arXiv research

Minimal submanifolds are stable in certain conformal spheres.

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 $S^{n+1}$, which is not totally geodesic and satisfies the α-structural hypothesis, has first stability…

Study on sphere immersions and their stability indices.

The paper finds new eigenfunctions for minimal immersions and their stability index.

Minimal surfaces in S3(2) linked to vector fields on punctured sphere.

New bounds and examples for sphere unknotting numbers.

The study of stable and index compact minimal submanifolds in Berger spheres.

The paper studies stability and instability of minimal submanifolds in complex Einstein spaces.

The paper proves stability of minimal embeddings in spheres and relates it to Yau's conjecture.

The paper studies eigenvalues and stability of hypersurfaces in spheres.

The paper studies stability of domains for the first eigenvalue on Riemannian manifolds.

New examples show scalar curvature's role in sphere stability.

Researchers prove a stability result for a 3-sphere inequality, extending previous work.

Optimizes sharp curvature inequality on spheres, proving near-minimizers are close to standard metric.

In this note, we study submanifold geometry of the Atiyah-Hitchin manifold, the double cover of the $2$-monopole moduli space. When the manifold is naturally identified as the total space of a line bundle over $S^2$, the zero section is a distinguished minimal $2$-sphere of considerable interest. In particular, there h…

The paper studies stability and index of biharmonic hypersurfaces in Riemannian manifolds.

We show that any non-minimal bridge decomposition of a torus knot is stabilized and that $n$-bridge decompositions of a torus knot are unique for any integer $n$. This implies that a knot in a bridge position is a torus knot if and only if there exists a torus containing the knot such that it intersects the bridge sphe…

The paper refines the stability index for a specific minimal hypersurface and verifies Yau's conjecture.

The study finds an infinite number of minimal surfaces in 3D spheres.

New optimal surfaces found in Heisenberg group defy Euclidean sphere optimality.

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…

The paper studies stability and minimizing properties of higher codimensional surfaces in Euclidean space.

The image of the Gauss map of any oriented isoparametric hypersurface of the unit standard sphere $S^{n+1}(1)$ is a minimal Lagrangian submanifold in the complex hyperquadric $Q_n({\mathbf C})$. In this paper we show that the Gauss image of a compact oriented isoparametric hypersurface with $g$ distinct constant princi…

Study stability and rigidity of axisymmetric marginally outer trapped surfaces.

The study shows that certain manifolds with positive curvature cannot contain specific geometric structures.

The paper explores $\alpha-$harmonic maps and their stability, proving key properties and conditions.

Proves stability of convex spheres with similar geodesic lengths.

The paper classifies decompositions of 3-sphere and lens spaces with handlebodies.

The paper examines stability of Sobolev inequalities on manifolds with Ricci curvature bounds.

We extend the results of our recent preprint [arXiv: 1811.00515] into higher dimensions $n \geq 4$. For minimizing harmonic maps $u\in W^{1,2}(Ω,\mathbb{S}^2)$ from $n$-dimensional domains into the two dimensional sphere we prove: (1) An extension of Almgren and Lieb's linear law, namely \[\mathcal{H}^{n-3}(\textrm{sin…

In this paper, we study complete oriented $f$-minimal hypersurfaces properly immersed in a cylinder shrinking soliton $(\mathbb{S}^n\times \mathbb{R}, \bar{g}, f)$. We prove that such hypersurface with $L_f$-index one must be either $\mathbb{S}^n\times\{0\}$ or $\mathbb{S}^{n-1}\times\mathbb{R}$, where $\mathbb{S}^{n-1…

We prove effective uniformization for nearly round 2-spheres and investigate their stability.

Proves conjecture about sphere widths under rotational symmetry.

In a seminal paper published in $1968$, J. Simons proved that, for $n\leq 5$, the Euclidean (minimal) cone $CM$, built on a closed, oriented, minimal and non totally geodesic hypersurface $M^n$ of $\mathbb S^{n+1}$ is unstable. In this paper, we extend Simons' analysis to {\em warped} (minimal) cones built over a close…

Small deformations of marginally (outer) trapped surfaces are considered by using their stability operator. In the case of spherical symmetry, one can use these deformations on any marginally trapped round sphere to prove several interesting results. The concept of 'core' of a black hole is introduced: it is a minimal …

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…

We study rigidity of minimal two-spheres $Σ$ that locally maximize the Hawking mass on a Riemannian three-manifold with a positive lower bound on its scalar curvature. After assuming strict stability of $Σ$, we prove that a neighborhood of it in $M$ is isometric to one of the deSitter-Schwarzschild metrics on $(- ε,ε)\…

Study on stability of mean curvature flow in hyperbolic space.

The round sphere is stable among spin manifolds with a specific scalar curvature bound.

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.

The Bartnik mass is a notion of quasi-local mass which is remarkably difficult to compute. Mantoulidis and Schoen [2016] developed a novel technique to construct asymptotically flat extensions of minimal Bartnik data in such a way that the ADM mass of these extensions is well-controlled, and thus, they were able to com…

Investigates harmonic self-maps' stability on cohomogeneity one manifolds.

Study phase transitions in noisy transformer dynamics on spheres.

The $n$-dimensional complex hyperquadric is a compact complex algebraic hypersurface defined by the quadratic equation in the $(n+1)$-dimensional complex projective space, which is isometric to the real Grassmann manifold of oriented 2- planes and is a compact Hermitian symmetric space of rank 2. In this paper we study…

In this paper we discuss the stability of geodesic spheres in $\mathbb{S}^{n+1}$ 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…

Let $M\subset S^{n+1}\subset\mathbb{R}^{n+2}$ be a compact minimal hypersurface of the $n$-dimensional Euclidean unit sphere. Let us denote by $|A|^2$ the square of the norm of the second fundamental form and $J(f)=-Δf-nf-|A|^2f$ the stability operator. It is known that the index (the number of negative eigenvalues of …

We prove that if a fibered knot $K$ with genus greater than one in a three-manifold $M$ has a sufficiently complicated monodromy, then $K$ induces a minimal genus Heegaard splitting $P$ that is unique up to isotopy, and small genus Heegaard splittings of $M$ are stabilizations of $P$. We provide a complexity bound in t…