Investigates deformations of Q-curvature on manifolds.
problem Deformation problems of Q-curvature on closed Riemannian manifolds.
method Uses Q-singular space and derived several results about geometry related to Q-curvature. result Showed that any smooth function can be realized as a Q-curvature on generic Q-flat manifolds.
Paper proves rigidity theorems for AE Q-singular spaces.
problem Analyzing Q-curvature on AE manifolds. method Introducing a fourth order energy and rewriting it in terms of a fourth order Ricci-like tensor.
result Yamabe positive J-flat AE manifolds are isometric to Euclidean space. A generalised Thurston-Bennequin invariant for a Q-singularity of a real algebraic variety is defined as a linking form on the homologies of the real link of the singularity. The main goal of this paper is to present a method to calculate the linking form in terms of the very good resolution graph of a real normal unib…
Defines a tensor related to Q-curvature on manifolds.
problem Understanding Q-curvature and its applications.
method Defines a symmetric 2-tensor J-tensor associated to Q-curvature.
result Establishes a relation between J-tensor and Q-curvature, akin to Ricci tensor and scalar curvature.
We show that the Reeb vector, and hence in particular the volume, of a Sasaki-Einstein metric on the base of a toric Calabi-Yau cone of complex dimension n may be computed by minimising a function Z on R^n which depends only on the toric data that defines the singularity. In this way one can extract certain geometric i…
Study on singularities of area-minimizing currents, focusing on frequency and branch points.
problem Understanding the nature of singular points in area-minimizing currents.
method Intrinsic frequency function and decomposition theorem for singular set.
result Established properties of the planar frequency function and decomposition of singular set.
Study of tangent spaces in diffeological spaces under Lie group actions.
problem Understanding tangent spaces in generalized spaces.
method Generalized tangent space construction and isomorphism proof.
result Internal tangent space isomorphic to stratified tangent space.
(1,1) non-L-space knots are foliar in 3D space.
problem Proving (1,1) non-L-space knots are foliar.
method Analyzing (1,1) non-L-space knots in S3 and lens spaces. result (1,1) non-L-space knots are persistently foliar.
Universal spaces for finite topological spaces simplify shape descriptions.
problem Describing shape properties of compact metric spaces.
method Inverse limits of finite spaces and Alexandroff extensions.
result Universal spaces simplify shape descriptions of compact metric spaces.
Defines strongly Gauduchon spaces and their properties.
problem Generalization of strongly Gauduchon manifolds in complex spaces.
method Definition and properties of strongly Gauduchon spaces and class SG.
result Similarities and properties of strongly Gauduchon spaces to Kahlerian spaces.
The paper extends Stone duality to topological convexity spaces.
problem Understanding the relationship between topological convexity spaces and sup-lattices.
method Extending Stone duality to topological convexity spaces using preconvexity spaces.
result An adjunction between topological convexity spaces and sup-lattices.
The paper extends tangent space theory for diffeological spaces and bundles.
problem Understanding tangent spaces of diffeological bundles and spaces.
method Introducing weakly filtered and filtered diffeological spaces, extending exact sequences, and defining Hector's tangent bundle.
result Tangent bundles of filtered diffeological spaces are diffeological vector spaces.
New findings on GRW space-times with constant scalar curvature.
problem Understanding GRW space-times in different subspaces.
method Analyzing orthogonal subspaces of Gray's decomposition.
result Generalized quasi-Einstein GRW space-times reduce to known types of space-times.
No Einstein hypersurfaces found in Damek-Ricci spaces.
problem Existence of Einstein hypersurfaces in symmetric spaces.
method Analyzing properties of Damek-Ricci spaces and proving no Einstein hypersurface exists.
result No Einstein hypersurfaces in Damek-Ricci spaces.
Generalizes balanced manifolds to complex spaces, studying their properties and relations.
problem Generalizing balanced manifolds to complex spaces.
method Defined class B and balanced spaces, compared with Kahlerian spaces, and studied properties and relations.
result Properties and relations between balanced spaces and class B are obtained.
The distance function ϱ(p,q) (or d(p,q)) of a distance space (general metric space) is not differentiable in general. We investigate such distance spaces over Rn, whose distance functions are differentiable like in case of Finsler spaces. These spaces have several good properties, yet they are no F…
Classifies g.o. spaces and studies geodesics in generalized Wallach spaces.
problem Classifying and studying geodesics in generalized Wallach spaces.
method Investigates homogeneous geodesics in generalized Wallach spaces for any invariant Riemannian metric.
result Provides classifications and examples of g.o. spaces and geodesics.
New kernels defined for various spaces, including measures.
problem Defining kernels on non-standard spaces like measures.
method Integrally strictly positive definite and characteristic kernels on Hilbert, Banach, and metric spaces.
result Explicit classes of kernels on Lp spaces and sets of measures. In a paper (math.DG/0403528) we obtained explicit examples of Moishezon twistor spaces of some compact self-dual four-manifolds admitting a non-trivial Killing field, and also determined their moduli space. In this note we investigate minitwistor spaces associated to these twistor spaces. We determine their structure, …
Metric spaces uniquely split into Hilbert and non-line-split parts.
problem Understanding the structure of metric spaces.
method Proved unique decomposition into Hilbert and non-line-split parts.
result Metric spaces have a unique decomposition into a Hilbert space and a non-line-split part.
A metric defined on body moduli space.
problem Defining a metric on the moduli space of bodies.
method Quotient space by integral affine transformations.
result Identifies moduli space of Delzant polytopes with symplectic toric manifolds.
Study geometry of tetrahedra in complex hyperbolic space and Hilbert spaces.
problem Understanding geometric relationships between complex hyperbolic spaces and Hilbert spaces.
method Use a complex analog of the cosine of a vertex angle as a novel technical tool.
result Describe possible triangular faces of tetrahedra in hyperbolic space and three-dimensional subspaces in Hilbert spaces with Pick kernels.
New curvature positivity helps classify spherical spaces and complex projective spaces.
problem Classifying spherical space forms and complex projective spaces.
method Introducing a new positivity notion for curvature.
result Characterizations for spherical space forms and complex projective spaces.
Study of complete space-like self-expanders in Minkovski space.
problem Characterize complete space-like self-expanders in Minkovski space.
method Use of maximum principle of Omori-Yau type to prove rigidity theorems.
result Classification of 2-dimensional complete space-like self-expanders with constant squared norm of the second fundamental form.
The abstract discusses the linear and smooth structures of mapping spaces.
problem The structure of mapping spaces in differential geometry.
method Proving diffeomorphisms and fibre bundle properties.
result Path spaces and base point preserving mapping spaces are Fréchet spaces.
Study on convergence of transformed metric spaces as dimensions grow.
problem Conditions for convergence of transformed metric spaces.
method Clarifying conditions for convergence of transformed spaces from original sequence and vice versa.
result Spheres and projective spaces converge to Gaussian space and its quotient as dimensions increase.
Berwald spaces with non-zero flag curvature are Riemannian.
problem Understanding the conditions under which Berwald spaces become Riemannian.
method Analyzing the flag curvature of Berwald spaces and proving their rigidity under certain conditions.
result Berwald spaces with non-zero flag curvature are Riemannian.
Characterizes when almost smooth spaces become RCD spaces.
problem Understanding conditions for almost smooth spaces to be RCD spaces.
method Characterizations via local volume doubling and Poincaré inequality.
result Characterizes Einstein 4-orbifolds.
Stability of Wasserstein spaces under various convergence types.
problem Stability and finiteness of Wasserstein spaces over singular and non-singular spaces.
method Gromov--Hausdorff convergence and equivariant Gromov--Hausdorff convergence.
result Analogue of Perelman's stability theorem on Wasserstein spaces.
New infinite families of flat spaces found from symmetric spaces.
problem Finding new flat homogeneous spaces.
method Starting from compact symmetric spaces, constructing infinite families of compact homogeneous spaces with invariant Bismut connections.
result Infinite families of compact homogeneous spaces with vanishing Ricci tensor.
New quasi space forms solve Thurston's geometrical space form problem.
problem Solving Thurston's geometrical space form problem.
method Introducing quasi space forms as non-real space forms with specific geometric properties.
result Quasi space forms offer a metrical, local geometrical solution to Thurston's problem.
Sphere theorems extended to RCD spaces and improved for Einstein stratified spaces.
problem Generalizing sphere theorems to new types of spaces.
method Proved sphere theorems for RCD(n-1, n) spaces and Einstein stratified spaces.
result Extended sphere theorems to RCD spaces and improved results for Einstein stratified spaces.
This work describes compactifications of metric spaces and vector spaces using asymmetric norms.
problem Compactifying metric spaces and vector spaces using asymmetric norms.
method Nonstandard methods, ultrapowers of the spaces at hand.
result Polyhedral compactifications of vector spaces with stratified structure.
The paper proves rigidity theorems for space-like λ-hypersurfaces in Lorentzian space.
problem Rigidity of space-like λ-hypersurfaces in Lorentzian space.
method Analyzing complete space-like λ-hypersurfaces in R1n+1. result Proves rigidity theorems for space-like self-shrinkers in R1n+1. Introduces new types of homogeneous spaces and their properties.
problem Defining and understanding new types of homogeneous spaces.
method Introducing and analyzing (strongly) (Θ-)discrete homogeneous spaces. result Discovers relationships between new and existing homogeneous space types.
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.
Estimates eigenvalues of Jacobi operator for harmonic spaces.
problem Estimating eigenvalues of Jacobi operator.
method Using density function of a harmonic space.
result Sharp estimates imply symmetric Osserman space.
Survey Bernstein-type theorems for graphical surfaces in Euclidean and Lorentz-Minkowski spaces.
problem Proving theorems for minimal and constant mean curvature graphs in Euclidean and Lorentz-Minkowski spaces.
method Explains several proofs and provides mean curvature estimates for graphs in Euclidean and Lorentz-Minkowski spaces.
result Bernstein-type theorems for constant mean curvature graphs in Euclidean 3-space and space-like graphs in Lorentz-Minkowski 3-space.
In this paper, we present a unified study of the moduli space of tropical curves and Outer space which we link via period maps to the moduli space of tropical abelian varieties and the space of positive definite quadratic forms. Our work is a first step towards exhibiting Outer space and the space of positive definite …
Higher rank rigidity theorem extended to Berwald spaces.
problem Generalizing rigidity theorem to Berwald spaces.
method Proving rigidity for Berwald spaces with specific curvature conditions.
result Berwald spaces with certain conditions are locally symmetric or Minkowski.
CAT(0) spaces with small volume growth are homeomorphic to Euclidean space.
problem Characterizing CAT(0) spaces with specific volume growth properties.
method Analyzing asymptotic topological regularity and volume growth.
result CAT(0) spaces with small volume growth are homeomorphic to Euclidean space.
Cantor Riemannium is a new type of space from holomorphic germs.
problem Defining a new type of space from holomorphic germs.
method Constructing the Cantor Riemannium by Borel monogenic continuation.
result The Cantor Riemannium is a metric, path connected, Gromov length space.
Study RKHS on manifolds, linking Sobolev and diffusion spaces.
problem Characterizing RKHS on manifolds and their properties.
method Analyzing Sobolev and diffusion spaces on Riemannian manifolds.
result Sobolev spaces are RKHS under certain conditions, and diffusion spaces are introduced.
Embeds finite Alexandrov spaces into Euclidean products.
problem Embedding Alexandrov spaces into Euclidean products.
method Canonical embedding into a product of Alexandrov and Euclidean spaces.
result Affine functions on X arise from Euclidean functions.
Study static Einstein-Maxwell space invariant by translation.
problem Classify static Einstein-Maxwell space invariant under translation.
method Analyze static Einstein-Maxwell space conformal to pseudo-Euclidean space.
result Complete classification of static Einstein-Maxwell space invariant under translation.
A "Chen space" is a set X equipped with a collection of "plots" - maps from convex sets to X - satisfying three simple axioms. While an individual Chen space can be much worse than a smooth manifold, the category of all Chen spaces is much better behaved than the category of smooth manifolds. For example, any subspace …
New spaces help connect manifold structures on equivariant Poincaré spaces.
problem Creating manifold structures on equivariant Poincaré spaces.
method Introducing semifree isovariant G-Poincaré spaces and gap conditions. result Space of isovariant structures on semifree G-Poincaré spaces is highly connected. The study explores properties of homologically locally connected spaces and their connections to other topological concepts.
problem Characterizing and understanding homologically locally connected spaces.
method Investigates properties and characterizations of homologically UVn-maps and lcGn-spaces, comparing them to existing concepts. result Identifies similarities and parallels between homologically locally connected spaces and other topological concepts.