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.
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.
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.
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.
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.
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.
Study heat trace expansion on manifolds with conic points.
problem Analyzing heat diffusion on manifolds with sharp corners.
method Using the Singular Asymptotics Lemma to derive an expansion.
result Detailed asymptotic expansion reveals geometric insights.
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 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. 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.
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 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.
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 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.
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.
The Yamabe flow preserves conical singularities under certain conditions.
problem Preserving conical singularities under the Yamabe flow.
method Using maximal Lq-regularity theory for conically degenerate operators. result Asymptotic expansion of the evolving metric near conical tips.
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.
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.
Study proves Hölder continuity for solutions to parabolic equations with conic singularities.
problem Proving Hölder continuity of solutions to parabolic equations with conic singularities.
method Using the results from previous works \cite{LSU} and \cite{FS}, the Hölder continuity is proven for linear parabolic equations of mixed type.
result Hölder continuity of solutions to parabolic equations with conic singularities is established.
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…
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…
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)).
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.
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.
Third in a series, this paper constructs non-trivial Cayley fibrations with conical singularities.
problem Constructing non-trivial Cayley fibrations with conical singularities.
method Using gluing methods and stability results for weak and usual fibrations.
result Construction of examples of Cayley fibrations on twisted connected sum G2 manifolds. In this note we discuss the problem of resolving conically singular cscK varieties to construct smooth cscK manifolds, showing a glueing result for (some) crepant resolutions of cscK varieties with discrete automorphism groups.
Geodesics near singularities either hit or wind around, with winding number dependent on singularity type.
problem Understanding geodesic behavior near singularities in Riemannian manifolds.
method Analytical study of geodesics on Riemannian manifolds near singularities.
result The winding number of geodesics around a singularity depends on the singularity type and approaches infinity as the singularity becomes cuspidal.
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.
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 conical Kaehler-Einstein metrics' curvature, proving regularity and geometric conditions.
problem Curvature of conical Kaehler-Einstein metrics on Kaehler manifolds.
method Proved regularity for Monge-Ampère equations on Kaehler manifolds with smooth divisors, applied to conical metrics.
result Geometric condition for boundedness of conical singularities.
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…
Compact Ricci-flat manifolds with conical singularities are found.
problem Finding compact Ricci-flat manifolds with conical singularities.
method Analyzing flat limits of smooth polarized Calabi-Yau manifolds and using the Calabi ansatz.
result The existence of compact Ricci-flat manifolds with non-orbifold isolated conical singularities.
The study finds metrics with constant scalar curvature on complex manifolds.
problem Finding Kähler metrics with constant scalar curvature on complex manifolds.
method Analyzing the Lichnerowicz operator and proving existence results for metrics with conic singularities.
result Existence of constant scalar curvature Kähler metrics with conic singularities.
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.
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.
We study a behavior of the conformal Laplacian operator Łg on a manifold with \emph{tame conical singularities}: when each singularity is given as a cone over a product of the standard spheres. We study the spectral properties of the operator Łg on such manifolds. We describe the asymptotic of a general solution …
Study on deformations of singular submanifolds in complex geometry.
problem Deformation theory of conically singular Cayley submanifolds.
method Proved expected dimension of moduli space and compared complex and Cayley deformations.
result Moduli space is smooth for 2D complex submanifolds of Calabi-Yau 4-folds.
Researchers prove existence of Kähler-Einstein metrics with conic singularities on Fano manifolds.
problem Existence of Kähler-Einstein metrics with conic singularities on Fano manifolds.
method Analyzing automorphisms and limits at various scales to prove the existence of metrics.
result Existence of Kähler-Einstein metrics with conic singularities for β>β∗ close to β∗. 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…
Using an approach based on the heat kernel we prove an Atiyah-Bott-Lefschetz theorem for the L2−Lefschetz numbers associated to an elliptic complex of cone differential operators over a compact manifold with conical singularities. We then apply our results to the case of the de Rham complex.
We study a particular class of open manifolds. In the category of Riemannian manifolds these are complete manifolds with cylindrical ends. We give a natural setting for the conformal geometry on such manifolds including an appropriate notion of the cylindrical Yamabe constant/invariant. This leads to a corresponding ve…
Compactifies Riemann surface metrics with conical singularities.
problem Constant curvature metrics on Riemann surfaces with conical singularities.
method Compactified spaces and study local deformation of metrics.
result Sharp regularity theorem for coalescing conic points.
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.
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 describes ALF instantons with conical singularities.
problem Understanding ALF instantons with conical singularities.
method Applied techniques from previous work.
result Only 4D subfamilies can be smoothly compactified with conical singularities.
Study Schauder estimates for conical singularities using Kähler metrics.
problem Derive interior Schauder estimates for equations with conical singularities.
method Use linear elliptic and parabolic equations with a Kähler metric of conical singularities.
result Prove short-time existence of conical Kähler-Ricci flow.