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. 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.
Constructs surfaces with conical singularities using variational methods.
problem Creating Hamiltonian Stationary Surfaces with specific singularities.
method Variational methods and convergence process similar to Ginzburg-Landau analysis.
result Obtained surfaces with prescribed conical singularities related to optimal Wente constants.
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.
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.
Survey on Ricci flow on spaces with conical singularities.
problem Analyzing Ricci flow on spaces with isolated conical singularities.
method Ricci de Turck flow preserving conical singularities, stability of Ricci flat metrics, preservation of positive scalar curvature under certain conditions.
result Ricci flat metrics with isolated conical singularities are stable and positive scalar curvature is preserved under the flow.
Developed theory of Perelman's W-functional on manifolds with conical singularities.
problem Analyzing manifolds with conical singularities using Perelman's W-functional.
method Theory development and mathematical analysis on manifolds with isolated conical singularities.
result Existence and asymptotic order of minimizers for the W-functional on manifolds with conical singularities.
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.
New special Lagrangian submanifolds with multiple conical singularities found.
problem Existence of special Lagrangian submanifolds with multiple conical singularities.
method Extending Caffarelli-Hardt-Simon perturbation argument to special Lagrangian setting and proving a bridge principle.
result Existence of conically singular special Lagrangian submanifolds with prescribed regular tangent cones.
Study finds all local models for flat metrics with isolated singularities.
problem Understanding isolated singularities in flat metrics on Riemann surfaces.
method Complex Analysis to find local models and polynomial growth conditions.
result An isolated singularity of a flat metric with finite area is a conical one.
The paper proves stability of Ricci de Turck flow near conical singularities.
problem Stability of Ricci de Turck flow near conical singularities.
method Constructing a Ricci de Turck flow starting close to a Ricci-flat metric with isolated conical singularities.
result The flow converges to a singular Ricci-flat metric under certain conditions.
The paper extends Perelman's λ-functional to manifolds with conical singularities.
problem Analyzing the spectrum of Schrödinger operators on manifolds with conical singularities.
method Proving spectral properties and extending Perelman's λ-functional.
result The spectrum of the Schrödinger operator on compact manifolds with conical singularities is discrete and asymptotic behavior of eigenfunctions near the singularity is obtained.
The paper analyzes Perelman's entropies on manifolds with conical singularities.
problem Analyzing manifolds with conical singularities using Perelman's entropies.
method Employing singular Ricci de Turck flow and Perelman's entropies to study manifolds with conical singularities.
result Entropy is monotone along the singular Ricci de Turck flow and helps in proving properties of Ricci solitons.
In this paper we study the short time existence problem for the (generalized) Lagrangian mean curvature flow in (almost) Calabi--Yau manifolds when the initial Lagrangian submanifold has isolated conical singularities modelled on stable special Lagrangian cones. Given a Lagrangian submanifold F0:L→M in an…
The paper proves weaker conditions for global smoothings of special Lagrangian submanifolds with conical singularities.
problem Conditions for global smoothings of special Lagrangian submanifolds with isolated conical singularities.
method Proof of weaker conditions for global smoothings.
result Global smoothings are possible under weaker hypotheses than previously known.
Smooth Ricci flow from conical singularities with curvature decay.
problem Ricci flow on manifolds with isolated conical singularities.
method Smooth Ricci flow starting from conical metrics with curvature decay.
result Existence of smooth Ricci flow with curvature decay and isolated singularities.
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…
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 spherical conic metrics on Riemann surfaces with isolated singularities.
problem Existence and deformation theory of spherical conic metrics.
method Extended configuration families of simple divisors and Friedrichs extension of the Laplacian.
result Smooth local moduli space of solutions possible when 2 lies in the spectrum of the Laplacian.
We prove that finite area isolated singularities of surfaces with constant positive curvature in R^3 are removable singularities, branch points or immersed conical singularities. We describe the space of immersed conical singularities of such surfaces in terms of the class of real analytic closed locally convex curves …
Extended a formula to higher dimensions with singularities.
problem Generalizing a formula to higher dimensions with singularities.
method Extended a formula to all even dimensions n≥4 for a class of conformally flat manifolds with singular points.
result First formula in dimensions higher than two with isolated conical singularities.
Uniqueness of conical flows helps understand singularities in surface flows.
problem Understanding singularities in surface flows.
method Analyzing asymptotically conical tangent flows.
result Uniqueness of multiplicity-one asymptotically conical tangent flows.
We discuss the bi-Lipschitz geometry of an isolated singular point of a complex surface which particular emphasis on when it is metrically conical.
J. Nitsche's theorem on conformal hyperbolic metrics is extended with new expressions for metrics near isolated singularities.
problem Characterizing isolated singularities of conformal hyperbolic metrics.
method Developing a complex coordinate map to analyze the metric near the singularity.
result New expressions for conformal hyperbolic metrics near isolated singularities are derived.
Proves well-posedness for elliptic problems on domains with singular points.
problem Elliptic boundary value problems on domains with singular points.
method Conformal changes of metric, differential geometry of manifolds with boundary and bounded geometry.
result Proves well-posedness and regularity results for elliptic boundary value problems.
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.
The paper studies curvature properties under a specific type of flow on spaces with conical singularities.
problem Preserving curvature properties (Ricci curvature and scalar curvature) under a flow with conical singularities.
method Ricci de Turck flow, preserving conical structure, additional assumptions for scalar curvature positivity.
result Positivity of scalar curvature is preserved under the flow with additional assumptions.
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…
Study on conical singularities in 2D surfaces, deriving Polyakov formulas.
problem Analyzing zeta-regularized determinants in surfaces with conical singularities.
method Demonstrated variational and integrated Polyakov formulas for conical singularities, circular sectors, and cones.
result Explicit formulas for the determinant of conical sectors and cones derived.
The paper extends the Cheeger-Müller theorem to spaces with conical singularities.
problem Analyzing the L2-analytic torsion and intersection torsion on spaces with conical singularities. method Developed a combinatorial cellular theory and spectral theory for Hodge-Laplace operator on spaces with conical singularities.
result The L2-analytic torsion coincides with the Ray-Singer intersection torsion under certain conditions. This is the first in a series of five papers math.DG/0211295, math.DG/0302355, math.DG/0302356, math.DG/0303272 studying 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 singularit…
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 survey what is known about singularities of special Lagrangian submanifolds (SL m-folds) in (almost) Calabi-Yau manifolds. The bulk of the paper summarizes the author's five papers math.DG/0211294, math.DG/0211295, math.DG/0302355, math.DG/0302356, math.DG/0303272 on SL m-folds X with isolated conical singularities.…
The paper discusses polynomial convergence to conical Kähler-Einstein metrics.
problem Understanding the convergence of Kähler-Einstein metrics to conical structures.
method Two-step degeneration theory and algebraic singularity analysis.
result Singular Kähler-Einstein metrics are conical if curvature grows quadratically near a point.
Explains mapping properties of elliptic operators in conical spaces.
problem Understanding mapping properties of geometric elliptic operators in conical spaces.
method Develops an approach based on B.-W. Schulze's work.
result Illustrates versatility of results in Geometric Analysis.
Polyhomogeneous expansions for Calabi-Yau metrics near singularities.
problem Analyzing metrics near conical singularities of Calabi-Yau conifolds.
method Weighted Melrose-type blow-ups, gluing, and solving complex Monge-Ampère equations.
result Polyhomogeneous expansions of smooth Calabi-Yau metrics on resolutions and smoothings.
We prove two gluing theorems for special Lagrangian (SL) conifolds in complex space C^m. Conifolds are a key ingredient in the compactification problem for moduli spaces of compact SLs in Calabi-Yau manifolds. In particular, our theorems yield the first examples of smooth SL conifolds with 3 or more planar ends and the…
Paper identifies conditions for singularity removal in equations with conical singularities.
problem Removability of singularities in equations with conical singularities.
method Identification of the Pohozaev constant to measure failure of Pohozaev identity.
result Vanishing Pohozaev constant is necessary and sufficient for singularity removal.
We define a very general "parametric connect sum" construction which can be used to eliminate isolated conical singularities of Riemannian manifolds. We then show that various important analytic and elliptic estimates, formulated in terms of weighted Sobolev spaces, can be obtained independently of the parameters used …
We exhibit infinitely many, explicit special Lagrangian isolated singularities that admit no asymptotically conical special Lagrangian smoothings. The existence/ nonexistence of such smoothings is an important component of the current efforts to understand which singular special Lagrangians arise as limits of smooth sp…
This paper proves Liouville theorems for conformally invariant fully nonlinear equations.
problem Positive entire solutions of certain fully nonlinear equations are unique.
method Derives necessary and sufficient conditions for Liouville-type theorems.
result Enhanced understanding of solutions near isolated singularities.
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…
Motivated by recent interest in the spectrum of the Laplacian of incomplete surfaces with isolated conical singularities, we consider more general incomplete m-dimensional manifolds with singularities on sets of codimension at least 2. With certain restrictions on the metric, we establish that the spectrum is discrete …
In this note we introduce the notion of a smooth structure on a conical pseudomanifold M in terms of C∞-rings of smooth functions on M. For a finitely generated smooth structure C∞(M) we introduce the notion of the Nash tangent bundle, the Zariski tangent bundle, the tangent bundle of M, and the …
Let X be a complex projective variety with only canonical singularities and with trivial canonical bundle. Let L be an ample line bundle on X. Assume that the pair (X,L) is the flat limit of a family of smooth polarized Calabi-Yau manifolds. Assume that for each singular point x∈X there exist a Kahler-Ein…
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.
This study addresses transitions in conically singular associative submanifolds and their desingularizations.
problem Counting closed associative submanifolds of G2-manifolds and understanding transitions arising from degenerations. method Analysis of moduli spaces, transversality results, and desingularization techniques for conically singular associative submanifolds.
result For generic co-closed G2-structures, there are no CS associative submanifolds with stability-index greater than 0 or 1. New conical metrics found on toric varieties with convex cones.
problem Finding conical metrics on toric affine varieties.
method Existence result for inhomogeneous Monge-Ampere equation, transversal a priori estimates.
result Existence of conical Ricci flat Kahler metrics on Q-Gorenstein affine toric varieties.