Surfaces In Sphere Research Articles

Biconservative surfaces in the 4-dimensional Euclidean sphere

Biconservative surfaces in the 4-dimensional Euclidean sphere

Willmore Surfaces in Spheres via Loop Groups IV: On Totally Isotropic Willmore Two-Spheres in S6

In this paper the author derives a geometric characterization of totally isotropic Willmore two-spheres in S6, which also yields to a description of such surfaces in terms of the loop group language. Moreover, applying the loop group method, he also obtains an algorithm to construct totally isotropic Willmore two-spheres in S6. This allows him to derive new examples of geometric interests. He first obtains a new, totally isotropic Willmore two-sphere which is not S-Willmore (i.e., has no dual surface) in S6. This gives a negative answer to an open problem of Ejiri in 1988. In this way he also derives many new totally isotropic, branched Willmore two-spheres which are not S-Willmore in S6.

Totally biharmonic hypersurfaces in space forms and 3-dimensional BCV spaces

A hypersurface is said to be totally biharmonic if all its geodesics are biharmonic curves in the ambient space. We prove that a totally biharmonic hypersurface into a space form is an isoparametric biharmonic hypersurface, which allows us to give the full classification of totally biharmonic hypersurfaces in these spaces. Moreover, restricting ourselves to the 3-dimensional case, we show that totally biharmonic surfaces into Bianchi–Cartan–Vranceanu spaces are isoparametric surfaces and we give their full classification. In particular, we show that, leaving aside surfaces in the 3-dimensional sphere, the only nontrivial example of a totally biharmonic surface appears in the product space [Formula: see text].

Minimal surfaces in spheres and a Ricci-like condition

We deal with minimal surfaces in spheres that are locally isometric to a pseudoholomorphic curve in a totally geodesic $${\mathbb {S}}^{5}$$ in the nearly Kahler sphere $${\mathbb {S}}^6$$ . Being locally isometric to a pseudoholomorphic curve in $${\mathbb {S}}^5$$ turns out to be equivalent to the Ricci-like condition $$\Delta \log (1-K)=6K,$$ where K is the Gaussian curvature of the induced metric. Besides flat minimal surfaces in spheres, direct sums of surfaces in the associated family of pseudoholomorphic curves in $${\mathbb {S}}^5$$ do satisfy this Ricci-like condition. Surfaces in both classes are exceptional surfaces. These are minimal surfaces whose all Hopf differentials are holomorphic, or equivalently the curvature ellipses have constant eccentricity up to the last but one. Under appropriate global assumptions, we prove that minimal surfaces in spheres that satisfy this Ricci-like condition are indeed exceptional. Thus, the classification of these surfaces is reduced to the classification of exceptional surfaces that are locally isometric to a pseudoholomorphic curve in $${\mathbb {S}}^5.$$ In fact, we prove, among other results, that such exceptional surfaces in odd dimensional spheres are flat or direct sums of surfaces in the associated family of a pseudoholomorphic curve in $${\mathbb {S}}^5$$ .

Generic conformally flat hypersurfaces and surfaces in 3-sphere

The aim of this paper is to verify that the study of generic conformally flat hypersurfaces in 4-dimensional space forms is reduced to a surface theory in the standard 3-sphere.The conformal structure of generic conformally flat (local-)hypersurfaces is characterized as conformally flat (local-)3-metrics with the Guichard condition.Then, there is a certain class of orthogonal analytic (local-)Riemannian 2-metrics with constant Gauss curvature -1 such that any 2-metric of the class gives rise to a one-parameter family of conformally flat 3-metrics with the Guichard condition.In this paper, we firstly relate 2-metrics of the class to surfaces in the 3-sphere:for a 2-metric of the class, a 5-dimensional set of (non-isometric) analytic surfaces in the 3-sphere is determined such thatany surface of the set gives rise to an evolution of surfaces in the 3-sphere issuing from the surfaceand the evolution is the Gauss map of a generic conformally flat hypersurface in the Euclidean 4-space. Secondly, we characterize analytic surfaces in the 3-sphere which give rise to generic conformally flat hypersurfaces.

Characterization of Clifford Torus in Three-Spheres

We characterize spheres and the tori, the product of the two plane circles immersed in the three-dimensional unit sphere, which are associated with the Laplace operator and the Gauss map defined by the elliptic linear Weingarten metric defined on closed surfaces in the three-dimensional sphere.

Toward a History of the Medial Regime: Force, Representation, and the Female Body

Toward a History of the Medial RegimeForce, Representation, and the Female Body Thomas Stubblefield (bio) In February 2014, Sports Illustrated celebrated the fiftieth anniversary of the swimsuit issue by photographing Kate Upton in a zero-gravity chamber. A modified Boeing 727 was enlisted to execute seventeen parabolas, which subjected the model/actress's body to a series of oscillating forces. On the downswing of the trajectory, the considerable g-force of the climb gave way to thirteen zero-gravity scenarios and four lunar-gravity events, prompting Upton to float majestically before the camera. For even the most casual art historian, there was something undeniably familiar about the female form that these free falls produced. The inflated cellular structure, free-floating extremities, and sinuous silhouette of this body reproduced the inventory of "extra vertebrae, … boneless feet, … helium-filled arms, and … malleable throats" that populate Baroque fantasies of the female body (Betzer, 12–13). Even the aeronautical mise-en-scène that inscribed this tentative, uncertain body seemed to resuscitate what Aby Warburg famously described as the anti-equilibrium of the Renaissance in which the human form is "caught up in a play of overwhelming forces … hair flowing, and garments blown back through exertion or by wind" (Michaud, 28). The uncanny nature of this elaborate restaging of historical codes of desirability relies upon transformations of the body that extend beyond the manipulation of the image or the intervention of the medium of photography. Clearly, the image has been "retouched" and the camera and its codes of looking continue to structure the scene, but if these were ends in themselves, a conventional "green screen" would have sufficed in the place of an actual zero-g flight. Instead, the shoot went to great lengths to engage the body in the image's coming into being. As the flesh becomes a medium for the processes of making visible, [End Page 130] the conventions of the history of art collapse with the natural functions of the body, producing a circular dynamic in which the female form is both the site for semiotic inscription and the seemingly autonomous origin of such signs. This corporeal enfolding is symptomatic of an enduring visual regime, which overwhelms the body's motor capacities with force (variously defined as intoxication, possession, unconscious drives, and mechanical energy) so as to prompt the flesh to produce recognizable signs of desirability. Targeting the automatic and nonintentional relations of the body, such scenarios posit the female body as both interface (a Platonic surface that opens onto an ideal signified) and medium (the means by which such a translation becomes possible), both object and instrument of desire. These relations constitute the recurring logic of what I will call the medial body, a dispositif whose extensive history stretches well beyond the sophisticated aeronautics of the Upton shoot or even the machines of the Industrial Revolution. Drawing upon a joint deployment of discourse and force, it appears in the rapture of the Renaissance nude and the throes of demonic possession during Romanticism before being recast in the nineteenth century in terms of the mechanical, electrical, and luminous energy that animated bodies at the Salpêtrière hospital in Paris. Most recently, it surfaces in the online sphere, where postindustrial formations of labor effectively transform the overtaken body into a medium for the production of surplus value. As individual instantiations of this regime interfuse processes of enfolding with histories of representation, the evolution the medial body charts less a developmental or progressivist narrative than a series of archeological knots in which the past circulates in the present as conditions of possibility. Accordingly, its mode of sign production is historical in a dual sense in that not only does the medial regime persist across a diverse set of contexts, but so do singular instances within this archive recirculate atavistic genres, narratives, and subject relations in order to facilitate the imaginary consumption of otherwise unavailable female bodies. Despite the familiarity of the images it produces, this automated body maintains something of an ambiguous relationship to contemporary theory. On one hand, the process of enfolding that undergirds the medial regime reiterates the rejection of Cartesian dualisms performed by prevailing theories of the...

New Characterizations of the Clifford Torus and the Great Sphere

In studying spherical submanifolds as submanifolds of a round sphere, it is more relevant to consider the spherical Gauss map rather than the Gauss map of those defined by the oriented Grassmannian manifold induced from their ambient Euclidean space. In that sense, we study ruled surfaces in a three-dimensional sphere with finite-type and pointwise 1-type spherical Gauss map. Concerning integrability and geometry, we set up new characterizations of the Clifford torus and the great sphere of 3-sphere and construct new examples of spherical ruled surfaces in a three-dimensional sphere.

Minimal surfaces in the three-dimensional sphere with high symmetry

Using the Lawson existence theorem of minimal surfaces and the symmetries of the Hopf fibration, we will construct symmetric embedded closed minimal surfaces in the three-dimensional sphere. These surfaces contain the Clifford torus, the Lawson’s minimal surfaces, and seven new minimal surfaces with genera [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text]. We will also discuss the relation between such surfaces and the maximal extendable group actions on subsurfaces of the three-dimensional sphere.

Minimal surfaces in a unit sphere pinched by intrinsic curvature and normal curvature

We establish a nice orthonormal frame field on a closed surfaceminimally immersed in a unit sphere $\mathbb{S}^{n}$, under which the shapeoperators take very simple forms. Using this frame field, we obtainan interesting property $K+K^{N}=1$ for the Gauss curvature $K$ andthe normal curvature $K^{N}$ if the Gauss curvature is positive.Moreover, using this property we obtain the pinching on theintrinsic curvature and normal curvature, the pinching on the normalcurvature, respectively.

Contact stationary Legendrian surfaces in 𝕊5

Let $(M^5,\alpha,g_\alpha,J)$ be a 5-dimensional Sasakian Einstein manifold with contact 1-form $\alpha$, associated metric $g_\alpha$ and almost complex structure $J$ and $L$ a contact stationary Legendrian surface in $M^5$. We will prove that $L$ satisfies the following equation \begin{eqnarray}\label{equ} -\Delta^\nu H+(K-1)H=0, \end{eqnarray} where $\Delta^\nu$ is the normal Laplacian w.r.t the metric $g$ on $L$ induced from $g_\alpha$ and $K$ is the Gauss curvature of $(L,g)$. Using equation \eqref{equ} and a new Simons' type inequality for Legendrian surfaces in the standard unit sphere $\mathbb{S}^5$, we prove an integral inequality for contact stationary Legendrian surfaces in $\mathbb{S}^5$. In particular, we prove that if $L$ is a contact stationary Legendrian surface in $\mathbb{S}^5$, $B$ is the second fundamental form of $L$, $S=|B|^2$, $\rho^2=S-2H^2$ and $$0\leq S\leq 2,$$ then we have either $\rho^2=0$ and $L$ is totally umbilic or $\rho^2\neq 0$, $S=2, H=0$ and $L$ is a flat minimal Legendrian torus.

Willmore surfaces in spheres via loop groups III: on minimal surfaces in space forms

The family of Willmore immersions from a Riemann surface into $S^{n+2}$ can be divided naturally into the subfamily of Willmore surfaces conformally equivalent to a minimal surface in $\mathbb{R}^{n+2}$ and those which are not conformally equivalent to a minimal surface in $\mathbb{R}^{n+2}$. On the level of their conformal Gauss maps into $Gr_{1,3}(\mathbb{R}^{1,n+3})=SO^+(1,n+3)/SO^+(1,3)\times SO(n)$ these two classes of Willmore immersions into $S^{n+2}$ correspond to conformally harmonic maps for which every image point, considered as a 4-dimensional Lorentzian subspace of $\mathbb{R}^{1,n+3}$, contains a fixed lightlike vector or where it does not contain such a ``constant lightlike vector''. Using the loop group formalism for the construction of Willmore immersions we characterize in this paper precisely those normalized potentials which correspond to conformally harmonic maps containing a lightlike vector. Since the special form of these potentials can easily be avoided, we also precisely characterize those potentials which produce Willmore immersions into $S^{n+2}$ which are not conformal to a minimal surface in $\mathbb{R}^{n+2}$. It turns out that our proof also works analogously for minimal immersions into the other space forms.

Constant mean curvature proper-biharmonic surfaces of constant Gaussian curvature in spheres

Constant mean curvature (CMC) surfaces in spheres are investigated under the extra condition of biharmonicity. From the work of Miyata, especially in the flat case, we give a complete description of such immersions and show that for any $h \in (0,1)$ there exist CMC proper-biharmonic planes and cylinders in ${\mathbb S}^5$ with $|H|=h$, while a necessary and sufficient condition on $h$ is found for the existence of CMC proper-biharmonic tori in ${\mathbb S}^5$.

A representation for pseudoholomorphic surfaces in spheres

We give a local representation for the pseudoholomorphic surfaces in Euclidean spheres in terms of holomorphic data. Similar to the case of the generalized Weierstrass representation of Hoffman and Osserman for minimal Euclidean surfaces, we assign such a surface in S 2 n \mathbb {S}^{2n} to a given set of n n holomorphic functions defined on a simply-connected domain in C \mathbb {C} .

Isometric deformations of isotropic surfaces

It was shown by Ramanathan (Manuscripta Math 60:417–422, 1988) that any compact oriented minimal surface in the three-dimensional round sphere admits at most a finite set of pairwise noncongruent minimal isometric immersions. In this paper, we extend this result to isotropic surfaces in spheres of higher dimension. The case of isotropic surfaces other than compact in space forms is also addressed.

On symmetric Willmore surfaces in spheres I: The orientation preserving case

In this paper we provide a systematic discussion of how to incorporate orientation preserving symmetries into the treatment of Willmore surfaces via the loop group method.In this context we first develop a general treatment of Willmore surfaces admitting orientation preserving symmetries, and then show how to induce finite order rotational symmetries. We also prove, for the symmetric space which is the target space of the conformal Gauss map of Willmore surfaces in spheres, the longstanding conjecture of the existence of meromorphic invariant potentials for the conformal Gauss maps of all compact Willmore surfaces in spheres. We also illustrate our results by some concrete examples.

Exceptional minimal surfaces in spheres

We study a class of exceptional minimal surfaces in spheres for which all Hopf differentials are holomorphic. Extending results of Eschenburg and Tribuzy (Rend Semin Mat Univ Padova, 79:185–202, 1998), we obtain a description of exceptional surfaces in terms of a set of absolute value type functions, the a-invariants, that determine the geometry of the higher order curvature ellipses and satisfy certain Ricci-type conditions. We show that the a-invariants determine these surfaces up to a multiparameter family of isometric minimal deformations, where the number of the parameters is precisely the number of non-vanishing Hopf differentials. We give applications to superconformal surfaces and pseudoholomorphic curves in the nearly Kahler sphere $${\mathbb{S}^{6}}$$ . Moreover, we study superconformal surfaces in odd dimensional spheres that are isometric to their polar and show a relation to pseudoholomorphic curves in $${\mathbb{S}^{6}}$$ .

Nontrivial minimal surfaces in a class of Finsler spheres of vanishing [formula omitted]-curvature

The minimal surface theory in Finsler geometry deserves to be well developed as that in Riemannian geometry. In this paper, we derive the mean curvature of submanifolds in a general (α,β)-manifold by considering the Busemann–Hausdorff measure and Holmes–Thompson measure respectively. We then study the rotationally invariant minimal surfaces, in the Finsler 3-sphere endowed with an (α,β)-metric F̃k=α̃kϕ(β̃k/α̃k), k>1, where ϕ is a smooth function, (S3,α̃k) is the Berger sphere E(4/k,1) and β̃k is a Killing one form of constant length along the Hopf fibers of S3. We define the energy of the minimal surfaces, and by using the volume ratio function introduced by the author and Y.-B. Shen, we give the explicit local expressions of the rotationally invariant BH-minimal and HT-minimal surfaces in such sphere, respectively. As a special case, we give a detailed study of the rotationally invariant HT-minimal surfaces in the 3-sphere with square metric.

Wintgen ideal submanifolds with a low-dimensional integrable distribution

Submanifolds in space forms satisfy the well-known DDVV inequality. A submanifold attaining equality in this inequality pointwise is called a Wintgen ideal submanifold. As conformal invariant objects, Wintgen ideal submanifolds are investigated in this paper using the framework of Mobius geometry. We classify Wintgen ideal submanfiolds of dimension m ⩽ 3 and arbitrary codimension when a canonically defined 2-dimensional distribution \(\mathbb{D}_2\) is integrable. Such examples come from cones, cylinders, or rotational submanifolds over super-minimal surfaces in spheres, Euclidean spaces, or hyperbolic spaces, respectively. We conjecture that if \(\mathbb{D}_2\) generates a k-dimensional integrable distribution \(\mathbb{D}_k\) and k < m, then similar reduction theorem holds true. This generalization when k = 3 has been proved in this paper.

Nontrivial minimal surfaces in a class of Finsler 3-spheres

In this paper, we study the rotationally invariant minimal surfaces in the Bao–Shen's spheres, which are a class of 3-spheres endowed with Randers metrics [Formula: see text] of constant flag curvature K = 1, where [Formula: see text] are Berger metrics, [Formula: see text] are one-forms and k &gt; 1 is an arbitrary real number. We obtain a class of nontrivial minimal surfaces isometrically immersed in the Bao–Shen's spheres, which is the first class of nontrivial minimal surfaces with respect to the Busemann–Hausdorff measure in Finsler spheres. Moreover, we also obtain a new class of explicit minimal surfaces in the classical Berger spheres [Formula: see text], which was expected to get in [F. Torralbo, Rotationally invariant constant mean curvature surfaces in homogeneous 3-manifolds, Differential Geom. Appl.28(5) (2010) 593–607].