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.
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.
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.
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.
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.
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.
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.
Generalizes halfspace theorems to higher dimensions for self-shrinkers.
problem Limitations of halfspace theorems in higher dimensions for self-shrinkers.
method Extends codimension 1 results to arbitrary codimension.
result Establishes new halfspace theorems for self-shrinkers in arbitrary codimension.
Generalized Fáry's theorem to higher dimensions.
problem No specific problem stated; generalization of Fáry's theorem.
method Proof of a higher-dimensional version of Fáry's theorem.
result Proved a generalization of Fáry's theorem in higher dimensions.
Study generalizes finiteness theorem using Lie theory.
problem Finiteness theorem in Hadwiger-Alesker theory.
method Generalization through Lie theory.
result Generalization of finiteness theorem.
A new proof of an extension theorem with bounded generators.
problem Extension theorems in complex analysis.
method Skoda-type L2 division theorem with bounded generators. result The new division theorem allows α to be 1 in the norm of the datum. 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.
Defines and proves generalized noncommutative residue theorems for specific dimensions.
problem Defining and proving residue theorems for noncommutative geometry.
method Defined generalized noncommutative residue of Dirac operator; proved Kastler-Kalau-Walze type theorems.
result Validated Kastler-Kalau-Walze type theorems for 4D and 6D compact 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.
The paper explores dualities in differential equations and their applications in Riemannian geometry.
problem Developing comparison theorems for mixed type differential equations.
method Utilizing dualities in differential equations and inequalities, and applying them to Riemannian geometry.
result Proves Hessian and Laplacian comparison theorems under various curvature assumptions.
In his celebrated paper "Generic projections", John Mather has given a striking transversality theorem and its applications on generic projections. On the other hand, in this paper, two transversality theorems on generic linearly perturbed Cr mappings are shown (r≥1). Moreover, some applications of the two the…
Proves another theorem for odd dimensional manifolds with boundary.
problem Spectral Einstein functional on odd dimensional manifolds with boundary.
method Proof of a theorem using the Dirac operator.
result Another general Dabrowski-Sitarz-Zalecki type theorem proved.
Proves Frankel theorem for generic submanifolds in Sasakian manifolds.
problem Intersection properties of generic submanifolds in Sasakian manifolds.
method Introduces a weaker notion of generic submanifolds and proves Frankel theorem under specific conditions.
result Derives topological information about generic submanifolds in Sasakian space forms.
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.
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.
The paper establishes a new sphere theorem for certain types of manifolds.
problem Finding conditions under which compact manifolds are spheres.
method Developed a generalized sphere theorem for manifolds with radial Ricci curvature.
result Established conditions for compact manifolds to be topologically spheres.
Proves a generalized vanishing theorem for quasi-smooth stacks, with applications in K-theory and birational geometry.
problem Vanishing theorems for quasi-coherent sheaves on derived blow-ups of quasi-smooth stacks.
method Derived blow-ups, intrinsic blow-up theory, Kiem-Li-Savvas blow-up theory, virtual localization theorem, desingularization theorem, resolution of diagonal.
result Generalized vanishing theorem for quasi-coherent sheaves on derived blow-ups of quasi-smooth stacks.
In this paper, we prove an equivariant Kastler-Kalau-Walze type theorem for spin manifolds without boundary. For 6 dimensional spin manifolds with boundary, we also give an equivariant Kastler-Kalau-Walze type theorem. Then we generalize this theorem to the general n dimensional manifold. An equivariant Kastler-Kal…
We formulate and establish a generalization of Kollár's injectivity theorem for adjoint bundles twisted by suitable multiplier ideal sheaves. As applications, we generalize Kollár's torsion-freeness, Kollár's vanishing theorem, and a generic vanishing theorem for pseudo-effective line bundles. Our approach is not Hodge…
Paper resolves Huisken's conjecture without strict genus drop theorem.
problem Huisken's genericity conjecture in mean curvature flow in R^3.
method Short density-drop theorem + Bamler-Kleiner multiplicity-one theorem for tangent flows.
result Fully resolves Huisken's conjecture without strict genus drop theorem.
Extends a theorem for first-order elliptic operators on manifolds.
problem Proving the relative index theorem for general first-order elliptic operators.
method Using boundary value problems and graphical decomposition of elliptically regular boundary conditions.
result Proves the relative index theorem for general first-order elliptic operators.
Generalizes Hopf degree theorem to nontrivial bundles.
problem Classifying maps from manifolds to spheres.
method Generalization of Hopf degree theorem to nontrivial bundles.
result Classifies sections of nontrivial n-sphere bundles. Generalizes Kodaira vanishing theorem to Kahler Lie algebroids.
problem Extending complex geometry results to Lie algebroids.
method Using local coordinate calculations to generalize Kahler identities.
result Kernel of Lie algebroid Laplace operator vanishes for sufficiently large p+q.
The paper proves new comparison theorems for sub-Laplacian in foliations with minimal leaves.
problem Proving comparison theorems for sub-Laplacian in Riemannian foliations with minimal leaves.
method Using Riemannian foliations with minimal leaves, the paper proves comparison theorems for the sub-Laplacian.
result The comparison theorems yield a Bonnet-Myers type theorem, stochastic completeness, and Lipschitz regularization property for the sub-Riemannian semigroup.
Proves positive mass theorem for non-spin weighted manifolds.
problem Proving the positive mass theorem for non-spin weighted manifolds.
method Establishing density theorem and generalizing Geroch conjecture.
result Proves positive weighted mass theorem for non-spin weighted manifolds.
We generalize classical large deviations theorems to the setting of complete Riemannian manifolds. We prove the analogue of Mogulskii's theorem for geodesic random walks via a general approach using visocity solutions for Hamilton-Jacobi equations. As a corollary, we also obtain the analogue of Cramér's theorem. The ap…
The Liouville theorem is proven for V T-harmonic map heat flow.
problem Proving Liouville theorems for V T-harmonic maps.
method Analyzing heat flow on manifolds with specific properties.
result Liouville theorems established for V T-harmonic maps.
The purpose of this paper is to study a complete orientable minimal hypersurface with finite index in an (n+1)-dimensional Riemannian manifold N. We generalize Theorems 1.5-1.6 (\cite{Seo14}). In 1976, Schoen and Yau proved the Liouville type theorem on stable minimal hypersurface, i.e., Theorem 1.7 (\cite{SchoenYa…
Paper generalizes scalar curvature theorem to weighted manifolds.
problem Generalizing scalar curvature rigidity theorem to weighted manifolds.
method Proves a refinement of Llarull's theorem for P-scalar curvature.
result Establishes a Llarull type theorem for SkimesTn−k. In this note we present various extensions of Obata's rigidity theorem concerning the Hessian of a function on a Riemannian manifold. They include general rigidity theorems for the generalized Obata equation, and hyperbolic and Euclidean analogs of Obata's theorem. Besides analyzing the full rigidity case we also chara…
Simplified proof for a theorem about graphs.
problem Proving a generalized Conway--Gordon--Sachs theorem for complete graphs.
method Provided a shorter proof over integers.
result A simpler proof of the generalized theorem.
The classical integral localization formula for equivariantly closed forms (Theorem 7.11 in [BGV]) is well-known and requires the acting Lie group to be compact. It is restated here as Theorem 2. In this article we extend this result to NONcompact groups. The main result is Theorem 20. Then, using this generalization, …
Generalizes Kastler-Kalau-Walze theorem to even-dimensional manifolds.
problem Proving a theorem for a specific type of Dirac operator on various manifolds.
method Extending previous results to even-dimensional almost product Riemannian spin manifolds.
result Established the general Kastler-Kalau-Walze type theorem for even-dimensional manifolds.
In this paper we prove a relative index theorem for pairs of generalized Dirac operators on orbifolds which are the same at infinity. This generalizes to orbifolds a celebrated theorem of Gromov and Lawson.
Proves generic surjectivity of vector bundles via degeneration.
problem Understanding generic surjectivity of vector bundles.
method Uses degeneration argument, Berndtsson's theorem, and Lempert's proof.
result Generalizes previous work on L2 division theorem. Paper introduces Lie algebroid index theory and a generalized Riemann-Roch theorem.
problem Generalizing Riemann-Roch theorem for manifolds with regular foliations.
method Developed Lie algebroid index theory and applied it to obtain a generalized Riemann-Roch theorem.
result Obtained a generalized Riemann-Roch theorem for manifolds with regular foliations.
Generalized Agol's theorem to 3-manifold groups.
problem Two-parabolic-generator subgroups in hyperbolic 3-manifold groups.
method Generalization of Agol's proof for 2-bridge link groups to 3-manifold groups.
result Refinement of Boileau-Weidmann's result.
New 'book links' generalize braids and plats, proving Markov's theorem.
problem Generalizing braid theory to new structures.
method Extending open book foliations to prove Markov's theorem for new objects.
result Proved Markov's theorem in a broader context of 'book links'.
The paper proves a generalized inverse function theorem for curved L∞ spaces.
problem Proving a generalized inverse function theorem for curved L∞ spaces. method Obstruction theory for L∞ homomorphisms and homotopy transfer theorem for curved L∞ algebras. result A morphism of curved L∞ spaces which is a quasi-isomorphism at a point has a local homotopy inverse. Generalized Huber's theorem for specific manifold curvature types.
problem Finite point conformal compactification on manifolds with certain curvature integrability.
method Generalization of Huber's theorem to higher dimensions with $L^ rac{n}{2}$ integrable Ricci curvatures.
result Validated finite point conformal compactification theorem for new class of manifolds.
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.
The paper studies λ-submanifolds in Gauss spaces and proves theorems for complete proper ones.
problem Understanding λ-submanifolds in Gauss spaces and their properties. method Using divergence type theorems and Simons' identities, the authors prove theorems for complete proper λ-submanifolds. result Proves halfspace and gap theorems for complete proper λ-submanifolds, generalizing previous results. There are proven few analogues of the Theorem of Moser using The Approximation Theorem of Artin.