Desingularizes conically singular Cayley submanifolds.
problem Constructing fibrations of compact Spin(7) manifolds by Cayley submanifolds.
method Desingularization through gluing rescaled asymptotically conical submanifolds.
result Conically singular Cayley submanifolds can be desingularized.
Study instantons on asymptotically conical Spin(7)-manifolds, identifying deformation spaces.
problem Deformation theory of instantons on specific Spin(7)-manifolds.
method Relating deformation complex to spinors, identifying kernel of twisted negative Dirac operator.
result Virtual dimension of moduli space calculated using index theorem and Dirac operator spectrum.
Study on potential behavior in special geometric spaces.
problem Understanding potential behavior in specific geometric spaces.
method Analyzing asymptotic behavior of p-capacitary potentials and weak Inverse Mean Curvature Flow. result Characterized the behavior of potentials in Asymptotically Conical manifolds.
The study shows that certain nearly G2 and nearly Kähler conifolds cannot be resolved by gluing asymptotically conical G2 and Calabi-Yau manifolds.
problem The inability to resolve nearly G2 and nearly Kähler conifolds by gluing asymptotically conical G2 and Calabi-Yau manifolds.
method Topological analysis of asymptotically conical G2 and Calabi-Yau manifolds to show conditions under which resolutions are impossible.
result For certain rates of the metric, the G2 4-form and Kähler form cannot be simultaneously exact, leading to non-existence of resolutions.
We show compactness in the locally smooth topology for certain natural families of asymptotically conical self-expanding solutions of mean curvature flow. Specifically, we show such compactness for the set of all two-dimensional self-expanders of a fixed topological type and, in all dimensions, for the set of self-expa…
We study coassociative 4-folds N in R^7 which are asymptotically conical to a cone C with rate lambda<1. If lambda is in the interval [-2,1) and generic, we show that the moduli space of coassociative deformations of N which are also asymptotically conical to C with rate lambda is a smooth manifold, and we calculate it…
Proves positive mass theorem for AF spin manifolds with conical singularities.
problem Proving the positive mass theorem for singular metrics on AF manifolds.
method Analyzes AF spin manifolds with isolated conical singularities, allowing topological singularities.
result Proves the positive mass theorem for AF spin manifolds with conical singularities.
We establish the existence of an integer degree for the natural projection map from the space of parameterizations of asymptotically conical self-expanders to the space of parameterizations of the asymptotic cones when this map is proper. As an application we show that there is an open set in the space of cones in the …
We show that the space of asymptotically conical self-expanders of the mean curvature flow is a smooth Banach manifold. An immediate consequence is that non-degenerate self-expanders -- that is, those self-expanders that admit no non-trivial normal Jacobi fields that fix the asymptotic cone -- are generic in a certain …
We study the asymptotics of the determinant of Laplacian on a translation surface (a compact Riemann surface equipped with a conformal flat conical metric with trivial holonomy) of genus g with 2g-2 conical points of angle 4πas two conical points collide.
Maps asymptotically embed conic transforms from circle bundles.
problem Embedding conic transforms from circle bundles.
method Asymptotic embeddings using equivariant Szegő projectors.
result Maps embed conic transforms from circle bundles.
Paper shows X-ray transform invertible on certain curved spaces.
problem Invertibility of X-ray transform on curved spaces.
method Introduced new pseudodifferential operator algebra for analysis.
result Stable invertibility near infinity on asymptotically conic manifolds.
Study self shrinkers with medium entropy in 4D space.
problem Analyzing self shrinkers with entropy bounds.
method Smooth asymptotically conical self shrinkers in R^4.
result Entropy bounded above by Λ_1.
We consider the mapping properties of generalized Laplace-type operators L=∇∗∇+R on the class of quasi-asymptotically conical (QAC) spaces, which provide a Riemannian generalization of the QALE manifolds considered by Joyce. Our main result gives conditions under which such opera…
In this article, we study the properties of the geodesic X-ray transform for asymptotically Euclidean or conic Riemannian metrics and show injectivity under non-trapping and no conjugate point assumptions. We also define a notion of lens data for such metrics and study the associated inverse problem.
New self-expander found between two given asymptotic ones.
problem Finding new self-expanders between given asymptotic ones.
method Developed a min-max theory for asymptotically conical self-expanders of mean curvature flow.
result Existence of a new asymptotically conical self-expander trapped between two given ones.
Study on deformations of Spin(7)-structures on manifolds.
problem Analyzing deformations of Spin(7)-structures on asymptotically conical manifolds.
method Examined the moduli space of torsion-free, asymptotically conical Spin(7)-structures, showing it is an orbifold for generic decay rates.
result Found that the classical Bryant-Salamon metric on positive spinors on S4 has no continuous deformations as an AC Spin(7)-metric. McLean studied the deformations of compact special Lagrangian submanifolds, showing in particular that they come in moduli spaces whose dimension depends only on the topology of the submanifold. In this article we study the analogous problem for non-compact, "asymptotically conical" SL submanifolds, with respect to var…
We use a Lagrangian perspective to show the limiting absorption principle on Riemannian scattering, i.e. asymptotically conic, spaces, and their generalizations. More precisely we show that, for non-zero spectral parameter, the `on spectrum', as well as the `off-spectrum', spectral family is Fredholm in function spaces…
Uniform elliptic theory for Dirac operators on orbifold resolutions.
problem Analyzing Dirac operators on orbifold resolutions.
method Viewing orbifolds as conically fibred singular spaces and resolving them by gluing asymptotically conical fibrations.
result Uniform index formula for Dirac operators on orbifold resolutions.
New one-parameter families of SU(2)2-invariant instantons found on Calabi-Yau 3-folds.
problem Behavior of Calabi-Yau instantons and monopoles with SU(2)2-symmetry. method Gauge theory on asymptotically conical Calabi-Yau 3-folds with SU(2)2 co-homogeneity one action. result New one-parameter families of invariant instantons found.
We present a method to desingularize a compact G_2 manifold with isolated conical singularities by cutting out a neighbourhood of each singular point and glueing in an asymptotically conical G_2 manifold. Controlling the error on the overlap glueing region enables us to use a result of Joyce to conclude that the result…
Given an associative 3-fold in R^7 which is asymptotically conical with generic rate less than 1, we show that its moduli space of deformations is locally homeomorphic to the kernel of a smooth map between smooth manifolds. Moreover, the virtual dimension of the moduli space is computed and shown to be non-negative for…
Proves mass theorem for AF manifolds with conical singularities.
problem Proving the positive mass theorem for specific types of manifolds.
method Conformal blow up technique applied to AF manifolds with isolated conical singularities.
result Positive mass theorem proven for the specified manifolds.
Study conic singular manifolds, proving Lipschitz normal embedding.
problem Understanding metric properties of conic singular manifolds.
method Analyzing interplay between conic and asymptotically conic behavior.
result Proves Lipschitz normal embedding for conic singular sub-manifolds.
Paper shows invertibility of tensor X-ray transform on certain manifolds.
problem Invertibility of tensor X-ray transform on asymptotically conic manifolds.
method Used 1-cusp pseudodifferential operator algebra and modified solenoidal gauge condition.
result Invertibility of tensor X-ray transform up to natural obstruction.
Study on how incomplete smooth metrics degenerate from asymptotically conical Ricci-flat Kähler metrics.
problem Degeneration of asymptotically conical Ricci-flat Kähler metrics.
method Analysis of Kähler class degeneration and convergence of metrics.
result Construction of singular Calabi-Yau metrics and their metric geometry.
The paper shows how to stabilize perturbed Kähler-Ricci solitons.
problem Stabilizing perturbed Kähler-Ricci solitons.
method Normalized Kähler-Ricci flow starting from perturbed metrics.
result The flow converges to an asymptotically conical gradient expanding Kähler-Ricci soliton.
Vortices on conical surfaces embedded in hyperbolic space.
problem Embedding Abelian vortices on conical surfaces in hyperbolic space.
method Analyzing elliptic sinh-Gordon and Tzitzeica equations, asymptotic analysis of Painleve III ODE.
result Radial solutions can be globally embedded in hyperbolic space.
Backwards uniqueness proved for flows with asymptotically conical singularities.
problem Proving uniqueness of mean curvature flows with specific singularities.
method Developed new global tools to handle singularities, asymptotic structure, and smooth parts of flows.
result Backwards uniqueness for mean curvature flows with asymptotically conical singularities proved.
In this paper, we develop the theory of Perelman's W-functional on manifolds with isolated conical singularities. In particular, we show that the infimum of W-functional over a certain weighted Sobolev space on manifolds with isolated conical singularities is finite, and the minimizer exists, if the scalar curvatur…
The article constructs Spin(7) metrics with Aloff--Wallach spaces as orbits.
problem Creating Spin(7) metrics with specific geometric properties.
method Continuous 1-parameter families of non-compact Spin(7) metrics with chiralities, focusing on Aloff--Wallach spaces.
result Construction of Spin(7) metrics with Aloff--Wallach spaces as principal orbits, including geometric transitions.
Ricci flow modelled on specific singularities on closed manifolds.
problem Analyzing singularities in Ricci flows.
method Closed manifold Ricci flow with singularity modeled on asymptotically conical shrinkers.
result Ricci flow solution forms a singularity that matches the given asymptotically conical shrinker.
We construct infinitely many new 1-parameter families of simply connected complete noncompact G_2-manifolds with controlled geometry at infinity. The generic member of each family has so-called asymptotically locally conical (ALC) geometry. However, the nature of the asymptotic geometry changes at two special parameter…
Study on topological rigidity of ALE vector bundles with specific conditions.
problem Classifying ALE vector bundles with asymptotically conical total spaces.
method Topological classification and geometric analysis of ALE vector bundles.
result Only 2-sphere, projective plane, and open contractible manifolds admit ALE tangent bundles.
Study of mean curvature flows with conical singularities using mathematical techniques.
problem Understanding the dynamics of mean curvature flows near conical singularities.
method Feynman-Kac formula and invariant cone method for noncompact settings.
result Generic initial perturbations avoid conical singularities in mean curvature flows.
The paper finds new Spin(7) metrics with specific orbits.
problem Existence of Spin(7) metrics with specified orbits. method Construction of three continuous families of non-compact Spin(7) metrics. result Existence of asymptotically conical and locally conical metrics.
Strict convexity proven for certain self-expanders in high dimensions.
problem Convexity of self-expanders in mean curvature flow.
method Investigation of convexity properties for asymptotically conical self-expanders.
result Strict convexity proven for n-dimensional self-expanders. We show that at the level of formal expansions, any compact Riemannian manifold is the sphere at infinity of an asymptotically conical gradient expanding Ricci soliton.
Proves existence of Yamabe metrics on conical manifolds with conical points and links.
problem Existence of Yamabe metrics on singular manifolds with conical points and links.
method Derives a counterpart of Aubin's result, uses conical links and Fourier analysis, adds lower-order correction to standard bubbles.
result Derives asymptotic expansions on the Yamabe quotient for generic type metrics.
Backward propagation rules for warped products under Ricci flow.
problem Understanding how warped product structures behave under Ricci flow.
method Establishing sufficient conditions for backward propagation of warped product structures.
result Asymptotically conical shrinkers are multiply-warped products over Einstein manifolds.
In this paper we define a new convergence called "asymptotically conic convergence" in which a smooth family of Riemannian metrics on a fixed compact manifold degenerate to a metric with isolated conic singularity. Our results are: convergence of the spectrum of the geometric Laplacians and uniform convergence of the c…
Proves smoothness of conical singularities in mean curvature flow.
problem Resolving singularities in mean curvature flow.
method Analyzes smooth hypersurfaces with isolated conical singularities.
result Smoothness of level set flow through asymptotically conical singularities.
Study linear differential operators on special manifolds.
problem Analyzing elliptic differential operators on specific types of manifolds.
method Examining a linear elliptic differential operator of the form Δ + V - λ on quasi-asymptotically conical manifolds.
result Established an isomorphism theorem for these operators.
New method recovers hyperkähler metrics from twistor models.
problem Constructing and recovering hyperkähler metrics from twistor models.
method Construction of principal twistor models and universality theorem.
result Universality theorem for recovering hyperkähler metrics from twistor spaces.
Proves unknottedness of certain 3D shapes with multiple ends.
problem Determining the structure of complex 3D shapes.
method Used mean curvature flow to analyze shapes with multiple ends.
result Proves unknottedness of shapes with multiple asymptotically conical ends.
The paper proves a Penrose inequality for 4-discs with hyperbolic ends and conic singularities.
problem Proving a Penrose inequality for conformal metrics on a unit 4-disc with hyperbolic ends and conic singularities.
method Defining a mass term and proving a Penrose type inequality with curvature condition.
result Proves a Penrose inequality for conformal metrics on a unit 4-disc with hyperbolic ends and conic singularities.
We study the existence and regularity of solutions to the Cauchy problem for the inhomogeneous heat equation on compact Riemannian manifolds with conical singularities. We introduce weighted Hölder and Sobolev spaces with discrete asymptotics and we prove existence and maximal regularity of solutions to the Cauchy prob…