Study estimates self-shrinker index with conical ends, proving index bound.
problem Estimating the index of self-shrinkers with asymptotically conical ends.
method Constructing Gaussian Harmonic forms and extending index estimates.
result Proves Morse index of self-shrinkers is at least (2g+r-1)/3.
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.
Study sharp asymptotic estimates for conical ends using frequency functions.
problem Proving sharp asymptotic estimates for almost eigenfunctions of drift Laplacians on conical ends.
method Used a weighted variant of Almgren's frequency functions.
result Obtained a purely elliptic proof of uniqueness of self-shrinkers and self-expanders of the mean curvature flow.
Mean curvature flow with a conical end becomes stable.
problem Stability of mean curvature flow with a conical end.
method Analysis of asymptotic behavior and entropy.
result Expanding solution is asymptotically stable.
Estimates ends of Ricci shrinkers, focusing on smooth and singular cases.
problem Understanding the structure of ends in Ricci shrinkers, especially singular ones.
method Analyzes general and asymptotically conical ends, applies to weak convergence.
result No new conical end can form in the limit of sequences of Ricci shrinkers.
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…
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.
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.
Existence proof of noncompact self-shrinkers with arbitrary genus.
problem Existence of noncompact self-shrinkers with arbitrary genus.
method Employing min-max techniques to rigorously prove existence.
result Confirmation of one asymptotically conical end for large genus self-shrinkers.
Differentiates conic optimization problems with unique solutions.
problem Conic optimization problems with unique solutions.
method Differentiability of the solution map under regular conditions.
result Derivative of the solution map is an abstract linear operator.
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.
We consider an inverse problem associated with some 2-dimensional non-compact surfaces with conical singularities, cusps and regular ends. Our motivating example is a Riemann surface M=Γ\H2 associated with a Fuchsian group of the 1st kind Γ containing parabolic elements. M is t…
The study creates Kähler-Einstein metrics with conical singularities.
problem Producing Kähler-Einstein metrics with specific singularities.
method Using the Calabi ansatz for metrics with conical singularities.
result Created regular Calabi-Yau cones and Kähler-Einstein metrics with negative Ricci.
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).
The study computes indicial roots and metric convergence orders for Ricci-flat conifolds.
problem Understanding the convergence behavior of Ricci-flat conifolds.
method Analyzing the Lichnerowicz Laplacian and tensor fields on cones, computing indicial roots and metric convergence orders.
result Lower bounds for metric convergence orders on Ricci-flat conifolds.
We prove functional identities for conic webs on del Pezzo surfaces.
problem Functional identities for conic webs on del Pezzo surfaces.
method Uniform approach based on Weyl group actions and classical results.
result Existence of hyperlogarithmic functional identities of weight 7-d.
Paper proves finite Morse index for certain self-shrinkers.
problem Finite Morse index of self-shrinkers.
method Sufficient condition for finite Morse index of complete properly self-shrinkers.
result Proves finite Morse index for self-shrinkers with finite asymptotically conical or cylindrical ends.
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.
The paper proves the existence of maxfaces with multiple swallowtails and planar ends.
problem Existence of maxfaces with specific geometric properties.
method Analyzes 1-parameter infinite genus families of maxfaces with swallowtails and planar ends.
result Existence of maxfaces with infinitely many swallowtails and planar ends.
The study proves Strichartz and spectral projection theorems on specific types of curved surfaces.
problem Proving Strichartz and spectral projection theorems on curved surfaces.
method Using large negative curvature neighborhoods, the study proves theorems on asymptotically conic and Euclidean ends surfaces.
result The study proves theorems without loss of interval on specific types of curved surfaces.
Using methods of A. Grigor'yan and L. Saloff-Coste we prove that on a manifold with a conical end the heat kernel has a Gaussian bound. This result is applied to asymptotically conical Kähler manifolds. It is a result of the author and R. Goto that a crepant resolution of a Ricci-flat Kähler cone admits a Ricci-flat Kä…
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.
Isometries of cones embed in those of asymptotic shrinking Ricci solitons.
problem Characterizing isometries of asymptotically conical shrinking Ricci solitons.
method Analyzing the embedding of isometries from the cone's cross-section into the entire shrinker.
result Isometries of the cone's cross-section embed in the entire shrinker.
A new capital adequacy test is proposed based on value-at-risk.
problem Regulator's need for a capital adequacy test that doesn't depend on firms' surplus or currency.
method Proving that the only surplus-invariant, law-invariant, and conic acceptance set is the set of positions with negative value-at-risk.
result The value-at-risk test is the only possible capital adequacy test under specified conditions.
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 on finite energy SU(2) monopoles on AC 3-manifolds, proving integrality of charge and curvature decay.
problem Understanding the asymptotic behavior of finite energy SU(2) monopoles on AC 3-manifolds.
method Analysis of critical points of the SU(2) Yang--Mills--Higgs energy on asymptotically conical 3-manifolds.
result Proves integrality of the monopole number and quadratic decay of curvature, among other findings.
Researchers create a parametrix for resolvents on manifolds with ends.
problem Essential self-adjointness of elliptic symmetric differential operators on manifolds with ends.
method Introduced semiclasical pseudodifferential operators compatible with the end structure.
result Essential self-adjointness of elliptic symmetric differential operators proved.
Constructs scalar-flat Kähler metrics on toric symplectic manifolds.
problem Creating scalar-flat Kähler metrics on toric symplectic manifolds.
method Explicit construction and alternative construction with conical singularity.
result Explicit construction of scalar-flat Kähler metrics on toric symplectic manifolds.
Researchers prove existence of self-shrinkers for a specific curvature flow.
problem Existence of self-shrinkers for a degree-one curvature flow with a conical end.
method Analyzing a symmetric homogeneous function and a rotationally symmetric cone to prove the existence of self-shrinkers.
result Proved the existence of a self-shrinker asymptotic to a given conical end.
New theorem shows noncompact self shrinkers are unknotted.
problem Understanding the structure of noncompact self shrinkers.
method Used mean curvature flow to extend theorem to noncompact cases.
result Noncompact self shrinkers without knotted components.
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…
Let C⊂Rn+1 be a regular cone with vertex at the origin. In this paper, we show the uniqueness for smooth properly embedded self-shrinking ends in Rn+1 that are asymptotic to C. As an application, we prove that not every regular cone with vertex at the origin has a smooth complete pro…
Constructs geodesic lines in curved surfaces using min-max methods.
problem Constructing geodesic lines in curved surfaces where minimization schemes fail.
method Employing min-max methods to construct uncountably many geometrically distinct geodesic lines.
result Proves the existence of uncountably many properly embedded geodesic lines with Morse index ≤ 1.
We show that if two gradient Ricci solitons are asymptotic along some end of each to the same regular cone, then the soliton metrics must be isometric on some neighborhoods of infinity of these ends. Our theorem imposes no restrictions on the behavior of the metrics off of the ends in question and in particular does no…
Uniqueness proven for specific types of geometric structures.
problem Proving uniqueness of asymptotically conical gradient shrinking solitons.
method Extends Kotschwar and Wang's argument for uniqueness of AC gradient shrinking Ricci solitons.
result G_2-structures are equivalent if asymptotically conical and asymptotic to the same closed G_2-cone.
The flow contracts cone divisors on Kähler surfaces to points.
problem Analyzing the conical Kähler-Ricci flow on Hirzebruch surfaces.
method Using the momentum construction of Calabi, the conical Kähler-Ricci flow is studied on Hirzebruch surfaces.
result The flow either converges to a sphere or a single point, or contracts the cone divisor to a single point.
For a torsion free Kleinian group Γ without parabolics, we consider the decomposition of the limit set L(Γ) into conical and ending limit sets and compare the Patterson-Sullivan measure with the harmonic measure on L(Γ) when L(Γ)=S∞2.
In many problems of PDE involving the Laplace-Beltrami operator on manifolds with ends, it is often useful to introduce radial or geodesic normal coordinates near infinity. In this paper, we prove the existence of such coordinates for a general class of manifolds with ends, which contains asymptotically conical and hyp…
We analyze the resolvent R(k)=(P+k2)−1 of Schrödinger operators P=Δ+V with short range potential V on asymptotically conic manifolds (M,g) (this setting includes asymptotically Euclidean manifolds) near k=0. We make the assumption that the dimension is greater or equal to 3 and that P has no L2 null …
We discuss the deformation theory of 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. This category allows for the simultaneous presence of conical singularities and of non-compact, asymptotic…
In this article, we examine complete, mean-convex self-expanders for the mean curvature flow whose ends have decaying principal curvatures. We prove a Liouville-type theorem associated to this class of self-expanders. As an application, we show that mean-convex self-expanders which are asymptotic to O(n)-invariant co…
The Hurwitz space is the moduli space of pairs (X,f) where X is a compact Riemann surface and f is a meromorphic function on X. We study the Laplace operator Δ∣df∣2 of the flat singular Riemannian manifold (X,∣df∣2). We define a regularized determinant for Δ∣df∣2 and study it as a functional on t…
We generalise the classical Chern-Gauss-Bonnet formula to a class of 4-dimensional manifolds with finitely many conformally flat ends and singular points. This extends results of Chang-Qing-Yang in the smooth case. Under the assumptions of finite total Q curvature and positive scalar curvature at the ends and at the si…
Paper develops compact formulations for optimization problems with rank-one convex functions and indicator variables.
problem Optimization problems involving rank-one convex functions with support constraints.
method Perspective reformulation techniques to exploit conic structure and establish convex hull results.
result Systematic perspective formulations for convex hull descriptions of sets with nonlinear separable or non-separable objective functions and combinatorial constraints.
The study examines asymptotic properties of G2-monopoles on nonparabolic G2-manifolds.
problem Investigating the asymptotic behavior of G2-monopoles on nonparabolic G2-manifolds.
method Proof of finite mass for monopoles with bounded curvature on nonparabolic G2-manifolds; sharp decay estimates and convergence to pseudo-Hermitian--Yang--Mills connection on asymptotically conical G2-manifolds.
result Sharp decay estimates and convergence to pseudo-Hermitian--Yang--Mills connection on asymptotically conical G2-manifolds.
This is the extended version of the paper "Special Lagrangian conifolds, I: Moduli spaces", which discusses the deformation theory of 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. The conif…
CMC-1 trinoids (i.e. constant mean curvature one immersed surface with three regular embedded ends) in hyperbolic 3-space H^3 are irreducible generically, and the irreducible ones have been classified. However, the reducible case has not yet been fully treated, so in this paper we give an explicit description of CMC-1 …
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.