A new definition of umbilic points at infinity for polynomial surfaces.
problem Defining umbilic points at infinity for homogeneous polynomial graphs.
method Proposed a stronger definition than Toponogov's, proving all are isolated and pairs, with geometric interpretation.
result All umbilic points at infinity are isolated and occur in pairs, being zeroes of the projective extension of the third fundamental form.
Degenerations of rank-two bundles on threefolds lead to isolated point singularities, with rigidity and bubbling properties.
problem Degenerations of rank-two vector bundles on complex threefolds to a rank-two torsion-free sheaf with an isolated point singularity.
method Proving a rigidity identity and using it to obtain smoothability obstructions and construct local smoothings.
result Smoothability obstructions and local smoothings are obtained, with a rigidity identity linking algebraic bubbling multiplicity and Ext-length.
Study limits of quasi-local angular momentum at infinity of gravitating systems.
problem Understanding limits of quasi-local angular momentum at infinity of gravitating systems.
method Based on optimal isometric embedding and quasilocal mass theory, the study defines and analyzes the limits of quasi-local angular momentum at spatial and null infinity.
result Limits of quasi-local angular momentum are discussed at spatial and null infinity of an isolated gravitating system.
Formula for umbilic points on polynomial surfaces, proving their isolated nature and topological type.
problem Understanding the global behavior of fields of principal directions on polynomial surfaces.
method Poincaré-Hopf type formula and projective extension analysis.
result Every umbilic point at infinity has index 1/2 and topological type a Lemon.
Characterizes contracting isometries in CAT(0) cube complexes and acylindrical hyperbolicity of diagram groups.
problem Characterizing contracting isometries in CAT(0) cube complexes and their relation to acylindrical hyperbolicity.
method Characterization of contracting isometries without local finiteness assumption, combinatorial boundary introduction, and application to diagram groups.
result Determine precise conditions for acylindrical hyperbolicity of diagram groups.
Uniqueness proven for stable hypersurface tangent cones.
problem Stability and uniqueness of tangent cones for stable hypersurfaces.
method Analysis of isolated singularities and tangent cones of stable minimal hypersurfaces.
result Uniqueness of tangent cones with integer multiplicities.
We show that the Riemannian Schwarzschild and the ``Taub-bolt'' instanton solutions are the only spaces (M,g) such that 1) M is a 4-dimensional, simply connected manifold with a Riemannian, Ricci-flat C^2-metric g which admits (at least) a 1-parameter group of isometries H without isolated fixed points on M. 2) The quo…
We describe how to compute topological objects associated to a polynomial map of several complex variables with isolated singularities. These objects are: the affine critical values, the affine Milnor numbers for all irregular fibers, the critical values at infinity, and the Milnor numbers at infinity for all irregular…
The study proves unique and isolated properties of Einstein 5-manifolds via gap theorems in 4 dimensions.
problem Understanding the structure and regularity of Einstein 5-manifolds.
method Analysis of tangent cones, gap theorems for 4-dimensional orbifolds, and careful metric analysis.
result Noncollapsed limits of Einstein 5-manifolds have unique and isolated tangent cones.
Study on unique solutions to one-phase free boundary problems.
problem One-phase free boundary problems with singularities.
method Analyzing solutions at singular points and at infinity using one-homogeneous functions.
result Uniqueness of blowups and rigidity results at infinity.
New obstruction found for smoothability of certain 4-manifolds.
problem Obstructing the smoothability of Riemannian metrics with non-positive sectional curvature.
method Extending Davis-Januszkiewicz-Lafont methods to construct examples of locally CAT(0) 4-manifolds with specific properties.
result Examples of locally CAT(0) 4-manifolds that do not have a Riemannian smoothing despite satisfying isolated flats condition.
Infinity-harmonic functions linked to IMCF clusters, revealing new properties in 2D.
problem Understanding properties of ∞-harmonic functions in 2D. method Relating ∞-harmonic functions to inverse mean curvature flow clusters and their po∞ limit. result New structural and regularity results for ∞-harmonic functions in 2D. We give a topological model for a polynomial map from $\C^n$ to $\C$ in the neighborhood of a fiber with isolated singularities. This is motivated out of the ``unfolding of links'' described earlier by the first author and Lee Rudolph. The topological model gives a useful encoding of the local and global monodromy for …
Path connectedness of boundaries for certain CAT(0) groups with isolated flats.
problem Conditions for the path connectedness of boundaries of CAT(0) groups.
method Study of CAT(0) groups with isolated flats acting on CAT(0) spaces.
result Visual boundaries of CAT(0) groups with isolated flats are path connected.
New foliation method for isolated systems in General Relativity.
problem Defining a robust center of mass for isolated systems in GR.
method Foliation by constant spacetime mean curvature (STCMC) 2-spheres.
result Unique STCMC-foliation exists near infinity of any asymptotically Euclidean initial data set.
New Calabi-Yau metrics found on complex symmetric spaces.
problem Finding Calabi-Yau metrics on complex symmetric spaces.
method Complete Calabi-Yau metrics with prescribed horospherical singular tangent cone.
result First examples of Calabi-Yau smoothings of singular tangent cones.
We consider a continuous family (fs), s∈[0,1] of complex polynomials in two variables with isolated singularities, that are Newton non-degenerate. We suppose that the Euler characteristic of a generic fiber is constant (or equivalently the sum of the affine Milnor number and the Milnor number at infinity $μ(s)+λ…
New proof of harmonic map uniqueness with analytic targets.
problem Uniqueness of energy-minimizing harmonic maps with analytic targets.
method Symmetric (log)-epiperimetric inequality for harmonic maps with analytic targets.
result Tangents at infinity of energy-minimizing harmonic maps are unique.
We give a global version of Le-Ramanujam mu-constant theorem for polynomials. Let f_t, (t in [0,1]), be a family of polynomials of n complex variables with isolated singularities, whose coefficients are polynomials in t. We consider the case where some numerical invariants are constant (the affine Milnor number, the Mi…
We construct examples of smooth 4-dimensional manifolds M supporting a locally CAT(0)-metric, whose universal cover X satisfy Hruska's isolated flats condition, and contain 2-dimensional flats F with the property that the boundary at infinity of F defines a nontrivial knot in the boundary at infinity of X. As a consequ…
The paper proves a rigidity theorem for null geodesic travel times in asymptotically AdS spacetimes.
problem Determining if a spacetime is conformally AdS based on null geodesic travel times.
method Analyzing all null geodesics from a point to its antipodal point, considering various spacetime conditions.
result The spacetime is conformally AdS if and only if all null geodesics from a point refocus at its antipodal point.
Proves conditions for positive scalar curvature on certain manifolds with conical singularities.
problem Conditions for positive scalar curvature on manifolds with isolated conical singularities.
method Analyzes isolated conical singularities and uses Geroch type results.
result No metric with positive scalar curvature on X#Tn with isolated conical singularity. New Calabi-Yau metrics constructed with detailed geometry at infinity.
problem Constructing complete Calabi-Yau metrics with specific properties.
method Weighted blow-up and Hölder spaces for Laplacian analysis.
result Examples of Calabi-Yau metrics with conical singularities and non-uniqueness of tangent cones.
ALE Kähler surfaces have finitely many types.
problem Classifying ALE Kähler surfaces.
method Resolution of quotient singularities and deformation theory.
result Only finitely many diffeomorphism types exist.
The paper classifies circle actions on 6D manifolds with isolated fixed points.
problem Classifying circle actions on 6D manifolds with isolated fixed points.
method Performing equivariant connected sums at fixed points with specific manifolds.
result A sequence of operations can reduce the fixed point data to the empty collection.
Researchers create a smooth family of metrics on a ball, including hyperbolic and complex hyperbolic metrics.
problem Constructing Poincaré-Einstein metrics on the ball.
method Gibbons-Hawking-type ansatz of Page and Pope.
result The family of metrics includes the hyperbolic metric and converges to complex hyperbolic at one end.
Paper improves anomaly detection by using non-uniform random choices in isolation forests.
problem Detecting clustered diverse outliers more effectively.
method Comparing different split guiding criteria in isolation forests.
result Non-uniform random choices improve outlier discrimination for certain outlier classes.
We proved a uniqueness theorem of tangent connections for a Yang-Mills connection with an isolated singularity with a quadratic growth of the curvature at the singularity. We also obtained controls over the rate of the asymptotic convergence of the connection to the tangent connection under assumptions that the connect…
We construct new examples of complete Einstein metrics on balls. At each point of the boundary at infinity, the metric is asymptotic to a homogeneous Einstein metric on a solvable group, which varies with the point at infinity.
Study localizes integrals at isolated degenerate zeros.
problem Localization of Futaki-Morita integrals at isolated degenerate zeros.
method Streamlined exposition in the spirit of Bott, localization procedure for a holomorphic vector field on CPn. result Essentially unique formula for Futaki-Morita integral invariants.
The study extends Lp-spectrum analysis to warped products and Kleinian groups.
problem Extending Lp-spectrum analysis to new types of manifolds. method Generalized to warped products and certain quotients of hyperbolic space.
result Proves the Lp-spectrum contains a parabolic region for specific manifolds. Study of G2-structures with isolated singularities and bounded torsion.
problem Understanding G2-structures with special torsion and isolated singularities. method Revisiting known examples, describing symmetries, and analyzing collapsing of circle fibres.
result Collapsing circle fibres at isolated points cannot produce G2-structures with bounded torsion. Study on 4D Riemannian manifolds solves curvature problem.
problem Resonant prescribed T-curvature problem on compact manifolds.
method Variational theory, energy and gradient estimates, Morse lemma, Liouville technique.
result New existence results for critical points at infinity.
We introduce a natural extension of the concept of gradient Ricci soliton: the Ricci almost soliton. We provide existence and rigidity results, we deduce a-priori curvature estimates and isolation phenomena, and we investigate some topological properties. A number of differential identities involving the relevant geome…
We propose a definition of center of mass for asymptotically flat manifolds satisfying Regge-Teitelboim condition at infinity. This definition has a coordinate-free expression and natural properties. Furthermore, we prove that our definition is consistent both with the one proposed by Corvino and Schoen and another by …
The paper calculates critical configurations and Morse indices for polygons on circles or ellipses.
problem Finding critical configurations and their properties for polygons on circles or ellipses.
method Computing Morse indices and gradient vector fields for isolated critical points, relating to eigenvalue questions.
result Computed Morse indices and relationships to eigenvalue questions for polygons on circles or ellipses.
The classical notion of center of mass for an isolated system in general relativity is derived from the Hamiltonian formulation and represented by a flux integral at infinity. In contrast to mass and linear momentum which are well-defined for asymptotically flat manifolds, center of mass and angular momentum seem less …
The study confirms a conjecture about critical points of smooth functions.
problem Understanding isolated critical points of smooth functions.
method Investigated cone-like, reasonable, and Rothe H hypothesis critical points.
result The conjecture holds true for certain critical points.
We use topological methods to study various semicontinuity properties of spectra of singular points of plane algebraic curves and of polynomials in two variables at infinity. Using Seifert forms and the Tristram--Levine signatures of links, we reprove (in a slightly weaker version) a result obtained by Steenbrink and V…
New Calabi-Yau metrics found on C^n for n>=3.
problem Finding Calabi-Yau metrics on complex manifolds.
method Constructing metrics with maximal volume growth and singular tangent cones.
result Infinitely many complete Calabi-Yau metrics on C^n.
The minimal number of critical points is studied for smooth functions on closed manifolds.
problem Determining the minimal number of critical points for smooth functions on closed manifolds.
method Investigates cylindrical ball neighborhoods and exotic critical points, proving the conjecture for certain types of critical points.
result The minimal number of critical points is the same for smooth functions without exotic critical points on closed manifolds of dimension at least 6.
This is the last in a series of five papers math.DG/0211294, math.DG/0211295, math.DG/0302355, math.DG/0302356 studying compact special Lagrangian submanifolds (SL m-folds) X in (almost) Calabi-Yau m-folds M with singularities x_1,...,x_n locally modelled on special Lagrangian cones C_1,...,C_n in C^m with isolated sin…
We show that a complete embedded maximal surface in the 3-dimensional Lorentz-Minkowski space L3 with a finite number of singularities is, up to a Lorentzian isometry, an entire graph over any spacelike plane asymptotic to a vertical half catenoid or a horizontal plane and with conelike singular points. We study the…
We study the problem of prescribing the Paneitz curvature on higher dimensional spheres. Particular attention is paid to the blow-up points, i.e. the critical points at infinity of the corresponding variational problem. Using topological tools and a careful analysis of the gradient flow lines in the neighborhood of suc…
Ecker's and Huisken's quantities agree for ancient mean curvature flows.
problem Understanding the finiteness of integral quantities for ancient mean curvature flows.
method Comparison of Ecker's and Huisken's integral quantities.
result Finiteness of Ecker's integral quantity implies finiteness of entropy at infinity.
In this paper we study the Milnor fibrations associated to real analytic map germs ψ:(Rm,0)→(R2,0) with isolated critical point at 0∈Rm. The main result relates the existence of called Strong Milnor fibrations with a transversality condition of a convenient family of analyti…
The study classifies area-maximizing hypersurfaces with singularities and exterior domains.
problem Classifying area-maximizing hypersurfaces with singularities and exterior domains.
method Complete classification for entire area maximizing hypersurfaces with isolated singularities. Construction of an example. Partial result on asymptotic behavior for exterior domains. Solvability of exterior Dirichlet problems.
result Complete classification and partial results on asymptotic behavior for area maximizing hypersurfaces.
Classifies multigraphs for torus actions on 6D manifolds with isolated fixed points.
problem Classifying torus actions on 6D manifolds with isolated fixed points.
method Associate multigraphs to fixed point data, study operations, and prove classification.
result Classifies multigraphs for 6D manifolds by converting them into the empty graph.