Proves multiplicity one for boundary minimal hypersurfaces in compact manifolds.
problem Proving multiplicity one for min-max free boundary minimal hypersurfaces in compact manifolds with boundary.
method Developed existence and regularity theory for free boundary hypersurfaces with prescribed mean curvature, including Morse index bounds.
result Proved multiplicity one theorem for min-max free boundary minimal hypersurfaces in compact manifolds with boundary.
We show that the theory of stable complex G-cobordisms, for a torus G, is embedded into the theory of stable complex G-cobordisms of not necessarily compact manifolds equipped with proper abstract moment maps. Thus the introduction of such non-compact cobordisms in the stable complex G-cobordism theory does not…
Coarse homotopy theory connects Euclidean cones to shape theory of compact spaces.
problem Establishing connections between coarse homotopy theory and shape theory.
method Using pointed shape invariants and inverse mapping telescopes.
result Proving two compact spaces are strong shape equivalent if their Euclidean cones are coarsely homotopy equivalent.
This is a detailed introductory survey of the cohomological dimension theory of compact metric spaces.
The paper establishes a duality between non-compact and compact symmetric pairs.
problem Understanding the relationship between non-compact and compact symmetric pairs.
method Developed a duality theorem between non-compact pseudo-Riemannian semisimple symmetric pairs and commutative compact semisimple symmetric triads.
result Explicit description of a one-to-one correspondence between non-compact and compact symmetric pairs.
Explores new perspectives in transverse index theory for Lie group actions.
problem Transverse index theory for compact Lie group actions.
method Kasparov's work on transverse index theory, connections to Berline-Vergne and Paradan-Vergne.
result Potential connections and new insights in transverse index theory.
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.
We develop a general structure theory for compact homogeneous Riemannian manifolds in relation to the co-index of symmetry. We will then use these results to classify irreducible, simply connected, compact homogeneous Riemannian manifolds whose co-index of symmetry is less or equal than three. We will also construct ma…
Study of mixed equation combining gauge theory and symplectic geometry.
problem Regularity and compactness of solutions to the mixed equation.
method Combining Uhlenbeck and Gormov compactness theorems.
result Moduli spaces of solutions to the mixed equation satisfy compactness properties.
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.
We prove in this paper that, under suitable coinditions on an initial data set, we can obtain Area and Curvature Estimates for simple marginally outer trapped surfaces (or MOTS). Using this estimates, we derive a Compactness Theorem for MOTS. Moreover, the Compactness Theorem will allow us to adapt the recent Degree Th…
This is one in a series of papers devoted to the foundations of Symplectic Field Theory sketched in [Y Eliashberg, A Givental and H Hofer, Introduction to Symplectic Field Theory, Geom. Funct. Anal. Special Volume, Part II (2000) 560--673]. We prove compactness results for moduli spaces of holomorphic curves arising in…
Derives stress-energy identities in Liouville theory on compact surfaces.
problem Stress-energy tensor correlation functions on compact Riemann surfaces.
method Varying correlation functions with respect to background metric, treating different types of variations separately.
result Stress-energy correlation functions expressed as differential operators acting on primary field correlation functions.
New Einstein manifolds split into symmetric and compact parts.
problem Understanding Einstein manifolds with unimodular isometry groups.
method Theory of polar actions, Lie-theoretic arguments, and maximum principles.
result Negative Einstein manifolds split into symmetric and compact parts.
Compactness theory for super Ricci flows provides convergence results.
problem Understanding convergence of super Ricci flows.
method Developed a compactness theory for super Ricci flows.
result Subsequential convergence to a metric flow under certain conditions.
In this thesis, we study deformations of compact holomorphic Poisson manifolds and algebraic Poisson schemes in the framework of Kodaira-Spencer's analytic deformation theory and Grothendieck's algebraic deformation theory.
Proves compact Cauchy horizons have constant surface gravity under null energy condition.
problem Proving compact Cauchy horizons have constant surface gravity.
method Combines ergodic theory, Hodge theory, and Riemannian flow theory.
result Compact Cauchy horizons admit a smooth lightlike tangent vector field of constant surface gravity.
The paper classifies fibrations of 3-dimensional flat orbifolds.
problem Classifying fibrations of compact flat 3-orbifolds.
method Developed a theory for classifying fibrations of compact flat n-orbifolds, applying it to 3-orbifolds. result All geometric fibrations of compact, connected, flat 3-orbifolds, over a 1-orbifold, up to affine equivalence.
Study on scalar curvature decay on non-compact manifolds linked at infinity.
problem Understanding scalar curvature decay on non-compact manifolds with topological linking at infinity.
method Analyzing polynomial decay, developing obstruction theory, using μ--bubble exhaustions, and index theory. result Topological linking at infinity forces polynomial decay of scalar curvature on manifolds of weakly bounded geometry.
Defines geometric quantization for non-compact Hamiltonian torus manifolds using index theory.
problem Geometric quantization for non-compact Hamiltonian torus manifolds.
method Deformation of Dirac operator along group orbits, localization to lattice points.
result Geometric quantization is independent of the choice of polarization.
Proves regularity of extremal function on compact Kähler manifolds.
problem Regularity of extremal function on compact Kähler manifolds.
method Local property analysis and equivalence of continuity and Hölder continuity.
result Equivalence of classical notions of local L-regularity and locally Hölder continuous property. Extends optimal regularity and Uhlenbeck compactness to non-Riemannian manifolds.
problem Establishing optimal regularity and compactness for connections on vector bundles over non-Riemannian manifolds.
method Proofs based on RT-equations for connections with Lp curvature, extending to non-compact gauge groups. result Removes singularities at GR shock waves, ensuring existence of geodesics and coordinates.
We develop the theory of Berezin-Toeplitz operator on any compact symplectic prequantizable manifold from scratch. Our main inspiration is the Boutet de Monvel-Guillemin theory, that we simplify in several ways to obtain a concise exposition. A comparison with the spin-c Dirac quantization is also included.
We investigate compact Hausdorff foliations on compact Riemannian manifolds in the context of the Gromov-Hausdorff distance theory. We give some sufficient conditions for such foliations to be separated in the Gromov-Hausdorff topology.
We propose a way to define and compute invariants of general smooth 4-manifolds based on topological twists of non-Lagrangian 4d N=2 and N=3 theories in which the problem is reduced to a fairly standard computation in topological A-model, albeit with rather unusual targets, such as compact and non-compact Gepner models…
Compactness theorem for 3-manifold Floer theory defined by Fueter sections.
problem Defining a new 3-manifold Floer theory with compact counts.
method Counting Fueter sections of hyperkähler bundles over 3-manifolds.
result Proved a compactness theorem for k=2.
The paper proves an index theorem for loop spaces of compact manifolds.
problem Defining an index theorem for loop spaces of compact manifolds.
method Formulated and proved an equivariant index theorem for non-compact manifolds with S1-actions, using a ring of formal power series. result Found an appropriate form of the index theorem for loop spaces.
With a view to constructing a Morse/Floer homology theory for CMC hypersurfaces, we prove a compactness result modulo broken trajectories for eternal mean curvature flows with forcing term in compact, hyperbolic manifolds.
Develops sublinear Morse theory in symmetric spaces.
problem Understanding sublinear Morse properties in symmetric spaces.
method Theory of sublinearly Morse boundary and lemma in higher rank symmetric spaces.
result Proves sublinear Morse lemma in higher rank symmetric spaces.
Compactness theory for biharmonic maps on degenerating Einstein manifolds.
problem Analyzing biharmonic maps on degenerating Einstein manifolds.
method Developed a compactness theory using asymptotic analysis over degenerating neck regions.
result Established a compactness theory for biharmonic maps with finitely many bubbles.
The paper studies deformations of cohesive modules on complex manifolds.
problem Deformation theory of cohesive modules on compact complex manifolds.
method Development of Kuranishi maps and obstructions for deformations of cohesive modules.
result Generalization of deformation theory for holomorphic vector bundles and coherent sheaves.
We develop a degree theory for compact immersed hypersurfaces of prescribed K-curvature immersed in a compact, orientable Riemannian manifold, where K is any elliptic curvature function. We apply this theory to count the (algebraic) number of immersed hyperspheres in various cases: where K is mean curvature; extr…
Proves elliptic operator images are closed on Hilbert bundles.
problem Closedness of images of elliptic operators on Hilbert bundles.
method Analyzes tensor product of elliptic operators and compares images.
result Establishes closedness of images with respect to natural topology.
Let (X,ω) be a compact Kähler manifold of dimension n and fix m∈N such that 1≤m≤n. We prove that any (ω,m)-sh function can be approximated from above by smooth (ω,m)-sh functions. A potential theory for the complex Hessian equation is also developed which generalizes the classical pluri…
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.
The study applies Riemannian flow theory to Lorentzian manifolds to understand horizons.
problem Understanding the geometry of horizons in Lorentzian manifolds.
method Importing results from Riemannian flows to Lorentzian horizons, clarifying the relation between isometric/geodesible flows and non-degeneracy conditions.
result Theorems on the dynamical structure of compact horizons without relying on degeneracy assumptions.
We extend the spectral theory of generalized Laplacians to integrable metrics on compact Riemann surfaces. As a consequence, we attach in a direct way, a holomorphic analytic torsion to any integrable metrics. We also provide a different approach to define the holomorphic analytic torsion. We prove that both approaches…
New insights into manifold properties using Seiberg-Witten and L2 harmonic theories.
problem Characterizing properties of 4-manifolds with specific geometric conditions.
method Combining Seiberg-Witten theory on compact manifolds and L2 harmonic theory on non-compact manifolds, with a new argument for asymptotic properties. result Found a pair of homeomorphic 4-manifolds with distinct geometric properties under Riemannian metrics.
Game theory applied to splitting surfaces of compact 2-manifolds.
problem Determining the winner in a game played on surfaces of compact 2-manifolds.
method Analyzing the game through Nim addition and series of G-values based on increasing genus. result The G-series determines the winner in the game played on compact 2-manifolds. 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.
Study curvatures on compact pseudo-Hermitian manifolds using special methods.
problem Prescribing Webster scalar curvatures on compact pseudo-Hermitian manifolds.
method Upper and lower solutions, perturbation theory of self-adjoint operators, CR conformal deformations.
result Described sets of Webster scalar curvature functions that can be realized.
In this paper, for a compact Lie group action,we prove the anomaly formula and the functoriality of the equivariant Bismut-Cheeger eta forms with perturbation operators when the equivariant family index vanishes. In order to prove them, we extend the Melrose-Piazza spectral section and its main properties to the equiva…
The study proves compactness and structure of Ricci flow limits.
problem Understanding the structure of Ricci flow limits.
method Weak compactness theorem and structure theory development.
result Ricci flow limit spaces have a regular part with smooth convergence and a singular set of high codimension.
Classifies compact spaces by shape, finite spaces by weak homotopy.
problem Classifying compact Hausdorff spaces and finite topological spaces.
method Constructs a category that classifies spaces by shape and weak homotopy.
result Classifies compact spaces by shape, finite spaces by weak homotopy.
We derive a selection of energy estimates for a generalisation of a critical equation on the unit disc in R2 introduced by Rivière. Applications include sharp regularity results and compactness theorems which generalise a large amount of previous geometric PDE theory, including some of the theory of harmoni…
In this paper, we provide a systematic and constructive description of Vaisman structures on certain principal elliptic bundles over complex flag manifolds. From this description we explicitly classify homogeneous l.c.K. structures on compact homogeneous Hermitian manifolds using elements of representation theory of co…
The paper finds closed hypersurfaces with prescribed mean curvature in non-compact manifolds.
problem Finding closed hypersurfaces with prescribed mean curvature in non-compact manifolds.
method One-parameter prescribed mean curvature min-max theory.
result Closed hypersurfaces with prescribed mean curvature are found in certain non-compact manifolds.
This paper classifies fibrations of flat orbifolds, advancing flat 4-manifold classification.
problem Classifying fibrations of compact flat orbifolds.
method Developed theory for classifying fibrations up to affine equivalence.
result Classified fibrations of compact flat 2-orbifolds.