arXiv research

Study shortest geodesics on flat cone spheres with conical singularities.

New flat surfaces found in 3D sphere space.

The paper provides uniform length estimates for trajectories on flat cone surfaces.

Geodesic nets on flat spheres are studied using Gauss-Bonnet theorem.

Paper connects hypersurfaces in 4D to surfaces in 3-sphere.

Locally flat 2-spheres in $\mathbb{C} P^2$ with knot group $\mathbb{Z}_2$ are ambiently isotopic if homologous.

New findings on minimal isometric immersions of flat n-tori into spheres.

The paper classifies minimal immersions of flat 3- and 4-tori in spheres by their first eigenfunctions.

Curved flats linked to pairs of Lie applicable surfaces.

Existence proved for static vacuum extensions near Schwarzschild spheres.

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…

Bounds on saddle connections on flat spheres with conical singularities.

E. Cartan proved that conformally flat hypersurfaces in S^{n+1} for n>3 have at most two distinct principal curvatures and locally envelop a one-parameter family of (n-1)-spheres. We prove that the Gauss-Codazzi equation for conformally flat hypersurfaces in S^4 is a soliton equation, and use a dressing action from sol…

The paper extends Siegel-Veech formula to convex flat cone spheres.

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 whether specific elements in the second homology of specific simply connected closed $4$-manifolds can be represented by smooth or topologically flat embedded spheres.

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.

We show that under some non-degeneracy assumption the only submersive harmonic morphism on a conformally flat $3-$sphere is the Hopf fibration. The proof involves an appropriate use the Chern-Simons functional.

Study on non-flat two-plectic geometry of six-sphere and its Hamiltonian dynamics.

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.

Study on group cocycles for volume-preserving diffeomorphisms.

CR 3-sphere rigidity proven through curvature invariant.

We show that the extrinsic diameter of immersed flat tori in the 3-sphere is $π$ under a certain topological condition for the projection of their asymptotic curves with respect to the Hopf fibration.

The paper proves a rigidity theorem for minimal submanifolds in spheres with flat normal bundle.

The paper proves conditions for Kähler-Einstein metrics on certain bundles.

Study biharmonic conformal immersions into a 3D flat space, finding new examples and classifications.

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

In this paper, helicoidal flat surfaces in the $3$-dimensional sphere $\mathbb{S}^3$ are considered. A complete classification of such surfaces is given in terms of their first and second fundamental forms and by linear solutions of the corresponding angle function. The classification is obtained by using the Bianchi-S…

Identifies a mod-$p$ triple cup product for rational homology 3-spheres with specific first homology.

We show that two of the Bryant-Salamon G_2-manifolds have a simple topology ; homeomorphic to the complement of some submanifolds of the 7-dimensional sphere. In this connection, we show there exists a complete Ricci-flat (non-flat) metric on the complement of an m-dimensional sphere in an n-dimensional sphere for some…

We give multiplicity results for the problem of prescribing the scalar curvature on Cauchy- Riemann spheres under Beta-flatness condition. To give a lower bound for the number of solutions, we use Bahri methods based on the theory of critical points at infinity and a Poincare-Hopf type formula.

We derive a gauge theoretic invariant of integral homology 3-spheres which counts gauge orbits of irreducible, perturbed flat SU(3) connections with sign given by spectral flow. To compensate for the dependence of this sum on perturbations, the invariant includes contributions from the reducible, perturbed flat orbits.…

We create a flat end foliation by critical spheres solving a Laplace-Beltrami problem.

We consider generalized Hodge-Laplace operators $αd δ+ βδd$ for $α, β> 0$ on $p$-forms on compact Riemannian manifolds. In the case of flat tori and round spheres of different radii, we explicitly calculate the spectrum of these operators. Furthermore, we investigate under which circumstances they are isospectral.

It is well-know that Hawking mass is nonnegative for a stable constant mean curvature ($CMC$) sphere in three manifold of nonnegative scalar curvature. R. Bartnik proposed the rigidity problem of Hawking mass of stable $CMC$ spheres. In this paper, we show partial rigidity results of Hawking mass for stable $CMC$ spher…

Study classifies metrics on a twice-punctured sphere, proving Delaunay metrics are complete.

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 …

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…

In a recent paper, the authors established the uniqueness of photon spheres in static vacuum asymptotically flat spacetimes by adapting Bunting and Masood-ul-Alam's proof of static vacuum black hole uniqueness. Here, we establish uniqueness of suitably defined sub-extremal photon spheres in static electro-vacuum asympt…

The paper examines conditions for Gromov-Hausdorff convergence of metric quotients and provides examples of conic-flat surfaces.

Sharp bounds for eigenfunctions on product spaces restricted to submanifolds.

New geometric variant of factorization homology for conformally flat manifolds.

The article proves $K5$ and $K_{3,3}$ are toroidal penny graphs.

It is proved, that if an almost Hermitian manifold satisfies the axiom of coholomorphic spheres, it is conformal flat.

We study the real Monge-Ampère equation in two and three dimensions, both from the point of view of the SYZ conjecture, where solutions give rise to semi-flat Calabi-Yau's and in affine differential geometry, where solutions yield parabolic affine sphere hypersurfaces. We find explicit examples, connect the holomorphic…

3D spheres with certain properties approach the round sphere.

The classical knot groups are the fundamental groups of the complements of smooth or piecewise-linear (PL) locally-flat knots. For PL knots that are not locally-flat, there is a pair of interesting groups to study: the fundamental group of the knot complement and that of the complement of the ``boundary knot'' that occ…

We develop techniques for computing the integer valued SU(3) Casson invariant. Our method involves resolving the singularities in the flat moduli space using a twisting perturbation and analyzing its effect on the topology of the perturbed flat moduli space. These techniques, together with Bott-Morse theory and the spl…