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.
We give a construction of the Floer homology of the pair of {\it non-compact} Lagrangian submanifolds, which satisfies natural continuity property under the Hamiltonian isotopy which moves the infinity but leaves the intersection set of the pair compact. This construction uses the concept of Lagrangian cobordism and ce…
We generalize the PL intersection product for chains on PL manifolds and for intersection chains on PL stratified pseudomanifolds to products of locally finite chains on non-compact spaces that are natural with respect to restriction to open sets. This is necessary to sheafify the intersection product, an essential ste…
In this work we prove the existence of embedded closed minimal hypersurfaces in non-compact manifolds containing a bounded open subset with smooth and strictly mean-concave boundary and a natural behavior on the geometry at infinity. For doing this, we develop a modified min-max theory for the area functional following…
A Heegaard splitting of an open 3-manifold is the partition of the manifold into two non-compact handlebodies which intersect on their common boundary. This paper proves several non-compact analogues of theorems about compact Heegaard splittings. The main theorem is: if N is a compact, connected, orientable 3-manifold …
We use a generalization of the Gibbons-Hawking ansatz to study the behavior of certain non-compact Calabi-Yau manifolds in the large complex structure limit. This analysis provides an intermediate step toward proving the metric collapse conjecture for toric hypersurfaces and complete intersections.
Joyce's criterion for sLag smoothings extended to non-compact, non-transverse intersections.
problem Existence of special Lagrangian smoothings for non-compact, non-transverse intersections.
method Leung-Yau-Zaslow transform, deformed Hermitian Yang-Mills connections, Calabi ansatz, mean curvature flow, Bridgeland stability conditions.
result Existence of sLag smoothings on stable loci with slope inequality.
We study geometric properties of complete non-compact bounded self-shrinkers and obtain natural restrictions that force these hypersurfaces to be compact. Furthermore, we observe that, to a certain extent, complete self-shrinkers intersect transversally a hyperplane through the origin. When such an intersection is comp…
We prove equality between the renormalized Ray-Singer analytic torsion and the intersection R-torsion on a Witt-manifold with cusps, up to an error term determined explicitly by the Betti numbers of the cross section of the cusp and the intersection R-torsion of a model cone. In the first step of the proof we compute e…
We prove two gluing theorems for special Lagrangian (SL) conifolds in complex space C^m. Conifolds are a key ingredient in the compactification problem for moduli spaces of compact SLs in Calabi-Yau manifolds. In particular, our theorems yield the first examples of smooth SL conifolds with 3 or more planar ends and the…
The abstract discusses constructing 3d N=2 gauge theories using surgeries and M5-branes.
problem Constructing geometrically 3d N=2 gauge theories.
method Using surgeries and M5-branes wrapping on plumbing three-manifolds.
result Various dualities of 3d theories can be interpreted as Kirby moves and equivalent surgeries.
In this thesis we study the topology and geometry of hyperkähler quotients, as well as some related non-compact Kähler quotients, from the point of view of Hamiltonian group actions. The main technical tool we employ is Morse theory with moment maps. We prove a Lojasiewicz inequality which permits the use of Morse theo…
Uniformises Kähler surfaces with positive curvature to complex plane.
problem Uniformisation of complete Kähler surfaces with positive sectional curvature.
method New approach using uniformly Lipschitz plurisubharmonic weight functions and weighted holomorphic functions.
result Proves any complete non-compact Kähler surface with positive sectional curvature is biholomorphic to C^2.
Given any connected, open 3-manifold U having finitely many ends, a non-compact 3-manifold M is constructed having the following properties: the interior of M is homeomorphic to U; the boundary of M is the disjoint union of finitely many planes; M is not almost compact; M is eventually end-irreducible; th…
We give the first rigorous construction of complete, embedded self-shrinking hypersurfaces under mean curvature flow, since Angenent's torus in 1989. The surfaces exist for any sufficiently large prescribed genus g, and are non-compact with one end. Each has 4g+4 symmetries and comes from desingularizing the inters…
Solves classification of compact Clifford-Klein forms for specific Lie groups.
problem Classification of standard compact Clifford-Klein forms of homogeneous spaces.
method Analysis of real Lie algebras and their subalgebras.
result Standard compact Clifford-Klein forms arise from specific Lie algebra triples.
We establish a Cheeger-Muller theorem for unimodular representations satisfying a Witt condition on a noncompact manifold with cusps. This class of spaces includes all non-compact hyperbolic spaces of finite volume, but we do not assume that the metric has constant curvature nor that the link of the cusp is a torus. We…
An end sum is a non-compact analogue of a connected sum. Suppose we are given two connected, oriented n-manifolds M1 and M2. Recall that to form their connected sum one chooses an n-ball in each Mi, removes its interior, and then glues together the two Sn−1 boundary components thus created by an orien…
Overview of algebraic geometry for almost complex manifolds.
problem Developing algebraic geometry for almost complex manifolds without genericity.
method Reviewing results based on pseudoholomorphic maps and intersection theory.
result Introduction of birational morphism between almost complex manifolds.
Adapts stereographic projection for ellipsoid and elliptic paraboloid.
problem Projecting quadric surfaces using stereographic method.
method Adapted stereographic projection for ellipsoid and elliptic paraboloid, analyzing geometric properties and challenges.
result Established results on eccentricities, curvatures, arc length, and areas of intersections and projections.
Invariant obstructs separating coassociative 4-folds.
problem Obstructing the separation of coassociative 4-folds.
method Defining a Z2-valued invariant for coassociative 4-folds with spin structures. result Invariant provides an obstruction to separating coassociative 4-folds.
Classifies π1-injective maps between non-compact surfaces.
problem Characterizing maps with injective fundamental groups.
method Proper homotopy classification of maps.
result All π1-injective proper maps are classified. Extending work of Bielawski-Dancer and Konno, we develop a theory of toric hyperkahler varieties, which involves toric geometry, matroid theory and convex polyhedra. The framework is a detailed study of semi-projective toric varieties, meaning GIT quotients of affine spaces by torus actions, and specifically, of Lawren…
Study Szegő kernel on non-compact CR manifolds with specific conditions.
problem Analyzing Szegő kernel on non-compact CR manifolds.
method Establish Szegő kernel asymptotic expansions on non-compact strictly pseudoconvex CR manifolds with transversal CR R-action under natural geometric conditions. result Szegő kernel asymptotic expansions established on non-compact CR manifolds.
Sharp lower bound for p-Laplacian eigenvalue on non-compact manifolds.
problem Estimating eigenvalues of p-Laplacian on non-compact manifolds. method Sharp lower bound established through domain properties and curvature conditions.
result Sharp lower bound for the first Dirichlet eigenvalue of p-Laplacian. Strong rigidity proven for non-compact surfaces.
problem Proving rigidity of non-compact surfaces.
method Generalized result showing proper homotopy to homeomorphism.
result All non-compact orientable surfaces, except plane and punctured plane, are topologically rigid.
The paper finds infinitely many magnetic geodesics on non-compact manifolds.
problem Existence and multiplicity of periodic orbits of magnetic flows.
method Morse theory applied to non-compact manifolds with energy levels above the Mañé critical value.
result Infinitely many noncontractible closed magnetic geodesics found.
Paper solves Minkowski problem for non-compact convex sets with asymptotic boundary conditions.
problem Solving Minkowski problem for non-compact convex sets with asymptotic boundary conditions.
method Combining covolume, Hadamard variational formula, and geometric interpretation.
result Solved Minkowski problem for non-compact convex sets under asymptotic conditions.
Paper calculates homotopy types of non-compact surfaces using groupoids.
problem Computing homotopy types of non-compact foliated surfaces.
method Application of van Kampen theorem for groupoids.
result Computation of homotopy types for specific non-compact surfaces.
Extends soliton theory to non-compact cases.
problem Generalized solitons in non-compact settings.
method Place conditions on vector field and curvature, use tensor properties.
result Non-compact q-solitons are stationary and q-flat. Paper proves non-compact inaudibility of symmetry and commutativity.
problem Proving inaudibility of symmetry and commutativity in non-compact settings.
method Using isospectral pairs of generalized Heisenberg groups.
result Proved inaudibility of weak symmetry and commutativity.
Study finds solutions for complex problems on non-compact manifolds.
problem Solving fully nonlinear Yamabe-type problems on non-compact manifolds.
method Existence results for a class of problems, considering both positive and negative cases.
result Explicit examples of manifolds satisfying the hypotheses of the theorems.
Study eta invariant on non-compact manifolds with positive scalar curvature.
problem Proving geometric formulas and index theorems for uniformly positive scalar curvature metrics.
method Using Dirac-Schrödinger operators and relative eta invariant.
result New geometric formula for spectral flow and index formula for uniformly positive scalar curvature metrics.
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.
The paper studies HKKN stratifications for non-compact spaces and proves convexity properties.
problem Proving convexity properties of moment maps for non-compact subsets.
method Algebraic and analytical study of HKKN stratifications for a vector space and compact Kähler manifold, then applying to non-compact subsets.
result Convexity properties of moment maps for invariant subsets are proven.
Compact curve solution emerges from non-compact curve.
problem Constructing solutions from non-compact curves.
method Slingshot solution to curve shortening flow.
result Compact embedded solution exists for a finite time.
We study integral geometric properties of non-compact harmonic spaces.
Paper proves geodesic ball maximizes second Robin eigenvalue in non-compact symmetric spaces.
problem Maximizing the second Robin eigenvalue in non-compact rank-1 symmetric spaces.
method Quantitative spectral inequality for the second Robin eigenvalue.
result Geodesic ball maximizes the second Robin eigenvalue among domains of the same volume.
Maps between non-compact surfaces can have geometric kernels under certain conditions.
problem Understanding when maps between non-compact surfaces have geometric kernels.
method Using Brown's proper fundamental group to establish sufficient conditions for geometric kernels.
result Characterization of conjugacy classes in the proper fundamental group and sufficient conditions for geometric kernels.
The paper studies spaces of non-compact real algebraic curves and their uniformisation.
problem Understanding the spaces of non-compact real algebraic curves and their uniformisation.
method Construction of spaces of non-compact real algebraic curves and description of their connected components using Fuchsian groups.
result Any connected component of the spaces of non-compact real algebraic curves is homeomorphic to a quotient of a finite-dimensional real vector space by a discrete group.
New examples of isoparametric families on non-compact symmetric spaces.
problem Constructing isoparametric families with non-austere focal sets.
method New extension method from Euclidean spaces to symmetric spaces of non-compact type.
result First examples of isoparametric families on non-compact symmetric spaces.
Goldman bracket distinguishes surface homeomorphisms.
problem Characterizing homeomorphisms between non-compact surfaces.
method Using the Goldman bracket to distinguish homeomorphisms.
result A homotopy equivalence is a homeomorphism if it preserves the Goldman bracket.
In non-compact manifolds, geodesic flowers exist.
problem Existence of geodesic flowers in non-compact manifolds.
method Proving the existence of non-trivial geodesic flowers in complete non-compact manifolds with locally convex ends.
result Non-trivial geodesic flowers exist in every complete non-compact manifold with locally convex ends.
We formulate a conjecture that arithmetic locally symmetric manifolds have simple homotopy type, and prove it for the non-compact case. More precisely, we show that, for any symmetric space S of non-compact type without Euclidean de Rham factors, there are constants a=a(S) and d=d(S) such that any non-compact arithmeti…
New local method solves Yamabe problems on compact and non-compact manifolds.
problem Yamabe problems on compact and non-compact manifolds.
method Local method for compact and non-compact manifolds.
result Generalizes Brezis and Nirenberg's nonlinear eigenvalue problem to subsets of manifolds.
The paper studies how hyperbolic surfaces degenerate along harmonic map rays.
problem The degeneration of hyperbolic surfaces along harmonic map rays.
method Using Teichmüller space and holomorphic quadratic differentials, the authors show convergence of rescaled distance functions to the intersection number with a vertical measured foliation.
result Hyperbolic surfaces along the ray converge to the dual R-tree of the vertical measured foliation in the sense of Gromov-Hausdorff.
Study vector fields on non-compact manifolds with group action.
problem Understanding vector fields on non-compact manifolds with group action.
method Established a Poincaré-Hopf theorem for bounded vector fields on non-compact manifolds.
result A vector field on a non-compact manifold with group action must have infinitely many zeros if the group is amenable and the manifold's quotient has non-zero Euler characteristic.
This paper extends Lusternik-Schnirelmann category to non-compact manifolds.
problem Extending Lusternik-Schnirelmann category to non-compact manifolds.
method Explanation and extension of Farber's results to non-compact manifolds.
result Farber's results hold equally well on non-compact manifolds, and new phenomena occur in gradient flows.