Relative form of Toponogov comparison theorem.
result New geometric space comparison theorem established.
Paper develops formulas and theorems in Hermitian geometry.
problem None explicitly stated in the abstract.
method Develops second variational formulas and index forms in Hermitian geometry.
result Establishes results analogous to classical theorems in Riemannian geometry.
The paper proves three circles theorems and Liouville type theorems for subharmonic and holomorphic functions.
problem Establishing theorems for subharmonic and holomorphic functions on specific geometric structures.
method Using subharmonic and holomorphic functions on Riemannian manifolds and gradient shrinking Ricci solitons.
result Proves Liouville type theorems as applications of the established theorems.
Revises a theorem by Thurston, finding a counter-example and a weaker version.
problem The bounded image theorem in Haken manifolds.
method Providing a counter-example and a weaker version of the second statement of Thurston's theorem.
result A counter-example and a weaker version of the second statement of Thurston's theorem are presented.
Proofs for Moon's theorem and its generalization.
problem Proving Moon's theorem and its generalization.
method Proofs based on key lemmas.
result Generalization of the four-vertex theorem.
Analyzes Saito vanishing theorem using L2 methods.
problem Proving the Saito vanishing theorem.
method Uses L2-methods to prove the theorem. result Analytic proof of the Saito vanishing theorem.
Investigates proving geometric theorems over complex and real numbers using tilings.
problem Proving incidence theorems over C and R using the master theorem.
method Formalizes tiling proofs and introduces a hierarchy of theorems based on topological spaces.
result Identifies which theorems can or cannot be proved over C and R.
Extends symplectic reduction and theorem to Lie algebroids.
problem Symplectic reduction and theorem for Lie algebroids.
method Extends Marsden-Weinstein reduction and Darboux-Moser-Weinstein theorems.
result Obtained coisotropic embedding theorem for symplectic Lie algebroids.
Paper generalizes complex Brunn-Minkowski theory and proves new extension theorems.
problem Complex Brunn-Minkowski theory and extension theorems.
method Hilbert bundle approach to complex Brunn-Minkowski theory.
result Generalizes Guan's sharp strong openness theorem and sharp Ohsawa-Takegoshi extension theorem.
Proves Thurston's bounded image theorem for Haken manifolds.
problem Proving Thurston's bounded image theorem for Haken manifolds.
method Using recent developments in Kleinian group theory.
result A proof of Thurston's original bounded image theorem.
Method upgrades limit theorems to mixing limit theorems for dynamical systems.
problem Improving limit theorems for dynamical systems.
method General method for upgrading limit theorems to mixing limit theorems.
result Mixing limit theorems for specific subbundles of the Kontsevich-Zorich cocycle.
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.
In LM, we proved a family version of the famous Witten rigidity theorems and several family vanishing theorems for elliptic genera. In this paper, we gerenalize our theorems LM in two directions. First we establish a family rigidity theorem for the Dirac operator on loop space twisted by general positive energy loop gr…
The paper explains the topological origin of the distinction between incidence theorems over division rings and fields.
problem Understanding the distinction between incidence theorems over division rings and fields.
method Extending the surface-graph approach to noncommutative settings, the paper analyzes the topological properties of graphs embedded on surfaces of different genera.
result Theorems associated with graphs on the sphere hold over any division ring, while those on surfaces of positive genus typically hold only if the ground ring is a field.
Proves two theorems on odd-dimensional manifolds with boundary.
problem Proving theorems on manifolds with boundaries.
method Proof of theorems using mathematical techniques.
result Proved the general Kastler-Kalau-Walze and Dabrowski-Sitarz-Zalecki type theorems.
Sharp convergence theorem for sphere submanifolds proved.
problem Sphere submanifolds in spheres.
method Proved a sharp convergence theorem.
result New differentiable sphere theorem for submanifolds in spheres.
INT benchmark tests theorem proving agents' ability to generalize to unseen theorems.
problem Evaluating theorem proving agents' ability to generalize to unseen theorems.
method INT benchmark based on a theorem generation and proof procedure with adjustable knobs for measuring 6 types of generalization.
result MCTS can help agents prove new theorems.
A homological selection theorem for C-spaces, as well as, a finite-dimensional homological selection theorem is established. We apply the finite-dimensional homological selection theorem to obtain fixed-point theorems for usco homologically UV^n set-valued maps.
Abstracts a theorem for non-smooth maps in infinite dimensions.
problem Generalizing inverse mapping theorem for non-smooth maps.
method Introduces property A and applies it to non-smooth maps.
result Generalized inverse mapping theorems for non-smooth maps.
Atiyah-Singer theorem links math fields, predicts topological insights.
problem Understanding the interplay between analysis, geometry, and topology.
method Analyzes and generalizes topological invariants in differential geometry.
result Predicts the index of elliptic operators based on topology.
Paper generalizes a theorem for real analytic singularities.
problem No specific problem stated; focuses on generalization.
method Generalization of a theorem for complex singularities.
result Generalized Join theorem for real analytic singularities.
The paper proves injectivity and vanishing theorems on compact Kahler manifolds.
problem Injectivity and vanishing theorems on compact Kahler manifolds.
method Hodge theory, Bochner-Kodaira-Nakano identity, analytic method, transcendental method, Demailly-Peternell-Schneider equisingular approximation theorem, Hormander L2 estimates.
result The main injectivity theorem implies several Nadel type vanishing theorems.
Several proofs of Fáry--Milnor theorem are presented.
problem Fáry--Milnor theorem
method Sketches several proofs
result Proofs of Fáry--Milnor theorem
Reidemeister's theorem proved using smooth functions and transversality.
problem Proving Reidemeister's theorem
method Using smooth functions and transversality
result Reidemeister's theorem proved
Proves an analytic Bertini theorem, generalizing previous work.
problem Generalizing previous results in algebraic geometry.
method Analytic Bertini theorem proof.
result Generalizes previous results in algebraic geometry.
This note explores comparison geometry concepts and theorems.
problem Exploring various comparison theorems in geometry.
method Analyzes Rauch and Toponogov theorems and their applications.
result Introduction of Gromov-Hausdorff convergence and Alexandrov Spaces.
Proof of Tait-Kneser theorem and related variations using Lorentzian geometry.
problem Proving variations of the Tait-Kneser theorem for different conics.
method Using Lorentzian geometry to prove the theorem and its variations.
result Proof of the theorem and its variations concerning different conics.
Proves Skoda's Division Theorem using degeneration and positivity of direct image bundles.
problem Division Theorem in Skoda's context
method Degeneration approach inspired by B. Berndtsson and L. Lempert's L2 extension theorem result Simplified and extended proof of L2 extension theorem We show how Latour's theorem can be understood as a natural generalization of the s-cobordism theorem for cohomology classes u∈H1(M;R). The s-cobordism theorem becomes a special degenerate case when u=0.
Generalizes symplectic reduction to cosymplectic groupoid actions.
problem Symplectic reduction for cosymplectic groupoid actions.
method Introduced cosymplectic groupoid actions and proved a theorem.
result Proved a theorem analogous to Mikami-Weinstein theorem.
Strong Frankel theorem for shrinkers in all dimensions.
problem Intersection of shrinkers in large balls.
method Proof using strong Bernstein theorem for stable Gaussian surfaces.
result Shrinkers are connected in all large balls.
The study establishes comparison theorems for weighted Finsler manifolds and spacetimes.
problem Analyzing weighted Finsler manifolds and spacetimes with curvature conditions.
method Using weight function and ε-range, the Bonnet-Myers theorem, Laplacian comparison theorem, and Bishop-Gromov volume comparison theorem are formulated. result New comparison theorems for weighted Finsler manifolds and spacetimes are derived, including those for weighted Riemannian manifolds.
Extends three circle theorem to almost Hermitian manifolds.
problem Three circle theorem for Kähler manifolds to almost Hermitian manifolds.
method General maximum principle established to prove three circle theorem.
result Sharp dimension estimates and Liouville theorems for holomorphic functions.
This is the second of three papers about the Compression Theorem. We give proofs of Gromov's theorem on directed embeddings [M Gromov, Partial differential relations, Springer--Verlag (1986); 2.4.5 C'] and of the Normal Deformation Theorem [The compression theorem I; 4.7], arxiv:math.GT/9712235.
The Bakry-Émery-Ricci tensor is extended and comparison theorems are proven.
problem Extending the Bakry-Émery-Ricci tensor and proving comparison theorems.
method Generalizations of the drifted Laplacian and Bakry-Émery-Ricci tensor, mean curvature comparison theorem, Myers-type theorem, Cheeger-Gromoll splitting theorem.
result Proved a version of the mean curvature comparison theorem and its consequences.