The paper proves a local existence theorem for a CR torsion flow in pseudohermitian manifolds.
problem Existence of solutions to the CR torsion flow in pseudohermitian manifolds.
method Analytic partial differential equations approach.
result A unique smooth solution exists in a small time interval for the CR torsion flow.
Vaisman's theorem extended to locally reducible Kähler spaces.
problem Existence of locally conformally Kähler metrics on compact Kähler spaces.
method Extended Vaisman's theorem to locally reducible Kähler spaces.
result Vaisman's theorem holds for compact Kähler spaces that are locally reducible.
Study local curvature estimates and existence of conformal metrics on noncompact manifolds.
problem Deriving local C0-estimates and existence of conformal metrics with prescribed curvature. method Utilizing Aviles-McOwen's result and its nonlinear extension, combined with asymptotic conditions.
result Proved existence of complete conformal metrics with prescribed curvature functions.
In this paper we prove a new version of the Schoenflies extension theorem for collared domains in Euclidean n-space: for 1 < p < n, locally bi-Lipschitz homeomorphisms between collared domains with locally p-integrable, second-order weak derivatives admit homeomorphic extensions of the same regularity. Moreover, the th…
The Abstract Boundary singularity theorem was first proven by Ashley and Scott. It links the existence of incomplete causal geodesics in strongly causal, maximally extended spacetimes to the existence of Abstract Boundary essential singularities, i.e., non-removable singular boundary points. We give two generalizations…
Study generalizes map properties between Hermitian manifolds preserving specific forms.
problem Understanding maps between Hermitian manifolds that preserve certain forms.
method Generalizing results from Chan-Yuan [2025] to new maps.
result Obtained further rigidity and non-existence theorems.
In this paper, we give the first detailed proof of the short-time existence of Deane Yang's local Ricci flow. Then using the local Ricci flow, we prove short-time existence of the Ricci flow on noncompact manifolds, whose Ricci curvature has global lower bound and sectional curvature has only local average integral bou…
The Darboux-Weinstein theorem is extended to LCS manifolds.
problem Extending classical results to locally conformally symplectic manifolds.
method Presentation of a version of the Darboux-Weinstein theorem in the LCS setting.
result Application to Lagrangian submanifolds.
A theorem proves integrability of Fréchet tangent distributions.
problem Integrability of Fréchet tangent distributions on manifolds.
method Introduced Condition W, applied variational approach, used differential forms.
result Existence and uniqueness of maximal foliations.
Proves compactness of geometric models for certain homogeneous spaces.
problem Existence and uniqueness of geometric models for locally homogeneous spaces.
method Proves existence and uniqueness of geometric models in the pointed C1,α-topology. result Compact set of geometric models for sectional curvature ≤ 1.
New theorem on geodesics on non-compact manifolds.
problem Existence of geodesics with specific properties.
method Analyzing geodesics with local homology in maximal degree.
result Infinitely many closed geodesics on non-compact manifolds.
We prove the existence of the flow by curvature of regular planar networks starting from an initial network which is non-regular. The proof relies on a monotonicity formula for expanding solutions and a local regularity result for the network flow in the spirit of B. White's local regularity theorem for mean curvature …
We give a new proof for the local existence of a smooth isometric embedding of a smooth 3-dimensional Riemannian manifold with nonzero Riemannian curvature tensor into 6-dimensional Euclidean space. Our proof avoids the sophisticated arguments via microlocal analysis used in earlier proofs. In Part 1, we introduce …
Recently, Cochran and Harvey defined torsion-free derived series of groups and proved an injectivity theorem on the associated torsion-free quotients. We show that there is a universal construction which extends such an injectivity theorem to an isomorphism theorem. Our result relates injectivity theorems to a certain …
The paper introduces a new concept of infinitesimal homogeneity for connections on bundles and applies it to prove known theorems and derive new results.
problem Understanding and proving theorems related to connections on bundles and their properties.
method Introducing infinitesimal homogeneity for sections in associated vector bundles and proving the existence of connections satisfying parallelism conditions.
result The existence of connections satisfying parallelism conditions and the ability to classify locally homogeneous and symmetric triples.
Slice theorem in infinite dimensions for Lie groups.
problem Generalizing slice theorem to infinite-dimensional settings.
method Developed slice theorem for locally convex Lie groups on locally convex manifolds using advanced theorems.
result Existence of orbit type stratification under slice condition.
In this paper, we prove an existence theorem of a local moduli space for geometric structures in a very general setting. Then to show the interest of this result, we apply it to the case of sasakian and Sasaki-Einstein structures.
The paper proves local isometric embeddings for singular metrics near a point.
problem Existence of local isometric embeddings for singular Riemannian metrics.
method Ramified local isometric embeddings using Leray's ramified Cauchy-Kovalevskaya Theorem.
result Existence of local analytic isometric embeddings into Euclidean space.
Localized min-max method proves minimal hypersurface existence.
problem Existence of minimal hypersurfaces in complete manifolds.
method Localized min-max approach to prove existence.
result Existence of complete embedded minimal hypersurface with index at most one.
Extends Whitney's theorem for functions on rough boundaries.
problem Global extension of manifold-valued functions on domains with rough boundaries.
method Using locally convex spaces of compactly-supported sections of vector bundles, proving the existence of an extension operator.
result The restriction map from everywhere-defined functions is a submersion, allowing local linear splittings.
The study proves local rigidity of minimal 2-spheres in electrovacuum spacetimes.
problem Proving local rigidity of minimal 2-spheres in electrovacuum spacetimes under certain conditions.
method Analyzing electrovacuum spacetimes and using constraints on charged Hawking mass and area minimization.
result Local rigidity of minimal 2-spheres in electrovacuum spacetimes, with isometric neighborhoods to specific spacetimes.
Universal functions and metrics with constant curvature on domains.
problem Finding universal functions and metrics with constant curvature.
method Proving Runge-type theorems and universality results for locally univalent functions, refining Heins' result.
result Existence of universal conformal metrics with constant curvature on hyperbolic domains.
Taking an elementary and straightforward approach, we develop the concept of a regular value for a smooth map f: O -> P between smooth orbifolds O and P. We show that Sard's theorem holds and that the inverse image of a regular value is a smooth full suborbifold of O. We also study some constraints that the existence o…
Paper proves mass theorems for nonnegative scalar curvature metrics.
problem Proving mass theorems for metrics with nonnegative scalar curvature.
method New local inverse mean curvature flow with quantitative stability.
result Existence of isoperimetric sets in low regularity metrics.
We present an elementary treatment of the Optional Decomposition Theorem for continuous semimartingales and general filtrations. This treatment does not assume the existence of equivalent local martingale measure(s), only that of strictly positive local martingale deflator(s).
Proves a synthetic Lorentzian Cartan-Hadamard theorem.
problem Formulates and proves a theorem for Lorentzian geometry.
method Uses an appropriate notion of local concavity for Lorentzian (pre-)length spaces.
result Establishes existence and uniqueness of timelike geodesics.
Kuranishi's proof of complex deformation theory revisited
problem Existence of complex deformations on compact complex manifolds
method Hamilton-Nash-Moser implicit function theorem
result Revisits classical proof with modern tools
Study non-Kahler symplectic manifolds, proving deformation and Torelli theorems.
problem Topology and deformation theory of non-Kahler holomorphically symplectic manifolds.
method Investigation of topology and deformation theory, proving local Torelli theorem and Fujiki formula.
result Holomorphically symplectic deformations of BG-manifolds are unobstructed, and the period map is locally a diffeomorphism.
The paper proves existence theorems for fractional Yamabe problem on conformal infinity.
problem Existence of solutions to fractional Yamabe problem on conformal infinity.
method Various geometric assumptions on the asymptotically hyperbolic manifold and its conformal infinity.
result Simpler geometric restrictions for existence results compared to previous works.
Quantum codes with optimal distance and dimension for n-dimensional space.
problem Designing efficient quantum codes in high dimensions.
method Combining asymptotically good codes, manifold construction, and embedding theorem.
result Optimal quantum codes with distance and dimension for n-dimensional space.
New proof of link factorization theorem, avoiding case exhaustion.
problem Proving every non-split link can be uniquely factored into prime links.
method Shorter proof using string links and ambient isotopy.
result Demonstrates existence of string links with no local knots.
The study finds static solutions in symplectic curvature flow in 4D.
problem Finding static solutions in symplectic curvature flow in 4D.
method Derived a local normal form for static solutions and used Cartan-Kahler theorem for solitons.
result Every complete static solution to symplectic curvature flow in 4D is Kahler-Einstein.
The author presents the generalized Stokes theorem for R-linear forms on Lie algebroids (which can be non-local). We apply the Stokes formula on forms to prove that two homotopic homomorphisms of Lie algebroids implies the existence of a chain operator joining their pullback operators.
Proves a local analytic Bertini theorem confirming a conjecture.
problem Analytic Bertini theorem in local generality.
method Full generality proof of the local analytic Bertini theorem.
result Confirms a conjecture by Boucksom in local analytic settings.
The classical Sturm-Hurwitz-Kellogg theorem asserts that a function, orthogonal to an n-dimensional Chebyshev system on a circle, has at least n+1 sign changes. We prove the converse: given an n-dimensional Chebyshev system on a circle and a function with at least n+1 sign changes, there exists an orientation preservin…
Survey discusses recent advances on Brakke flows and their properties.
problem Existence and regularity of Brakke flows.
method Analyzes recent progress including new theorems and proofs.
result Proof that branching singularities trigger dynamical instability.
We prove an existence theorem for Spin(7)-instantons, which are highly concentrated near a Cayley submanifold; thus giving a partial converse to Tian's foundational compactness theorem. As an application, we show how to construct Spin(7)-instantons on Spin(7)-manifolds with suitable local K3 Cayley fibrations. This rec…
The paper proves a unique conformal measure for Anosov groups and shows local mixing.
problem Proving the uniqueness of conformal measures for Anosov groups.
method Analogue of Sullivan's theorem for Anosov subgroups of semisimple groups.
result Uniqueness of conformal measures and local mixing for Anosov groups.
Article strengthens initial data rigidity theorem to show unique spacetime extension.
problem Initial data rigidity in spacetime geometry.
method Showed initial data sets carry a lightlike parallel vector field, leading to unique spacetime extension.
result Local uniqueness of spacetimes extending initial data sets under dominant energy condition.
Paper proves a Cohen-Dimca-Orlik type theorem for Z-local systems of hyperplane arrangements.
problem Proving a Cohen-Dimca-Orlik type theorem for Z-local systems. method Analyzing local system cohomology groups of hyperplane arrangements complements.
result Proves a Cohen-Dimca-Orlik type theorem for Z-local systems. In his monograph "Leçons sur les systèmes orthogonaux et les coordonnées curvilignes. Principes de géométrie analytique", 1910, Darboux stated three theorems providing local existence and uniqueness of solutions to first order systems of the type \[\partial_{x_i} u_α(x)=f^α_i(x,u(x)),\quad i\in I_α\subseteq\{1,\dots,n\…
In this paper, we introduce new methods for solving the vacuum Einstein constraints equations: the first one is based on Schaefer's fixed point theorem (known methods use Schauder's fixed point theorem) while the second one uses the concept of half-continuity coupled with the introduction of local supersolutions. These…
We generalize classical theorems due to Lichnerowicz and Hitchin on the existence of Riemannian metrics of positive scalar curvature on spin manifolds to the case of foliated spin manifolds. As a consequence, we show that there is no foliation of positive leafwise scalar curvature on any torus, which generalizes the fa…
Enhances Ricci flow theorem with scalar curvature bound.
problem Improving no-local-collapsing theorem of Ricci flow.
method Derives improved theorem under scalar curvature bound condition.
result Refines Perelman's no-local-collapsing theorem.
The goal of the present paper is to establish some kind of regularity of an energy minimizer map between Riemannian polyhedra. More precisely, we will show the hölder continuity of local energy minimizers between Riemannian polyhedra with the target spaces without focal points. With this new result, we also complete ou…
The abstract proves the existence and regularity of Brakke flows starting from a given set.
problem Existence and regularity of Brakke flows starting from a given set.
method Proves the existence and regularity of Brakke flows using a closed countably 1-rectifiable set in R^2.
result For almost all time, the flow locally consists of a finite number of embedded curves of class W^{2,2} whose endpoints meet at junctions with angles of 0, 60, or 120 degrees.
New method proves Alexandrov theorem for curved spacetimes.
problem Proving Alexandrov theorem for curved surfaces with conical singularities.
method Adapting Volkov's variational method to handle Lorentzian angles.
result Existence of a locally Minkowski 3-manifold with conical singularities isometric to a given surface.
Proves a local version of Myers-Steenrod theorem for specific manifolds.
problem Generalization of Myers-Steenrod theorem to local topological groups.
method Proof for local topological groups of isometries acting on specific manifolds.
result New regularity result for locally homogeneous Riemannian metrics.