Positive mass theorem for non-smooth metrics on flat manifolds with corners.
problem Proving a positive mass theorem for non-smooth metrics on asymptotically flat manifolds with non-compact boundary.
method Proves a positive mass theorem for metrics that are only continuous across a compact hypersurface.
result Obtains a positive mass theorem on manifolds with non-compact corners.
Variant of previous work on smooth algebraic functions with compact and non-compact preimages.
problem Constructing smooth algebraic functions with specific preimage properties.
method Explicit construction of smooth real algebraic functions with controlled preimage compactness.
result New results in singularity theory and real algebraic geometry.
New heat dispersion laws established for smooth compact manifolds.
problem Understanding heat dispersion in smooth compact manifolds.
method Established new heat dispersion laws through Theorem 1.1 and explored them further with Propositions 3.1 and 3.2.
result New heat dispersion laws for smooth compact manifolds.
Paper proves structure for compact Kähler manifolds with pseudo-effective tangent bundles.
problem Compact Kähler manifolds with pseudo-effective tangent bundles.
method Smooth or locally constant rationally connected fibration onto a quotient of a compact complex torus.
result Compact Kähler manifolds with pseudo-effective tangent bundles admit a fibration structure.
The compact-open topology is minimal on diffeomorphism and homeomorphism groups of most smooth manifolds.
problem The minimality of compact-open topology on diffeomorphism and homeomorphism groups.
method Analyzing the compact-open topology on diffeomorphism and homeomorphism groups of smooth manifolds.
result The compact-open topology is minimal on diffeomorphism and homeomorphism groups of most smooth manifolds.
Smooth manifolds can be triangulated with graphs of bounded twin-width.
problem Understanding the structure of triangulations of smooth manifolds.
method Using Whitney's triangulation method and bounding the twin-width of specific graphs.
result Compact smooth manifolds have triangulations with graphs of bounded twin-width.
Proves smooth solution uniqueness and long-term existence for a parabolic equation on a complex manifold.
problem Existence and uniqueness of solutions to a parabolic equation on compact complex manifolds.
method Uses parabolic Donaldson's equation to prove existence and uniqueness of smooth solutions.
result Smooth solutions to the parabolic Donaldson's equation on compact complex manifolds exist and are unique for all time.
Incompressible fluid dynamics on special manifolds.
problem Fluid dynamics on specific geometric manifolds.
method Analyzes G-invariant vector fields on compact manifolds. result Shows existence of smooth solutions to Euler equations.
Compact hyperbolic complex manifolds are rigid under deformation.
problem Studying the deformation behavior of compact hyperbolic complex manifolds.
method Analyzing smooth families of compact complex manifolds over the unit disk and compact Riemann surfaces.
result The H-locus is either at most a discrete subset or the whole domain, depending on the family structure. The paper shows that certain Einstein orbifolds cannot be limits of smooth Einstein metrics.
problem Understanding the limits of smooth Einstein metrics on compact Einstein orbifolds.
method Analyzing sequences of compact Einstein manifolds and their limits, providing an explicit obstruction for certain orbifolds.
result Explicit obstruction for negative Einstein orbifolds appearing as limits of compact Einstein manifolds, which does not vanish for hyperbolic orbifolds.
Improved proofs for topological and smooth pseudo-isotopies of simply connected 4-manifolds.
problem Proving topological and smooth pseudo-isotopies of simply connected 4-manifolds.
method Provided different arguments that bypass the replacement criterion, thus completing Quinn's proofs.
result Corrected and completed Quinn's proofs of both topological and stable smooth pseudo-isotopy theorems.
Study smooth mappings between manifolds and their properties.
problem Understanding mappings between manifolds with various dimensions and boundaries.
method Analyzes mappings Cα in terms of iterated directional derivatives and smooth structures. result Establishes a canonical smooth manifold structure for mappings under certain conditions.
We introduce a new generalization of Gompf nuclei and give applications. We construct infinitely many exotic smooth structures for a large class of compact 4-manifolds with boundary, regarding topological invariants. We prove that a large class of closed 3-manifolds (including disjoint unions of Stein fillable 3-manifo…
Local smoothing of metrics with small curvature, removing Ricci curvature condition.
problem Establishing local smoothing of metrics with curvature concentration.
method Local mollification, removing Ricci curvature condition, Sobolev constants and volume growth.
result Compactness of manifolds with small curvature concentration under Ahlfors regularity and Sobolev constant.
Gauduchon's theorem extended to singular spaces with smoothing.
problem Extending Gauduchon's theorem to singular spaces.
method Using smoothing techniques for singular spaces.
result Existence of conformally equivalent metrics on singular spaces.
We study the space of smooth Riemannian structures on compact three-manifolds with boundary that satisfies a critical point equation associated with a boundary value problem, for simplicity, Miao-Tam critical metrics. We provide an estimate to the area of the boundary of Miao-Tam critical metrics on compact three-manif…
We show how to construct absolutely exotic smooth structures on compact 4-manifolds with boundary, including contractible manifolds. In particular, we prove that any compact smooth 4-manifold W with boundary that admits a relatively exotic structure contains a pair of codimension-zero submanifolds homotopy equivalent t…
Constructs complete Calabi-Yau metrics from smoothed Calabi-Yau intersections.
problem Creating complete Calabi-Yau metrics on non-compact manifolds.
method Extending Székelyhidi's work, constructing metrics with varying complex structures and possible singularities.
result Produces Calabi-Yau metrics with fibers having varying complex structures and possibly isolated singularities.
Smooth bundles with rough data maintain Hodge kernel isomorphism.
problem Maintaining Hodge kernel isomorphism for smooth bundles with non-smooth geometric data.
method Analyzing nilpotent differential operators and Hodge-Dirac-type operators under perturbations of geometric data.
result Kernels of Hodge-Dirac operators remain isomorphic under uniform perturbations of geometric data.
This work is a continuation of the former paper in which principal bundles are given by compact spin toric manifolds and compact connected semisimple Lie groups. In this paper, ambient manifolds are assumed to be compact toric manifolds and Lie groups are compact connected. The main result is that locally smooth manifo…
Constructs real algebraic functions with both compact and non-compact preimages.
problem Finding real algebraic functions with specific preimage properties.
method Explicit construction of real algebraic functions.
result Demonstrates real algebraic functions on non-compact manifolds with non-compact preimages.
Proves rigidity for specific initial data sets under the dominant energy condition.
problem Rigidity of initial data sets with boundary and convex polytopes.
method Solution of boundary value problems for Dirac operators and approximations by manifolds with smooth boundary.
result Proves rigidity for compact smooth spin manifolds and convex polytopes under the dominant energy condition.
Researchers prove existence of cscK metrics on smooth minimal models.
problem Existence of constant scalar curvature Kähler metrics on compact Kähler manifolds.
method Direct proof showing existence on smooth minimal models and blowups.
result Compact Kähler manifolds with nef canonical bundle always admit cscK metrics.
Study shows only two topological configurations for Spin(7)-manifold fibrations, ruling out smooth Cayley fibrations.
problem Understanding smooth fibrations of compact Spin(7)-manifolds by Cayley submanifolds.
method Geometric and topological constraints from Spin(7)-structure, spinnability criterion, gauge-theoretic input.
result Rules out smooth Cayley fibrations on all known compact torsion-free Spin(7)-manifolds.
In an earlier paper we showed that the space of deformations of a smooth, compact, orientable Harvey-Lawson submanifold HL in a G2 manifold M can be identified with the direct sum of the space of smooth functions and closed 2-forms on HL. In that paper, we also introduced a new class of Lagrangian-type 4-dimensional su…
We show that, up to scaling, the complex Monge-Ampere equation on compact Hermitian manifolds always admits a smooth solution.
It is known that every compact Stein 4-manifolds can be embedded into a simply connected, minimal, closed, symplectic 4-manifold. By using this property, we discuss a new method of constructing corks. This method generates a large class of new corks including all the previously known ones. We prove that every one of th…
The article defines and compares two types of quantizations on compact manifolds.
problem Quantization on arbitrary compact smooth manifolds.
method Embedding into CP^n and inducing quantizations from there.
result Generalizations of earlier quantization methods.
Study on diffeotopy groups of non-compact 4-manifolds, extending previous results.
problem Understanding diffeotopy groups of non-compact 4-manifolds.
method Analyzing smoothings of punctured 4-manifolds and proving properties of diffeotopy groups.
result Uncountable diffeotopy groups for certain non-compact 4-manifolds.
The Morse complex is shown to be an infinite functor.
problem Understanding the structure of Morse complexes as infinite functors.
method Showed the Morse complex of a compact Lie monoid can be given the structure of an f-bialgebra and defined an ∞-functor.
result Obtained two other ∞-functors mapping manifolds and actions to their Morse complexes.
Study on lens spaces bounding 4-manifolds with specific Betti numbers.
problem Which lens spaces can bound 4-manifolds with second Betti number one?
method Construction of specific 4-manifolds and analysis of lens space boundaries.
result Infinite families of lens spaces can bound 4-manifolds with second Betti number one, but not all.
Confirming operator characterization on smooth manifolds.
problem Characterizing an operator on smooth sections of tangent bundles.
method Using algebraic axioms and H^1(M, R) = {0} assumption.
result Operator can be characterized universally for any smooth manifold.
We study the possibility of realizing exotic smooth structures on punctured simply connected 4-manifolds as leaves of a codimension one foliation on a compact manifold. In particular, we show the existence of uncountably many smooth open 4-manifolds which are not diffeomorphic to any leaf of a codimension one trans…
In terms of Turaev's shadows, we provide a sufficient condition for a compact, smooth, acyclic 4-manifold with boundary the 3-sphere to be diffeomorphic to the standard 4-ball. As a consequence, we prove that if a compact, smooth, acyclic 4-manifold with boundary the 3-sphere has shadow-complexity at most 2, then it is…
In this paper, we solve a logarithmic ∂ˉ-equation on a compact Kähler manifold associated to a smooth divisor by using the cyclic covering trick. As applications, we discuss the closedness of logarithmic forms, injectivity theorems and obtain a kind of degeneration of spectral sequence at E1, and we al…
The paper investigates exotic smooth structures on manifolds with group actions.
problem Existence of homeomorphic but not diffeomorphic smooth manifolds with shared basic spectra.
method Investigates Riemannian Laplacian eigenvalues and eigenfunctions on manifolds with compact Lie group actions.
result Establishes the existence of homeomorphic yet not diffeomorphic manifolds with shared basic spectra.
In this paper, we prove that, a compact complex manifold X admits a smooth Hermitian metric with positive (resp. negative) scalar curvature if and only if KX (resp. KX−1) is not pseudo-effective. On the contrary, we also show that on an arbitrary compact complex manifold X with complex dimension ≥2, …
For a 4-manifold represented by a framed knot in S3, it has been well known that the 4-manifold admits a Stein structure if the framing is less than the maximal Thurston-Bennequin number of the knot. In this paper, we prove either the converse of this fact is false or there exists a compact contractible oriented smo…
Using the Cartan-Kahler theory, and results on real algebraic structures, we prove two embedding theorems. First, the interior of a smooth, compact 3-manifold may be isometrically embedded into a G_2-manifold as an associative submanifold. Second, the interior of a smooth, compact 4-manifold K, whose double has a trivi…
Any bounding compact smooth manifold bounds a compact manifold with a spine consisting of transversely intersecting codimension one submanifolds. This paper provides details for a picture proof given in previous papers with S. Akbulut.
Compact Kähler manifolds with positive curvature are projective and rationally connected.
problem Characterizing compact Kähler manifolds with positive curvature.
method Proving properties of compact Kähler manifolds with quasi-positive second Chern-Ricci curvature.
result Compact Kähler manifolds with quasi-positive second Chern-Ricci curvature are projective and rationally connected.
In 1960s, Almgren initiated a program to find minimal hypersurfaces in compact manifolds using min-max method. This program was largely advanced by Pitts and Schoen-Simon in 1980s when the manifold has no boundary. In this paper, we finish this program for general compact manifold with nonempty boundary. As a result, w…
The paper finds singular isoperimetric regions in high-dimensional spaces.
problem Existence of singular isoperimetric regions in high-dimensional manifolds.
method Construction of specific manifolds with singular isoperimetric regions.
result Construction of manifolds with singular isoperimetric regions.
Let X be a smooth manifold belonging to one of these three collections: acyclic manifolds (compact or not, possibly with boundary), compact connected manifolds (possibly with boundary) with nonzero Euler characteristic, integral homology spheres. We prove that Diff(X) is Jordan. This means that there exists a const…
Study finds topological restrictions on 4-manifolds with uniformly positive scalar curvature.
problem Understanding which 4-manifolds can have metrics with uniformly positive scalar curvature.
method Topological obstructions and metric constructions on specific 4-manifolds.
result Existence of uncountably many exotic R4's without such metrics and topological uniqueness of certain metrics. The paper studies stability of CR structures on compact manifolds.
problem Stability of CR structures on compact manifolds.
method Smooth deformations and projective embeddings.
result Nearby structures still admit projective CR embeddings.
We show that if a holomorphic n dimensional compact torus action on a compact connected complex manifold of complex dimension n has a fixed point then the manifold is equivariantly biholomorphic to a smooth toric variety.
The problem of minimal distortion bending of smooth compact embedded connected Riemannian n-manifolds M and N without boundary is made precise by defining a deformation energy functional Φ on the set of diffeomorphisms $\diff(M,N)$. We derive the Euler-Lagrange equation for Φ and determine smooth minimizers o…