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.
Conic singular sub-manifolds are Lipschitz Normally Embedded in compact non-Euclidean manifolds.
problem Understanding the Lipschitz geometry of conic singular sub-manifolds.
method Analyzing the metric properties of conic singular sub-manifolds in compact non-Euclidean manifolds.
result Connected conic singular sub-manifolds are Lipschitz Normally Embedded.
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.
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.
Proves positive mass theorem on conical manifolds with small angles.
problem Proving the positive mass theorem on conical manifolds with small cone angles.
method Analyzes conical manifolds with small cone angles, assuming spin structure and locally conformal flatness.
result Proves the positive mass theorem under specified conditions.
Proves Morse index theorem for geodesics in conic Finsler manifolds.
problem Geodesic index theorem in conic Finsler manifolds with variable endpoints.
method Proves Morse index theorem for geodesics connecting submanifolds in a C7 manifold with a C6 conic pseudo-Finsler metric. result Establishes the Morse index theorem for geodesics in conic Finsler manifolds.
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.
The paper proves the existence of Sasaki-Einstein metrics on specific Sasakian manifolds.
problem Proving the existence of conic Sasaki-Einstein metrics on log Fano Sasakian manifolds of dimension five.
method Deriving uniform L^{4}-bounds and analyzing the conic Sasaki-Ricci flow.
result Existence of conic Sasaki-Einstein metrics on log Fano Sasakian manifolds of dimension five.
Study finds existence of Q-curvature metrics on even-dimensional manifolds with conical singularities.
problem Existence of Q-curvature metrics on manifolds with conical singularities. method Blow-up analysis of a 2mth-order PDE and variational min-max argument. result First existence result for supercritical conic manifolds (except spheres).
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.
Proves curvature comparison theorem for manifolds with conical singularities.
problem Comparing scalar mean curvature of manifolds with conical singularities.
method Uses Dirac operator and index theory to prove curvature comparison theorem.
result Proves curvature comparison theorem without knowing the index of the twisted Dirac operator.
Defines Perelman's functionals on manifolds with non-isolated conical singularities.
problem Defining functionals on manifolds with non-isolated conical singularities.
method Starting from a spectral point of view for the Perelman's λ-functional, defining the spectrum of Schrödinger operator and proving the existence of discrete eigenvalues.
result Proves the existence of the infimum of W-functional and obtains asymptotic behavior of eigenfunctions.
The paper classifies isoparametric hypersurfaces in conic Finsler spaces.
problem Identifying new isoparametric hypersurfaces in conic Finsler spaces.
method Introduced isoparametric functions and hypersurfaces in conic Finsler spaces, classified them in specific spaces.
result Found additional isoparametric hypersurfaces in conic Minkowski spaces, such as helicoids.
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.
Proves existence of Yamabe metrics on conical 4-manifolds using min-max method.
problem Existence of Yamabe metrics on conical 4-manifolds with singular points.
method Min-max scheme adapted to singular setting, leveraging recent positive mass theorems.
result Existence of Yamabe metrics on conical 4-manifolds with finitely-many singular points.
Extends existence results for scalar curvature on conical manifolds.
problem Existence of metrics with positive scalar curvature on conical manifolds.
method Extends Kazdan-Warner and Cruz-Vitório results to conical manifolds with isolated singularities using index theory.
result Any bounded and smooth negative function on a conical manifold is the scalar curvature of some conical metric.
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. 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…
In this paper, we study the long-term behavior of the conical Kähler-Ricci flow on Fano manifold M. First, based on our work of locally uniform regularity for the twisted Kähler-Ricci flows, we obtain a long-time solution to the conical Kähler-Ricci flow by limiting a sequence of these twisted flows. Second, we study…
The paper solves navigation problems on conic Kropina manifolds and establishes curvature relationships.
problem Navigation problems on conic Kropina manifolds.
method Analyzes the solution of navigation problems and establishes curvature relationships.
result The solution to navigation problems on conic Kropina manifolds must be either a Randers metric or a Kropina metric.
We give criterions for the existence of toric conical Kahler-Einstein and Kahler-Ricci soliton metrics on any toric manifold in relation to the greatest Ricci and Bakry-Emery-Ricci lower bound. We also show that any two toric manifolds with the same dimension can be joined by a continuous path of toric manifolds with c…
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.
Deformations of singular Cayley submanifolds studied.
problem Constructing fibrations of compact Spin(7) manifolds.
method Deformation theory of conically singular and asymptotically conical Cayley submanifolds.
result Detailed description of the deformation theory.
In this paper, we study the stability of the conical Kähler-Ricci flows on Fano manifolds. That is, if there exists a conical Kähler-Einstein metric with cone angle 2πβ along the divisor, then for any β′ sufficiently close to β, the corresponding conical Kähler-Ricci flow converges to a conical Kähler-Einstein me…
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.
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.
We consider the constant Q-curvature metric problem in the given conformal class on conic 4-manifolds and study related differential equations.
Smoothness proven for conical Calabi-Yau potentials on Fano cones.
problem Smoothness of conical Calabi-Yau potentials on Fano cones.
method Pluripotential theory on degenerate Sasakian manifolds.
result Locally bounded conical Calabi-Yau potentials are smooth on the regular locus.
Study spectral properties on manifolds with conical singularities, proving new inequalities.
problem Analyzing spectral properties and geometric inequalities on manifolds with conical singularities.
method Develops new inequalities for manifolds with conical singularities, not covered by existing methods.
result Proves a Bakry-Émery inequality, Hardy inequality, and spectral gap estimate.
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 goal of this article is to generalise the Witten deformation to even dimensional conic manifolds and a class of functions called admissible Morse functions.
In this paper, we prove that on a compact manifold with isolated conical singularity the spectrum of the Schrödinger operator −4Δ+R consists of discrete eigenvalues with finite multiplicities, if the scalar curvature R satisfies a certain condition near the singularity. Moreover, we obtain an asymptotic behavior fo…
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.
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.
In this paper, we prove the conic version of YTD conjecture on log Fano manifolds.
We consider the unnormalized Yamabe flow on manifolds with conical singularities. Under certain geometric assumption on the initial cross-section we show well posedness of the short time solution in the Lq-setting. Moreover, we give a picture of the deformation of the conical tips under the flow by providing an asym…
We prove that on one Kähler-Einstein Fano manifold without holomorphic vector fields, there exists a unique conical Kähler-Einstein metric along a simple normal crossing divisor with admissible prescribed cone angles. We also establish a curvature estimate for conic metrics along a simple normal crossing divisor which …
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…
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.
We partially confirm a conjecture of Donaldson relating the greatest Ricci lower bound R(X) to the existence of conical Kahler-Einstein metrics on a Fano manifold X. In particular, if D∈∣−KX∣ is a smooth simple divisor and the Mabuchi K-energy is bounded below, then there exists a unique conical Kahler-Eins…
Study finds conditions for conformal deformations to constant scalar curvature in conic metrics.
problem Finding conditions for conformal deformations to constant scalar curvature in conic metrics.
method Analyzes conformal deformations within a class of incomplete Riemannian metrics that generalize conic orbifold singularities.
result Determines sufficient conditions for the existence of a conformal deformation to a conic metric with constant scalar curvature -1.
Study conically singular instantons over SU(3)-manifolds, proving existence and dimension formulas.
problem Existence and dimension of moduli spaces of conically singular instantons.
method Develop Fredholm deformation theory, investigate cokernel of instanton operator.
result Formula for virtual dimension of moduli space of conically singular instantons with structure group P(U(n)).
The study examines conical Kähler-Einstein metrics on specific Fano manifolds and their behavior at the limit angle.
problem Analyzing conical Kähler-Einstein metrics on rank one horosymmetric Fano manifolds.
method Investigates the convergence of these metrics as the cone angle approaches zero, focusing on the behavior on the complement of a codimension one orbit.
result The metrics converge to the Kähler-Einstein metric on the basis and Stenzel's Ricci flat Kähler metrics on the fibers.
In this paper we discuss Perelman's Lambda-functional, Perelman's Ricci shrinker entropy as well as the Ricci expander entropy on a class of manifolds with isolated conical singularities. On such manifolds, a singular Ricci de Turck flow preserving the isolated conical singularities exists by our previous work. We prov…
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.
The aim of this paper is to extend the definition of geodesics to conical manifolds, defined as submanifolds of Rn with a finite number of singularities. We look for an approach suitable both for the local geodesic problem and for the calculus of variation in the large
We derive a detailed asymptotic expansion of the heat trace for the Laplace-Beltrami operator on functions on manifolds with conic singularities, using the Singular Asymptotics Lemma of Jochen Bruening and Robert T. Seeley [BS]. In the subsequent paper we investigate how the terms in the expansion reflect the geometry …
We study the problem of desingularizing coassociative conical singularities via gluing, allowing for topological and analytic obstructions, and discuss applications. This extends the author's earlier work on the unobstructed case. We interpret the analytic obstructions geometrically via the obstruction theory for defor…