arXiv research

Stable compact minimal submanifolds of the product of a sphere and any Riemannian manifold are classified whenever the dimension of the sphere is at least three. The complete classification of the stable compact minimal submanifolds of the product of two spheres is obtained. Also, it is proved that the only stable comp…

The paper shows how stable minimal spheres emerge in certain 3D spaces under Ricci flow.

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

Minimal submanifolds are stable in certain conformal spheres.

In the 1-parameter family of Berger spheres S^3(a), a > 0 (S^3(1) is the round 3-sphere of radius 1) we classify the stable constant mean curvature spheres, showing that in some Berger spheres (a close to 0) there are unstable constant mean curvature spheres. Also, we classify the orientable compact stable constant mea…

New minimal 2-spheres found in hyperkähler 4-manifolds, unstable and not holomorphic.

Let $M$ be a Riemannian 3-manifold of nonnegative Ricci curvature, Ric $\geq 0.$ We suppose that $M$ is conformally flat and simply connected or more generally that it admits a conformal immersion into the standard 3-sphere. Let $Σ$ be a compact connected and orientable surface immersed in $M$ which is a stable constan…

Heat equation on projective spaces leads to stable minimal Morse functions.

We show that there exists a metric with positive scalar curvature on S2xS1 and a sequence of embedded minimal cylinders that converges to a minimal lamination that, in a neighborhood of a strictly stable 2-sphere, is smooth except at two helicoid-like singularities on the 2-sphere. The construction is inspired by a rec…

The study proves local rigidity of minimal 2-spheres in electrovacuum spacetimes.

The author proves that there is an open non empty set of metrics on any 3-manifold such that there exists a family of stably embedded minimal 2-spheres whose area is unbounded. This generalizes the work of T. Colding and W. Minicozzi who have shown an analogous result for the torus and B. Dean who showed the positive g…

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

Turaev's shadow can be seen locally as the Stein factorization of a stable map. In this paper, we define the notion of stable map complexity for a compact orientable 3-manifold bounded by (possibly empty) tori counting, with some weights, the minimal number of singular fibers of codimension 2 of stable maps into the re…

New minimal surfaces in spheres with complex topologies from capillarity.

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

In this paper, we prove a classification theorem for the stable compact minimal submanifolds of the Riemannian product of an $m_1$-dimensional ($m_1\geq3$) hypersurface $M_1$ in the Euclidean space and any Riemannian manifold $M_2$, when the sectional curvature $K_{M_1}$ of $M_1$ satisfies $\frac{1}{\sqrt{m_1-1}}\leq K…

Paper proves every stable 4-sphere has a unique diffeomorphism class.

As discussed in the paper, in a matter-filled spacetime, perhaps with positive cosmological constant, a stable marginally outer trapped 2-sphere must satisfy a certain area inquality. Namely, its area must be bounded above by $4π/c$, where $c > 0$ is a lower bound on a natural energy momentum term. In this note we cons…

In a matter-filled spacetime, perhaps with positive cosmological constant, a stable marginally outer trapped 2-sphere must satisfy a certain area inequality. Namely, as discussed in the paper, its area must be bounded above by $4π/c$, where $c > 0$ is a lower bound on a natural energy-momentum term. We then consider th…

Classifies non-linear Fredholm maps linking to stable homotopy groups of spheres.

New invariant prevents minimal submanifolds in curved spaces.

A major breakthrough in the theory of topological algorithms occurred in 1992 when Hyam Rubinstein introduced the idea of an almost normal surface. We explain how almost normal surfaces emerged naturally from the study of geodesics and minimal surfaces. Patterns of stable and unstable geodesics can be used to character…

Explicitly constructs moduli spaces of stable parabolic bundles.

Proves intersection properties of minimal hypersurfaces in various spaces.

Authors create stable proper biharmonic maps from unit ball to spheres.

The stabilisation height of a fibre surface in the 3-sphere is the minimal number of Hopf plumbing operations needed to attain a stable fibre surface from the initial surface. We show that families of fibre surfaces related by iterated Stallings twists have unbounded stabilisation height.

Researchers found unique large stable spheres in specific 3D space.

New bounds on singular set size for harmonic maps into 2-sphere in higher dimensions.

Constructs stable maps from 3-manifolds to surfaces without cusps.

A well known conjecture of Yau states that the first eigenvalue of every closed minimal hypersurface $M^n$ in the unit sphere $S^{n+1}(1)$ is just its dimension $n$. The present paper shows that Yau conjecture is true for minimal isoparametric hypersurfaces. Moreover, the more fascinating result of this paper is that t…

We establish an interesting connection between Morin singularities and stable homotopy groups of spheres. We apply this connection to computations of cobordism groups of certain singular maps. The differentials of the spectral sequence computing these cobordism groups are given by the composition multiplication in the …

Stable 2-lobed Delaunay tori found in 3-sphere.

Counterexample disproves conjecture about 3-manifolds.

The study shows that certain metrics on spheres prevent stable tangent cones for area-minimizing boundaries.

Study finds existence and non-existence of large stable CMC spheres in asymptotically flat 3-manifolds.

Round spheres remain stable under slight entropy changes.

Stable generalized complex structures on certain surfaces are constant.

Stable solutions to Yang-Mills-Higgs equations on spheres and tori identified.

New actions found on exotic spheres using group theory.

The paper proves partial rigidity of Hawking mass for stable CMC spheres in specific manifolds.

The flow of symmetric spheres converges to a round sphere.

We study the isoperimetric structure of asymptotically flat Riemannian 3-manifolds (M,g) that are C^0-asymptotic to Schwarzschild of mass m>0. Refining an argument due to H. Bray we obtain an effective volume comparison theorem in Schwarzschild. We use it to show that isoperimetric regions exist in (M, g) for all suffi…

The paper proves the existence of stable spheres in asymptotically flat 3-manifolds.

The Ricci flow on the 2-sphere with marked points is shown to converge in all three stable, semi-stable, and unstable cases. In the stable case, the flow was known to converge without any reparametrization, and a new proof of this fact is given. The semi-stable and unstable cases are new, and it is shown that the flow …

New method detects Kaehler scalar flat metrics and minimal hypersurfaces.

We investigate random complex dynamics of rational or polynomial maps on the Riemann sphere. We show that regarding random complex dynamics of polynomials, generically, the chaos of the averaged system disappears at any point in the Riemann sphere due to the automatic coopeartion of many kinds of maps in the system, ev…

Study on stable maps from 3-sphere to 3-space, focusing on singularities and invariants.

Paper proves no stable Yang-Mills fields on spheres.