Constant Extrinsic Curvature Research Articles

Taut almost cosymplectic hyperbolas and almost bi-contact metric structures on three-manifolds

Cappelletti-Montano et al. [6] introduced and studied taut cosymplectic circles/spheres, which are the analogues in the cosymplectic setting of taut contact circles/spheres introduced and studied by Geiges-Gonzalo [11], [13], [14]. The main purpose of this paper is to introduce and study, in dimension three, the hyperbolic analogue in the cosymplectic setting. We give a characterization of a taut almost cosymplectic hyperbola, classify compact three-manifolds which admit a cosymplectic hyperbola, and give a complete classification of 3D Lie groups which admit a left invariant taut cosymplectic hyperbola/circle/hyperboloid/sphere. Besides, we introduce the notion of almost bi-contact metric structure. In particular, in dimension three, an almost bi-contact structure defines a taut almost cosymplectic hyperbolas or a taut almost cosymplectic circle. Then we characterize, in dimension three, the existence of an almost bi-contact metric structure, in particular with the associated 1-forms closed, and thus a complete classification of almost bi-α-coKähler three-manifolds with constant extrinsic curvature is established.

On quasicomplete k$k$‐surfaces in 3‐dimensional space‐forms

AbstractIn the study of immersed surfaces of constant positive extrinsic curvature in space‐forms, it is natural to substitute completeness for a weaker property, which we here call quasicompleteness. We determine the global geometry of such surfaces under the hypotheses of quasicompleteness. In particular, we show that, for , the only quasicomplete immersed surfaces of constant extrinsic curvature equal to in the 3‐dimensional space‐form of constant sectional curvature equal to are the geodesic spheres. Together with earlier work of the author, this completes the classification of quasicomplete immersed surfaces of constant positive extrinsic curvature in 3‐dimensional space‐forms.

On the asymptotic geometry of finite-type $k$-surfaces in three-dimensional hyperbolic space

For 0<k<1 , a finite-type k - surface in 3 -dimensional hyperbolic space is a complete, immersed surface of finite area and of constant extrinsic curvature equal to k . In Smith (2021), we showed that such surfaces have finite genus and finitely many cusp-like ends. Each of these cusps is asymptotic to an immersed cylinder of exponentially decaying radius about a complete geodesic and terminates at an ideal point which we call the extremity of the cusp. We show that every cusp of any finite-type k -surface has a well-defined axis, which we will call the Steiner geodesic of the cusp. One of the end-points of this axis is the extremity, and we will call the other, which constitutes new geometric data, the Steiner point of the cusp. We prove a new identity involving extremities and Steiner points in terms of Möbius invariant vector fields over the Riemann sphere. We define two new functionals over the space of finite-type k -surfaces. The first, which will be called the generalized volume , is defined by the integral of a certain well-chosen form, and extends to the non-embedded case the concept of volume of the set bounded by the surface. The second, which will be called the renormalized energy , is related to the integral of the mean curvature of the surface, and is well-defined up to a choice of Busemann function. Upon describing natural parametrizations of the strata of the space of finite-type k -surfaces by open complex manifolds, we prove a new Schläfli-type formula relating the extremities and Steiner points to the first order variations of the generalized volume and the renormalized energy. In particular, Möbius invariance of this formula yields the aforementioned identity. We conclude by studying some applications of this identity and Schläfli-type formula.

Graph Surfaces Invariant by Parabolic screw Motions with Constant Curvature in $ \: \mathbb H^2 \times \mathbb R$

In this work we study vertical graph surfaces invariant by parabolic screw motions with pitch $\ell >0$ and constant Gaussian curvature or constant extrinsic curvature in the product space $\mathbb H^2 \times \mathbb R$. In particular, we determine flat and extrinsically flat graph surfaces in $\mathbb H^2 \times \mathbb R$. We also obtain complete and non-complete vertical graph surfaces in $\mathbb H^2 \times \mathbb R$ with negative constant Gaussian curvature and zero extrinsic curvature.

Willmore surfaces and Hopf tori in homogeneous 3-manifolds

Some classification results for closed surfaces in Berger spheres are presented. On the one hand, a Willmore functional for isometrically immersed surfaces into an homogeneous space {mathbb {E}}^{3}(kappa ,tau ) with isometry group of dimension 4 is defined and its first variational formula is computed. Then, we characterize Clifford and Hopf tori as the only Willmore surfaces satisfying a sharp Simons-type integral inequality. On the other hand, we also obtain some integral inequalities for closed surfaces with constant extrinsic curvature in {mathbb {E}}^3(kappa ,tau ), becoming equalities if and only if the surface is a Hopf torus in a Berger sphere.

Rigidity of Surfaces with Constant Extrinsic Curvature in Riemannian Product Spaces

The present paper deals with complete surfaces having constant extrinsic curvature in a Riemannian product space $$M^{2}(c)\times {\mathbb {R}}$$ , where $$M^{2}(c)$$ is a space form with constant sectional curvature $$c\in \{-1,1\}$$ . In such setting, we find a Simons-type formula for Cheng-Yau’s operator which is used to prove that such surfaces are isometric to a cylinder $${\mathbb {H}}^{1}\times {\mathbb {R}}$$ , when $$c=-1$$ or isometric to a slice $${\mathbb {S}}^{2}\times \{t_{0}\}$$ for some $$t_{0}\in {\mathbb {R}}$$ when $$c=1$$ . Finally, we extend the result, when $$c=-1$$ , for the Weingarten linear case.

The Gauss map and second fundamental form of surfaces in a Lie group

In this article, we give the integrability conditions for the existence of an isometric immersion from an orientable simply connected surface having prescribed Gauss map and positive extrinsic curvature into some unimodular Lie groups. In particular, we discuss the case when the Lie group is the euclidean unit sphere $$\mathbb {S}^3$$ and establish a correspondence between simply connected surfaces having extrinsic curvature K, K different from 0 and $$-1$$ , immersed in $$\mathbb {S}^3$$ with simply connected surfaces having non-vanishing extrinsic curvature immersed in the euclidean space $$\mathbb {R}^3$$ . Moreover, we show that a surface isometrically immersed in $$\mathbb {S}^3$$ has positive constant extrinsic curvature if, and only if, its Gauss map is a harmonic map into the Riemann sphere.

Surfaces of Rotation with Constant Extrinsic Curvature in a Conformally Flat 3-Space

In this work, we relate the extrinsic curvature of surfaces with respect to the Euclidean metric and any metrics that are conformal to the Euclidean metric. We introduce the space \({\mathbb{E}_3}\) —the 3-dimensional real vector space equipped with a conformally flat metric that is a solution of the Einstein equation. We characterize the surfaces of rotation with constant extrinsic curvature in the space \({\mathbb{E}_3}\) . We obtain a one-parameter family of two-sheeted hyperboloids that are complete surfaces with zero extrinsic curvature in \({\mathbb{E}_3}\) . Moreover, we obtain a one-parameter family of cones and show that there exists another one-parameter family of complete surfaces of rotation with zero extrinsic curvature in \({\mathbb{E}_3}\) . Moreover, we show that there exist complete surfaces with constant negative extrinsic curvature in \({\mathbb{E}_3}\) . As an application we prove that there exist complete surfaces with Gaussian curvature K ≤ − e < 0, in contrast with Efimov’s Theorem for the Euclidean space, and Schlenker’s Theorem for the hyperbolic space.

Complete surfaces with positive extrinsic curvature in product spaces

We prove that every complete connected immersed surface with positive extrinsic curvature K in ℍ^2 × ℝ must be properly embedded, homeomorphic to a sphere or a plane and, in the latter case, study the behavior of the end. Then we focus our attention on surfaces with positive constant extrinsic curvature ( K -surfaces). We establish that the only complete K -surfaces in \mathbb{S}^2 × ℝ and ℍ^2 × ℝ are rotational spheres. Here are the key steps to achieve this. First height estimates for compact K -surfaces in a general ambient space \mathbb{M}^2 × ℝ with boundary in a slice are obtained. Then distance estimates for compact K -surfaces (and H -surfaces) in ℍ^2 × ℝ with boundary on a vertical plane are obtained. Finally we construct a quadratic form with isolated zeroes of negative index.

Linear and nonlinear optics in curved space

Starting from Maxwell equations in curved space, we derive the field equations of light when constrainted to a curved surface via a possible nonlinear waveguide. We show that surfaces with constant extrinsic curvature, in particular minimal surfaces, allow solutions with a well-defined polarization. We derive a generalized nonlinear Schr\odinger equation and solve the linear propagation for surfaces of revolution with constant positive and negative Gaussian curvature.

Perturbative evaluation of the zero-point function for self-interacting scalar field on a manifold with boundary

The character of quantum corrections to the gravitational action of a conformally invariant field theory for a self-interacting scalar field on a manifold with boundary is considered at third loop-order in the perturbative expansion of the zero-point function. Diagramatic evaluations and higher loop-order renormalization can be best accomplished on a Riemannian manifold of positive constant curvature accommodating a boundary of constant extrinsic curvature. The associated spherical formulation for diagramatic evaluations reveals a non-trivial effect which the topology of the manifold has on the vacuum processes and which ultimately dissociates the dynamical behaviour of the quantized field from its behaviour in the absence of a boundary. The first surface divergence is evaluated and the necessity for simultaneous renormalization of volume and surface divergences is shown.

Spherical formulation for diagrammatic evaluations on a manifold with boundary

The mathematical formalism necessary for the diagrammatic evaluation of quantum corrections to a conformally invariant field theory for a self-interacting scalar field on a curved manifold with boundary is considered. The evaluation of quantum corrections to the effective action past one-loop necessitates diagrammatic techniques. Diagrammatic evaluations and higher loop-order renormalization can be best accomplished on a Riemannian manifold of constant curvature accommodating a boundary of constant extrinsic curvature. In such a context, the stated evaluations can be accomplished through a consistent interpretation of the Feynman rules within the spherical formulation of the theory which the method of images allows. To this effect, the mathematical consequences of such an interpretation are analysed and the spherical formulation of the Feynman rules on the bounded manifold is, as a result, developed.

AdS/CFT and gravity

The radiation-dominated k=0 FRW cosmology emerges as the induced metric on a codimension one hypersurface of constant extrinsic curvature in the five-dimensional AdS-Schwarzschild solution. That we should get FRW cosmology in this way is an expected result from AdS/CFT in light of recent comments regarding the coupling of gravity to "boundary" conformal field theories. I remark on how this calculation bears on the understanding of Randall and Sundrum's "alternative to compactification." A generalization of the AdS/CFT prescription for computing Green's functions is suggested, and it is shown how gravity emerges from it with a strength G_4 = 2 G_5/L. Some numerical bounds are set on the radius of curvature L of AdS_5. One of them comes from estimating the rate of leakage of visible sector energy into the CFT. That rate is connected via a unitarity relation to deviations from Newton's force law at short distances. The best bound on L obtained in this paper comes from a match to the parameters of string theory. It is L < 1 nm if the string scale is 1 GeV. Higher string scales imply a tighter bound on L.

Radiative contributions to the effective action ofself-interacting scalar field on a manifold withboundary

The effect of quantum corrections to a conformally invariantfield theory for a self-interacting scalar field on a curvedmanifold with boundary is considered. The analysis is mosteasily performed in a space of constant curvature the boundaryof which is characterized by constant extrinsic curvature. Anextension of the spherical formulation in the presence of aboundary is attained through use of the method of images.Contrary to the consolidated vanishing effect in maximallysymmetric spacetimes the contribution of the massless `tadpole'diagram no longer vanishes in dimensional regularization. As aresult, conformal invariance is broken due to boundary-relatedvacuum contributions. The evaluation of 1-loop contributions tothe two-point function suggests an extension, in the presence ofmatter couplings, of the simultaneous volume and boundaryrenormalization in the effective action.

Initial-value constraints and generation of isometries and homothetic motions.

We discuss the generation of spacetime symmetries from an initial-value point of view. After reviewing the effective set of initial-value constraints (a combination of the familiar gravitational constraints and an additional set arising from projections of ${\mathcal{L}}_{\ensuremath{\xi}}g=\ensuremath{\alpha}g$ on the initial surface), we focus our attention on the construction of initial data generating spacetimes admitting homothetic motion. From the analysis of the constrained equations we show that there are only two classes of homothetic, spherically symmetric spacetimes ($M, g$) where (a) $M$ admits a flat Cauchy hypersurface $\ensuremath{\Sigma}$, (b) $g$ is a perfect-fluid solution of Einstein's equation with $P=k\ensuremath{\rho}$, $0\ensuremath{\le}k\ensuremath{\le}1$ as the equation of state and with an initial fourvelocity parallel to the normal of $\ensuremath{\Sigma}$, and (c) the projection of the infinitesimal homothetic generator $\ensuremath{\xi}$ along the normal of $\ensuremath{\Sigma}$ is everywhere a nonvanishing constant. The first class is the spatially flat Friedmann-Robertson-Walker (FRW) class of metrics. The second is a particular class of the marginally bound Tolman class of metrics. In combination with recent work by Eardley et al. [which shows the absence of spacetimes ($M, g$) with $g$ a nonvacuum homothetic solution of Einstein's equations and $M$ admitting a compact Cauchy hypersurface of constant extrinsic curvature], these results appear to suggest that only a "few" metrics (a) are compatible with Einstein's (nonvacuum) equations, and (b) allow the existence of a homothetic symmetry.