Study finds existence and non-existence of large stable CMC spheres in asymptotically flat 3-manifolds.
problem Existence and non-existence of large stable CMC spheres in asymptotically flat 3-manifolds.
method Extends Lyapunov-Schmidt analysis to 'far-off-center' regime and general Schwarzschild asymptotics.
result Sharp existence and non-existence results for large stable CMC spheres.
Riemannian manifolds can be sphere at infinity of certain solitons.
problem Understanding the structure of asymptotically conical expanding Ricci solitons.
method Formal expansions to show any compact manifold can be a sphere at infinity of a soliton.
result Any compact Riemannian manifold is the sphere at infinity of an asymptotically conical gradient expanding Ricci soliton.
Existence proved for static vacuum extensions near Schwarzschild spheres.
problem Proving existence of static vacuum extensions near Schwarzschild spheres.
method Existence and local uniqueness of static vacuum extensions for Bartnik data on a sphere near a Schwarzschild sphere.
result Existence of static vacuum extensions near Schwarzschild spheres.
Study Goeritz groups of link decompositions, focusing on their asymptotic behavior.
problem Understanding the asymptotic behavior of Goeritz groups for link decompositions.
method Defined Goeritz groups for link decompositions, analyzed their properties, and discussed their asymptotic behavior.
result Discussed the asymptotic behavior of minimal pseudo-Anosov entropies and related it to Goeritz groups of Heegaard splittings.
For an integral homology 3-sphere embedded asymptotically flatly in an Euclidean space, we find a natural framing extending the standard trivialization on the asymptotically flat part.
In this note we study constant mean curvature surfaces in asymptotically flat 3-manifolds. We prove that, in an asymptotically flat 3-manifold with positive mass, stable spheres of given constant mean curvature outside a fixed compact subset are unique. Therefore we are able to conclude that there is a unique foliation…
The paper proves the existence of stable spheres in asymptotically flat 3-manifolds.
problem Existence of stable spheres in asymptotically flat 3-manifolds.
method Lyapunov-Schmidt reduction
result Existence of an asymptotic foliation of (M,g) by stable constant mean curvature spheres. Flat tori in 3-sphere have π extrinsic diameter under specific conditions.
problem Determining the extrinsic diameter of immersed flat tori in the 3-sphere.
method Analyzing asymptotic curves and Hopf fibration projections.
result The extrinsic diameter is π under certain topological conditions.
Researchers found unique large stable spheres in specific 3D space.
problem Characterizing large stable spheres in specific types of 3D space.
method Unconditional characterization of spheres using Riemannian geometry.
result Global uniqueness of large stable spheres in asymptotically flat Riemannian three-manifolds.
In this paper I study the constant mean curvature surface in asymptotically flat 3-manifolds with general asymptotics. Under some weak condition, I prove that outside some compact set in the asymptotically flat 3-manifold with positive mass, the foliation of stable spheres of constant mean curvature is unique.
In hyperbolic 3-space, H-planes can be formed around curves with smooth points.
problem Forming H-planes in hyperbolic 3-space.
method Analyzing Jordan curves in the asymptotic sphere.
result Embedded H-planes can be created for any H in [0,1) around curves with smooth points.
Study on area-constrained Willmore spheres in asymptotic Schwarzschild manifolds.
problem Existence of area-constrained Willmore spheres with non-negative Hawking mass and inner radius.
method Analysis of scalar curvature and asymptotic properties of 3-manifolds.
result No large area-constrained Willmore spheres exist under certain conditions.
The article provides formulas for homological blocks of Seifert fibered homology 3-spheres.
problem Calculating Witten-Reshetikhin-Turaev invariants for Seifert fibered homology 3-spheres.
method Explicit modular transformation formulas of homological blocks.
result New proof of Witten asymptotic conjecture for Seifert fibered homology 3-spheres.
Motivated by the foliation by stable spheres with constant mean curvature constructed by Huisken-Yau, Metzger proved that every initial data set can be foliated by spheres with constant expansion (CE) if the manifold is asymptotically equal to the standard [t=0]-timeslice of the Schwarzschild solution. In this paper, w…
The paper constructs unstable and outlying CMC spheres in specific manifolds.
problem Stability of CMC spheres in certain manifolds.
method Non-linear ODE method to construct CMC surfaces in Riemannian manifolds with symmetry.
result Existence of unstable CMC spheres, indicating the stability condition cannot be removed.
Adapting Israel's proof of static black hole uniqueness, we show that the Schwarzschild spacetime is the only static vacuum asymptotically flat spacetime that possesses a suitably defined photon sphere.
Study finds many nonplanar minimal spheres in elongated ellipsoids.
problem Existence of nonplanar minimal spheres in elongated ellipsoids.
method Global bifurcation techniques to establish existence and quantify number.
result Arbitrarily many nonplanar minimal spheres exist in elongated ellipsoids.
New methods prove floating areas on spheres and hyperbolic spaces.
problem Existence of floating areas on curved spaces.
method Weighted floating bodies and polytopal approximation.
result New asymptotic approximation results on curved spaces.
Study ancient Ricci flow solutions, proving unique asymptotic behavior.
problem Understanding unique asymptotics of compact ancient solutions to 3D Ricci flow.
method Analyzing noncollapsed compact ancient solutions, proving asymptotic behavior.
result Proves unique asymptotic behavior for compact ancient solutions.
In this paper, we will show that the limit of some quasilocal mass integrals of the coordinate spheres in an asymptotically hyperbolic (AH) manifold is the mass integral of the AH manifold. This is the analogue of the well known result that the limit of the Brown-York mass of coordinate spheres is the ADM mass in an as…
Uniqueness proven for photon spheres in higher-dimensional spacetimes.
problem Proving uniqueness of photon spheres in higher-dimensional electrovacuum spacetimes.
method Combining ideas from earlier works with newer techniques.
result Proven uniqueness of subextremal Reissner-Nordström manifolds with positive mass.
The paper studies translation lengths on sphere complexes and related cones.
problem Understanding the translation lengths of monodromies in fibered manifolds.
method Defined the generalized fibered cone and related cones, and proved their properties.
result Proved the generalized fibered cone is a rational slice of Fried's cone, providing bounds for asymptotic translation lengths.
The paper calculates the Kobayashi-Royden metric on a punctured sphere and finds rational coefficients.
problem Calculating the Kobayashi-Royden metric on a punctured sphere.
method Explicit formula and asymptotic expansion using exponential Bell polynomials.
result The coefficients in the asymptotic expansion of the Kobayashi-Royden metric on the punctured sphere are rational numbers.
Derives precise asymptotic expansion of Kähler-Einstein metric on punctured sphere.
problem Calculating the complete Kähler-Einstein metric on a punctured Riemann sphere.
method Uses Schwarzian derivative and modular forms to determine coefficients.
result Explicitly determines coefficients for 3 to 12 omitting points.
The study classifies singularities in discrete improper affine spheres.
problem Classifying singularities in discrete improper affine spheres.
method Analysis of discrete improper affine spheres based on asymptotic nets, distinguishing singular edges and vertices.
result First step in classifying singularities of discrete nets.
The paper proves partial rigidity of Hawking mass for stable CMC spheres in specific manifolds.
problem Rigidity of Hawking mass for stable CMC spheres in asymptotic flat and hyperbolic manifolds.
method Mean-field equation and monotonicity of Hawking mass, combined with Shi's rigidity results.
result If the Hawking mass of a nearly round stable CMC surface vanishes, the surface must be a standard sphere in R^3 and the interior is flat.
Sharp curvature estimates for mean curvature flow in spheres.
problem Understanding the behavior of surfaces evolving under mean curvature flow in spheres.
method Proving asymptotically sharp curvature pinching estimates and using them to derive derivative and convexity estimates.
result Partial classification of singularity models and new rigidity results for ancient solutions.
New G2-instantons found on 3-sphere's spinor bundle.
problem Classifying and constructing G2-instantons. method Used SU(2)3-symmetries and asymptotically conical, co-homogeneity one G2-metric. result Found new examples of G2-instantons with obstructed deformations. Researchers prove uniqueness of photon spheres in electro-vacuum spacetimes.
problem Proving the uniqueness of photon spheres in static electro-vacuum spacetimes.
method Adapting Masood-ul-Alam's proof for static vacuum black hole uniqueness to electro-vacuum spacetimes.
result Ruling out the existence of multiple electrically charged bodies and sub-extremal black holes.
Homological blocks match Witten-Reshetikhin-Turaev invariants for Seifert fibered 3-spheres.
problem Matching homological blocks with WRT invariants for specific 3-manifolds.
method Developed an asymptotic formula and vanishing result of coefficients.
result Radial limits of homological blocks match Witten-Reshetikhin-Turaev invariants.
We show that the spheres in Hilbert geometry have the same volume growth entropy as those in the Lobachevsky space. We give the asymptotic estimates for the ratio of the volume of metric ball to the area of the metric sphere in Hilbert geometry. Derived estimates agree with the well-known fact in the Lobachevsky space
Proves gap estimate for convex domains on sphere, extending previous work.
problem Estimating the gap between the first two eigenvalues of Laplacian on convex domains of sphere.
method Extends previous results to 2D, uses asymptotic expansion for eigenvalues.
result Establishes gap estimate of 3π²/D² for convex domains on sphere, generalizing to all dimensions.
It is well-known that the Ricci flow of a closed 3-manifold containing an essential minimal 2-sphere will fail to exist after a finite time. Conversely, the Ricci flow of a complete, rotationally symmetric, asymptotically flat manifold containing no minimal spheres is immortal. We discuss an intermediate case, that of …
For any asymptotically conical self-shrinker with entropy less than or equal to that of a cylinder we show that the link of the asymptotic cone must separate the unit sphere into exactly two connected components, both diffeomorphic to the self-shrinker. Combining this with recent work of Brendle, we conclude that the r…
In this paper, the existence and uniqueness of foliations by constant mean curvature spheres on asymptotically flat manifolds of nonzero ADM mass in all dimensions were established. (A similar result in the case of positive mass was obtained independently by G. Huisken and S. T. Yau, see the introduction of this paper …
We study some asymptotic properties of the sequences of symplectic Lefschetz pencils constructed by Donaldson. In particular we prove that the vanishing spheres of these pencils are, for large degree, conjugated under the action of the symplectomorphism group of the fiber. This implies the non-existence of homologicall…
We establish in this paper an upper bound on the second eigenvalue of n-dimensional spheres in the conformal class of the round sphere. This upper bound holds in all dimensions and is asymptotically sharp as the dimension increases.
We show that the constant mean curvature hypersurfaces in the hyperbolic n-space spanning the boundary of a star shaped C^{1,1} domain in the asymptotic sphere give a foliation of the hyperbolic n-space. We also show that if C is a closed codimension-1 C^{2,a} submanifold in the asymptotic sphere bounding a unique cons…
The paper explores harmonic and asymptotically harmonic Finsler manifolds and their properties.
problem Characterizing and understanding harmonic Finsler manifolds.
method Investigation of various types of harmonic Finsler manifolds, characterizations via mean curvature and Laplacian, and construction techniques.
result Certain harmonic Finsler manifolds are of Einstein type and examples of non-Riemannian Finsler harmonic manifolds are provided.
The paper examines ancient solutions of mean curvature flow in space forms with curvature pinching conditions.
problem Investigating rigidity of ancient solutions of mean curvature flow in space forms.
method Sharp asymptotic pointwise curvature pinching conditions and asymptotic integral curvature pinching conditions.
result Ancient solutions in a sphere are either a shrinking spherical cap or a totally geodesic sphere, and in a hyperbolic space, they are a family of shrinking spheres.
We create a flat end foliation by critical spheres solving a Laplace-Beltrami problem.
problem Foliation of an asymptotically flat end by critical hypersurfaces.
method Constructing hypersurfaces as critical points of a functional, solving an over-determined boundary value problem.
result Solutions to the Laplace-Beltrami operator over a foliation of critical spheres.
Simple proof for sphere mass calculation.
problem Computing the ADM mass of static sphere extensions.
method Uses mass formula for static asymptotically flat manifolds.
result Validated mass formula for small spheres.
Study on black holes and photon surfaces in 4D spacetimes, proving uniqueness theorems.
problem Uniqueness of black hole and photon surfaces in 4D spacetimes.
method Potential theory approach, self-contained proofs for known and new cases.
result Proves new results for connected photon spheres and photon surfaces in the extremal case, and super-extremal case.
This article explains how to construct immersed Lagrangian submanifolds in C^2 that are asymptotic at large distance from the origin to a given braid in the 3-sphere. The self-intersections of the Lagrangians are related to the crossings of the braid. These Lagrangians are then used to construct immersed Lagrangians in…
Study examines Hawking energy along null asymptotically flat hypersurfaces.
problem Analyzing the asymptotic behavior of Hawking energy on null hypersurfaces.
method Obtained the limit of Hawking energy for a broad class of foliations on null hypersurfaces with weak asymptotic flatness.
result The Hawking energy converges to the Bondi energy under certain conditions.
We give a simple topological argument to show that the number of solutions of the asymptotic Plateau problem in hyperbolic space is generically unique. In particular, we show that the space of codimension-1 closed submanifolds of sphere at infinity, which bounds a unique absolutely area minimizing hypersurface in hyper…
The paper proves compactness of metrics with isolated singularities on a sphere.
problem The moduli space of metrics with constant Q-curvature and positive scalar curvature on a sphere with punctures.
method Defined asymptotic necksize and radial Pohozaev invariant, proved sequential compactness.
result Any bounded set in the moduli space is sequentially compact.
Study on quantum state entanglement using Kaehler manifolds.
problem Quantum state entanglement on Kaehler manifolds.
method Semiclassical asymptotics and pure states on spheres.
result Entropy analysis of quantum states on spheres.