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By refining the volume estimate of Heintze and Karcher [11], we obtain a sharp pinching estimate for the genus of a surface in S 3 , which involves an integral of the norm of its traceless second fundamental form. More specifically, we show that if g is the genus of a closed orientable surface Σ in a 3-dimensional orientable Riemannian manifold M whose sectional curvature is bounded below by 1, then 4 π 2 g (Σ) ≤ 2 2 π 2 - | M | + ∫ Σ f (| A ∘ |) , where A ∘ is the traceless second fundamental form and f is an explicit function. As a result, the space of closed orientable embedded minimal surfaces Σ with uniformly bounded ‖ A ‖ L 3 (Σ) is compact in the C k topology for any k ≥ 2 . [ABSTRACT FROM AUTHOR]

The Minkowski product of unit quaternion sets is introduced and analyzed, motivated by the desire to characterize the overall variation of compounded spatial rotations that result from individual rotations subject to known uncertainties in their rotation axes and angles. For a special type of unit quaternion set, the spherical caps of the 3-sphere S3 in R4, closure under the Minkowski product is achieved. Products of sets characterized by fixing either the rotation axis or rotation angle, and allowing the other to vary over a given domain, are also analyzed. Two methods for visualizing unit quaternion sets and their Minkowski products in R3 are also discussed, based on stereographic projection and the Lie algebra formulation. Finally, some general principles for identifying Minkowski product boundary points are discussed in the case of full-dimension set operands.

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We show that the disk complex of a genus g > 1 Heegaard surface for the 3-sphere is homotopy equivalent to a wedge of (2 g − 2) -dimensional spheres. This implies that genus g > 1 Heegaard surfaces for the 3-sphere are topologically minimal with index 2 g − 1. [ABSTRACT FROM AUTHOR]

We present a steady Euler ow on the round 3-sphere S³ whose velocity vector fleld has the remarkable property of having two independent first integrals, being tangent to the flbres of an almost submersion onto the 2-sphere. This submersion turns out to be a critical point for the quartic Faddeev-Skyrme model with a standard potential. [ABSTRACT FROM AUTHOR]

We introduce non-Abelian Kuramoto model on S3 in the most general form. Following an analogy with the classical Kuramoto model (on the circle S1), we study some interesting variations of the model on S3 that are obtained for particular coupling functions. As a partial case, by choosing "standard" coupling function one obtains a previously known model, that is referred to as Kuramoto-Lohe model on S3.We briefly address two particular models: Kuramoto models on S3 with frustration and with external forcing. These models on higher dimensional manifolds have not been studied so far. By choosing suitable values of parameters we observe different nontrivial dynamical regimes even in the simplest setup of globally coupled homogeneous population.Although non-Abelian Kuramoto models can be introduced on various symmetric spaces, we restrict our analysis to the case when underlying manifold is the 3-sphere. Due to geometric and algebraic properties of this specific manifold, variations of this model are meaningful and geometrically well justified. [ABSTRACT FROM AUTHOR]

We study consensus and anti-consensus on the 3-sphere as the global optimization problems. The corresponding gradient descent algorithm is a dynamical systems on S 3 , that is known in Physics as non-Abelian Kuramoto model. This observation opens a slightly different insight into some previous results and also enables us to prove some novel results concerning consensus and balancing over the complete graph. In this way we fill some gaps in the existing theory. In particular, we prove that the anti-consensus algorithm over the complete graph on S 3 converges towards a balanced configuration if a certain mild condition on initial positions of agents is satisfied. The form of this condition indicates an unexpected relation with some important constructions from Complex Analysis. [ABSTRACT FROM AUTHOR]

In the present work, we study the decompositions of codimension-one transitions that alter the singular set the of stable maps from S 3 to R 3 , the topological behaviour of the singular set and the singularities in the branch set that involves cuspidal curves and swallowtails that alter the singular set. We also analyse the effects of these decompositions on the global invariants with prescribed branch sets. [ABSTRACT FROM AUTHOR]

Topological surgery in dimension 3 is intrinsically connected with the classification of 3 -manifolds and with patterns of natural phenomena. In this, mostly expository, paper, we present two different approaches for understanding and visualizing the process of 3 -dimensional surgery. In the first approach, we view the process in terms of its effect on the fundamental group. Namely, we present how 3 -dimensional surgery alters the fundamental group of the initial manifold and present ways to calculate the fundamental group of the resulting manifold. We also point out how the fundamental group can detect the topological complexity of non-trivial embeddings that produce knotting. The second approach can only be applied for standard embeddings. For such cases, we give new visualizations of 3 -dimensional surgery as rotations of the decompactified 2 -sphere. Each rotation produces a different decomposition of the 3 -sphere which corresponds to a different visualization of the 4 -dimensional process of 3 -dimensional surgery. [ABSTRACT FROM AUTHOR]

Following the Euclidean results of Varopoulos and Pankka- Rajala, we provide a necessary topological condition for a sub-Riemannian 3-manifold M to admit a nonconstant quasiregular mapping from the sub- Riemannian Heisenberg group H. As an application, we show that a link complement S3\L has a sub-Riemannian metric admitting such a mapping only if L is empty, an unknot or Hopf link. In the converse direction, if L is empty, a specific unknot or Hopf link, we construct a quasiregular mapping from H to S3\L. The main result is obtained by translating a growth condition on p1(M) into the existence of a supersolution to the 4-harmonic equation, and relies on recent advances in the study of analysis and potential theory on metric spaces. [ABSTRACT FROM AUTHOR]

In this paper, helicoidal flat surfaces in the 3‐dimensional sphere S3 are considered. A complete classification of such surfaces, that generalizes a classification of rotational flat surfaces, is given in terms of the first and second fundamental forms for asymptotic parameters. The result consists in a relation between helicoidal flat surfaces and linear solutions of the corresponding homogeneous wave equation for the angle function. [ABSTRACT FROM AUTHOR]

Abstract We introduce and analyze several models of swarm dynamics on the sphere S 3 with adaptive (state-dependent) interactions between agents. The equations describing the interaction dynamics are variations of the classical Hebbian principle from Neuroscience. We study asymptotic behavior in models with various realizations of Hebbian and anti-Hebbian learning rules. The swarm with the Hebbian rule and strictly nonnegative (attractive) interactions evolves towards consensus. If the Hebbian rule allows both attractive and repulsive interactions the swarm converges to bipolar configuration. The most interesting is the model with anti-Hebbian learning rule with both attractive and repulsive interactions. This model displays a rich variety of dynamical regimes and stationary formations, depending on the number of agents and system parameters. We prove that the model with such anti-Hebbian rule evolves towards a stable stationary configuration if the system parameter is above a certain bifurcation threshold. Finally, some simulation results are presented demonstrating how these theoretical results can be applied to coordination of rotating bodies in 3D space; this is done by mapping the trajectories from S 3 to special orthogonal group SO(3). [ABSTRACT FROM AUTHOR]

We consider an embedding of a 2-dimensional CW complex into the 3-sphere, and construct its dual graph. Then we obtain a homogeneous system of linear equations from the 2-dimensional CW complex in the first homology group of the complement of the dual graph. By checking that the homogeneous system of linear equations does not have an integral solution, we show that some 2-dimensional CW complexes cannot be embedded into the 3-sphere. [ABSTRACT FROM AUTHOR]

We define two transforms of non-conformal harmonic maps from a surface into the 3-sphere. With these transforms one can construct, from one such harmonic map, a sequence of harmonic maps. We show that there is a correspondence between harmonic maps into the 3-sphere, H-surfaces in Euclidean 3-space and almost complex surfaces in the nearly Kähler manifold S³ × S³. As a consequence we can construct sequences of H-surfaces and almost complex surfaces. [ABSTRACT FROM AUTHOR]

We present two transforms of non-conformal harmonic maps from a surface into the 3-sphere. With these transforms one can construct from one non-conformal harmonic map a sequence of non-conformal harmonic maps. We also discuss the correspondence between non-conformal harmonic maps into the 3-sphere, H-surfaces in Euclidean 3-space and almost complex surfaces in the nearly Kähler manifold S3 × S3. [ABSTRACT FROM AUTHOR]

In this article, we demonstrate methods for the local removal and modification of complex tangents to embeddings of S3 into C3. In particular, given any embedding of S3 and a neighborhood of the complex tangents of the embedding, we show that there exists a (C0-close) totally real embedding which agrees with the original embedding outside the given neighborhood of the complex tangents. We also demonstrate that given any knot type K ⊂ S3, either there exists an embedding of S3 which assumes nondegenerate complex tangents exactly along K or there exists a nondegenerate embedding complex tangent along two unlinked copies of K (both cases may hold). We also note possible directions of future investigations. [ABSTRACT FROM AUTHOR]

In this article, we demonstrate methods for the local removal and modification of complex tangents to embeddings of S3 into C3. In particular, given any embedding of S3 and a neighborhood of the complex tangents of the embedding, we show that there exists a (C0-close) totally real embedding which agrees with the original embedding outside the given neighborhood of the complex tangents. We also demonstrate that given any knot type K ⊂ S3, either there exists an embedding of S3 which assumes nondegenerate complex tangents exactly along K or there exists a nondegenerate embedding complex tangent along two unlinked copies of K (both cases may hold). We also note possible directions of future investigations. [ABSTRACT FROM AUTHOR]

Perturbing the classical metric on the round 3-sphere $$S^3$$ by the Killing vector fields tangent to Hopf fibers, one gets a class of Finsler metrics of Randers type with constant flag curvature $$\mathbf{K}=1$$, depending on one parameter $$k>1$$, called Bao-Shen's (J Lond Math Soc 66:453-467, ) metrics. The corresponding spheres will be called Bao-Shen's spheres, which are proper candidates of positively curved Finsler space forms. In this paper, we study the minimal surfaces in Bao-Shen's spheres. We first study submanifolds isometrically immersed in a Randers manifold by the method of Zermelo's navigation. Then we give a clear formula of the mean curvature of the surface in a Bao-Shen's sphere by introducing the volume ratio function to show its relation with the mean curvature of the surface in round 3-sphere. As an application, we find an interesting family of minimal surfaces with respect to Busemann-Hausdorff volume form in Bao-Shen's sphere called helicoids. This family contains the compact minimal surfaces $$\tau _{m,n}$$ in round 3-sphere constructed by Lawson (Ann Math 92(3):335-374, ), including great 2-spheres, Clifford torus, Klein bottles, etc. Moreover, two rigidity results are given. [ABSTRACT FROM AUTHOR]

The Bishop invariant is a powerful tool in the analysis of real submanifolds of complex space that associates to every (nondegen-erate) complex tangent of the embedding a nonnegative real number (or infinity). It is a biholomorphism invariant that gives information regarding the local hull of holomorphy of the manifold near the complex tangent. In this paper, we derive a readily applicable formula for the computation of the Bishop invariant for graphical embeddings of 3-manifolds into C³. We then exhibit some examples over S³ demonstrating the different possible configurations of the Bishop invariant along complex tangents to such embeddings. We will also generate a few more results regarding the behavior of the Bishop invariant in certain situations. We end our paper by analyzing some of the different possible outcomes of the perturbation of a degenerate complex tangent. [ABSTRACT FROM AUTHOR]

In the mid-1980's, M. Gromov used his machinery of the h- principle to prove that there exists totally real embeddings of S³ into C³. Subsequently, Patrick Ahern and Walter Rudin explicitly demonstrated such a totally real embedding. In this paper, we consider the generic situation for such embeddings, namely where complex tangents arise as codimension-2 subspaces. We first consider the Heisenberg group H and generate some interesting results therein. Then, by using the biholomorphism of H with the 3-sphere minus a point, we demonstrate that every homeomorphism-type of knot in S³ may arise precisely as the set of complex tangents to an embedding S³ → C³. We also make note of the (nongeneric) situation where complex tangents arise along surfaces. [ABSTRACT FROM AUTHOR]

Using quaternions, we give a concise derivation of the Ricci tensor for homogeneous spaces with topology of the 3-dimensional sphere. We derive explicit and numerical solutions for the Ricci flow PDE and discuss their properties. In the collapse (or expansion) of these models, the interplay of the various components of the Ricci tensor are studied. [ABSTRACT FROM AUTHOR]

Enomoto, Weiner and the first author showed the rigidity of the Clifford torus amongst the class of embedded flat tori in S3. In the proof of that result, an estimate of extrinsic diameter of flat tori plays a crucial role. It is reasonable to expect that the same rigidity holds in the class of immersed flat tori in S3. In this paper, we give a new method for characterizing immersed flat tori in S3 with extrinsic diameter π, which is a somewhat similar technique to the proof of the 6-vertex theorem for certain closed plane curves given by the second author. As an application, we show that the Clifford torus is rigid in the class of immersed flat tori whose mean curvature functions do not change sign. Recently, the global behaviour of flat surfaces in H3 and R 3 regarded as wave fronts has been studied. We also give here a formulation of flat tori in S3 as wave fronts. As an application, we shall exhibit a flat torus as a wave front whose extrinsic diameter is less than π. [ABSTRACT FROM AUTHOR]

The volume of the orientation workspace is measured based on two invariant principles: It must be invariant with respect to the ground frame and invariant with respect to the orientation description. A method which is based on quaternions and differential geometry is developed for the measurement of the volume correctly. The method is extended for an arbitrary orientation description by means of a mapping theorem proposed for the first time. An example of a serial spherical wrist shows that the volumes that are obtained by the proposed method are consistent with the two invariant principles. [ABSTRACT FROM AUTHOR]

In this paper, we study volume-preserving great circle flows on the 3-sphere S3, and introduce a moduli space for these, consisting of distance-decreasing holomorphic maps from connected open subsets of S2 to S2; one consequence is that the only volume-preserving great circle flows defined on the entire 3-sphere are those tangent to Hopf fibrations. [ABSTRACT FROM AUTHOR]