Study on sphere immersions and their stability indices.
problem Analyzing the stability of sphere immersions.
method Calculation of Morse indices and stability indices for specific sphere immersions.
result Bounds on stability index of associative cone in R7. Proves stability of convex spheres with similar geodesic lengths.
problem Stability of convex spheres with specific geodesic properties.
method Proves C^0 Cheeger-Gromov closeness to the round sphere.
result Strictly convex 2-spheres are close to the round sphere.
The paper classifies decompositions of 3-sphere and lens spaces with handlebodies.
problem Classifying decompositions of 3-manifolds with handlebodies.
method Studied decompositions of 3-sphere and lens spaces with three handlebodies, using stabilizations.
result Determined whether decompositions are stabilized.
New bounds and examples for sphere unknotting numbers.
problem Comparing unknotting numbers for 2-spheres in 4-space.
method Algebraic and geometric techniques.
result Stabilization number is bounded above by one more than Casson-Whitney number.
We prove effective uniformization for nearly round 2-spheres and investigate their stability.
problem Proving effective uniformization for nearly round 2-spheres and their stability.
method Utilizing an identity related to the third-order differential of the conformal factor, and an isometric embedding of a round sphere into Euclidean space using an orthogonal basis of the first eigenspace of the Laplacian operator.
result We provide a simplified proof of effective uniformization and its stability.
Proves conjecture about sphere widths under rotational symmetry.
problem Width stability of rotationally symmetric metrics.
method Proof of conjecture and extensions to higher dimensions.
result Stability of min-max width under rotational symmetry.
Minimal submanifolds are stable in certain conformal spheres.
problem Stability of minimal submanifolds in conformal spheres.
method Analyzing n-dimensional Riemannian spheres with specific curvature conditions. result Closed stable minimal submanifolds are not found in δ-pinched conformal spheres. It is known that the totally umbilical hypersurfaces in the (n+1)-dimensional spheres are characterized as the only hypersurfaces with weak stability index 0. That is, a compact hypersurface with constant mean curvature, cmc, in S^{n+1}, different from an Euclidean sphere, must have stability index greater than or equa…
Study on stability of mean curvature flow in hyperbolic space.
problem Stability of volume preserving mean curvature flow in hyperbolic space.
method Analysis of initial conditions and flow behavior in hyperbolic space.
result The flow converges exponentially to an umbilical sphere under certain conditions.
The paper studies stability of domains for the first eigenvalue on Riemannian manifolds.
problem Stability of extremal domains for the first eigenvalue of the Laplacian operator.
method Second variation of the first Dirichlet eigenvalue, stability criterion, classification of stable domains.
result Classification of stable extremal domains in spheres and topological bounds for general compact surfaces.
The round sphere is stable among spin manifolds with a specific scalar curvature bound.
problem Stability of the round sphere among spin manifolds with scalar curvature below a certain threshold.
method Showed that if the scalar curvature is bounded from below by n(n−1)−ε, the manifold is C0-close to a finite number of spheres outside a small bad set. result The spherical stability problem is completely solved.
Any 2-bridge knot in the 3-sphere has a bridge sphere from which any other bridge surface can be obtained by stabilization, meridional stabilization, perturbation and proper isotopy.
Hyperbolic groups act on spheres, and nearby actions are semi-conjugate.
problem Stability of group actions on spheres.
method Topological stability in dynamical sense.
result Nearby actions are semi-conjugate to the standard boundary action.
Investigates harmonic self-maps' stability on cohomogeneity one manifolds.
problem Stability of harmonic self-maps on cohomogeneity one manifolds.
method Systematic study of Jacobi equation for harmonic self-maps.
result Explicit solutions for specific cases show identity map's stability.
In this short note we extend an estimate due to J. Simons on the first stability eigenvalue of minimal hypersurfaces in spheres to the singular setting. Specifically, we show that any singular minimal hypersurface in Sn+1, which is not totally geodesic and satisfies the α-structural hypothesis, has first stability…
In this paper we discuss the stability of geodesic spheres in Sn+1 under constrained curvature flows. We prove that under some standard assumptions on the speed and weight functions, the spheres are stable under perturbations that preserve a volume type quantity. This extends results by Escher and Simonet…
New examples show scalar curvature's role in sphere stability.
problem Characterizing sphere stability through scalar curvature.
method Improving Gromov-Lawson tunnel construction and sewing techniques.
result Constructs sequences demonstrating sphere stability under scalar curvature.
Study on stability of harmonic maps with sub-Riemannian geometry.
problem Stability of exponentially subelliptic harmonic maps.
method Derived first and second variation formulas, applied to prove stability under certain conditions.
result Exponentially subelliptic harmonic maps are stable if the target manifold has nonpositive curvature.
Researchers prove a stability result for a 3-sphere inequality, extending previous work.
problem Quantitative stability of nonlinear Yamabe-type inequalities on the 3-sphere.
method Proved a two-term refinement of the Schur lemma inequality in the conformal class of the 3-sphere.
result Deduced quantitative stability of an entire family of nonlinear Yamabe-type inequalities.
The paper finds new eigenfunctions for minimal immersions and their stability index.
problem Finding new eigenfunctions for minimal immersions and their stability index.
method Explicitly showed new eigenfunctions for the stability operator.
result The stability index of minimal immersions is at least kℓ+3k+3ℓ+8. Optimally stabilizes Möbius group maps in spheres across dimensions.
problem Ensuring optimal stability of Möbius group maps in spheres of all dimensions.
method Proves stability estimate involving optimal λ and φ transformations.
result Compact sequences with vanishing deficit under Möbius group action.
We prove a homological stability theorem for moduli spaces of high-dimensional, highly connected manifolds, with respect to forming the connected sum with the product of spheres Sp×Sq, for p<q<2p−2. This result is analogous to recent results of S. Galatius and O. Randal-Williams regarding the homo…
The study examines stability and isoperimetry of CMC spheres in hyperbolic and spherical manifolds.
problem Stability and isoperimetry of constant mean curvature spheres in hyperbolic and spherical manifolds.
method Analyzes rotational CMC spheres in HnimesR and SnimesR, proving stability and instability properties. result Rotational CMC spheres in HnimesR are always stable, while those in SnimesR with large mean curvature are stable and those with small mean curvature are unstable. Leon Green obtained remarkable rigidity results for manifolds of positive scalar curvature with large conjugate radius and/or injectivity radius. Using Ck,α convergence techniques, we prove several differentiable stability and sphere theorem versions of these results and apply those also to the study of Einstein m…
Paper proves no stable Yang-Mills fields on spheres.
problem Existence of stable Yang-Mills fields on spheres.
method Analyzes C2 neighborhoods of Euclidean sphere metrics and warped product manifolds. result No nontrivial weakly stable Yang-Mills connections in specified neighborhoods.
The paper studies stability and index of biharmonic hypersurfaces in Riemannian manifolds.
problem Analyzing stability and index of biharmonic hypersurfaces.
method Using a second variation formula for biharmonic hypersurfaces, computing stability index, and proving non-existence of unstable hypersurfaces.
result Proves non-existence of unstable proper biharmonic hypersurfaces in Euclidean space or hyperbolic space.
Geodesic spheres in certain symmetric spaces are quantitatively stable under small perturbations.
problem Stability of geodesic spheres in symmetric spaces under perturbations.
method Quantitative stability analysis using spectral gap of the Laplacian on geodesic spheres.
result Geodesic spheres are uniformly stable with respect to small C1-volume preserving perturbations. The paper studies eigenvalues and stability of hypersurfaces in spheres.
problem Finding eigenvalues and stability of hypersurfaces in spheres.
method Derives equations for mean curvature and uses numerical methods to compute eigenvalues.
result Numerical computation of eigenvalues and stability indices for specific hypersurfaces.
Free surface-links are shown to be ribbon links.
problem Characterizing surface-links as ribbon links.
method Four proofs are provided, including a stabilization approach.
result Every free surface-link is a ribbon surface-link.
We show that the of 2-lobed Delaunay tori are stable as constrained Willmore surfaces in the 3-sphere.
3D spheres with certain properties approach the round sphere.
problem Flexibility of Llarull's Theorem in dimension 3.
method Proof based on spacetime harmonic functions.
result 3D spheres with bounded Cheeger isoperimetric constant and scalar curvatures tending to 6 approach the round sphere.
New theorem shows nearly spherical manifolds can be mapped from spheres.
problem Generalizing Caffarelli's theorem to nearly spherical manifolds.
method Optimal transport map on the sphere, stability result.
result Every nearly spherical manifold can be mapped from a sphere.
Minimal surfaces in S3(2) linked to vector fields on punctured sphere.
problem Connecting minimal surfaces in S3(2) to vector fields on a punctured sphere.
method Established a correspondence between minimal surfaces and area-minimizing vector fields.
result Stability relation for Lawson cylinders in S3(2).
The study of stable and index compact minimal submanifolds in Berger spheres.
problem Stability and index of compact minimal submanifolds in Berger spheres.
method Analyzing stability and index properties of compact minimal submanifolds in Berger spheres.
result Stable compact minimal submanifolds exist in Berger spheres for specific values of τ, and their classification is provided.
We show that if a closed surface in R3 has entropy near to that of the unit two-sphere, then the surface is close to a round two-sphere in the Hausdorff distance.
Study on representations of four-punctured sphere group in hyperbolic spaces.
problem Understanding representations of the four-punctured sphere group.
method Investigation into simple-stable and Bowditch representations in Gromov-hyperbolic spaces.
result Simple-stable representations and Bowditch representations are equivalent.
The homology groups of the automorphism group of a free group are known to stabilize as the number of generators of the free group goes to infinity, and this paper relativizes this result to a family of groups that can be defined in terms of homotopy equivalences of a graph fixing a subgraph. This is needed for the sec…
Define the 1-handle stabilization distance between two surfaces properly embedded in a fixed 4-dimensional manifold to be the minimal number of 1-handle stabilizations necessary for the surfaces to become ambiently isotopic. For every nonnegative integer m we find a pair of 2-knots in the 4-sphere whose stabilization…
We establish various stability results for symplectic surfaces in symplectic 4−manifolds with b+=1. These results are then applied to prove the existence of representatives of Lagrangian ADE-configurations as well as to classify negative symplectic spheres in symplectic 4−manifolds with κ=−∞. This involve…
This is a follow-up of our paper \cite{KS-Kerr1} on the construction of general covariant modulated (GCM) spheres in perturbations of Kerr, which we expect to play a central role in establishing their nonlinear stability. We reformulate the main results of that paper using a canonical definition of ℓ=1 modes on a …
We study the stability of critical maps from (or into) spheres with respect to the symplectic Dirichlet and σ2 energies which are the fourth power terms in Skyrme type sigma-models.
Sharp stability of Möbius group among sphere-valued maps proved in arbitrary dimensions.
problem Proving a sharp quantitative form of Liouville's theorem for sphere-valued maps.
method New arguments and an inequality from Sobolev inequality proof.
result Sharp stability estimate for weakly conformal maps of arbitrary dimensions.
We prove that the boundary of a (not necessarily connected) bounded smooth set with constant nonlocal mean curvature is a sphere. More generally, and in contrast with what happens in the classical case, we show that the Lipschitz constant of the nonlocal mean curvature of such a boundary controls its C2-distance fro…
The paper examines stability of the Sobolev inequality in metric spaces with curvature dimension conditions.
problem Investigating stability of the Sobolev inequality in metric spaces with curvature dimension conditions.
method Assuming almost the same optimal constant, the paper shows that the cumulative distribution of almost extremal functions is close to that of an Aubin-Talenti bubble on the round sphere.
result Quantitative stability with sharp exponent for the Sobolev inequality in various curvature and dimension assumptions.
Here we discuss an example of topologically isotopic but smoothly non-isotopic pair of 2-spheres in a simply connected 4-manifold, which become smoothly isotopic after stabilizing by connected summing with S^2 x S^2.
Exotic submanifolds in 4-manifolds remain exotic after stabilizations.
problem Constructing exotic codimension-1 submanifolds in 4-manifolds.
method Constructing pairs of exotic codimension-1 submanifolds with diffeomorphic complements and showing they remain exotic after stabilizations.
result Exotic submanifolds remain exotic after any number of stabilizations.
Existence and instability of biharmonic maps from balls to spheres.
problem Existence and stability of biharmonic maps between balls and spheres.
method Existence proof and instability analysis using bienergy.
result Existence of two proper biharmonic maps and instability in low dimensions.
Study stability and rigidity of axisymmetric marginally outer trapped surfaces.
problem Stability and rigidity of axisymmetric marginally outer trapped surfaces.
method Refined results from initial data sets with Killing vector fields, using new foliation lemma.
result Conditions for the stability of axisymmetric MOTS and new foliation lemma.