Flat Manifolds Research Articles

Harmonic Functions and the Mass of 3-Dimensional Asymptotically Flat Riemannian Manifolds

An explicit lower bound for the mass of an asymptotically flat Riemannian 3-manifold is given in terms of linear growth harmonic functions and scalar curvature. As a consequence, a new proof of the positive mass theorem is achieved in dimension three. The proof has parallels with both the Schoen–Yau minimal hypersurface technique and Witten’s spinorial approach. In particular, the role of harmonic spinors and the Lichnerowicz formula in Witten’s argument is replaced by that of harmonic functions and a formula introduced by the fourth named author in recent work, while the level sets of harmonic functions take on a role similar to that of the Schoen–Yau minimal hypersurfaces.

A σ2 Penrose inequality for conformal asymptotically hyperbolic 4-discs

In this paper, we consider conformal flat metrics on R4 with an asymptotically hyperbolic (AH) end and possible isolated conic singularities. We define a mass term of the AH end. If the σ2 curvature has lower bound σ2≥32, we prove an inequality relating the mass and contributions from singularities. We also classify sharp cases, which is the standard hyperbolic 4-space H4 when no singularity occurs. It is worth noting that our curvature condition implies non-positive energy density.

Convergence of the Ricci flow on asymptotically flat manifolds with integral curvature pinching

We prove a curvature pinching result for the Ricci flow on asymptotically flat manifolds: if an asymptotically flat manifold of dimension $n\geq 3$ has scale-invariant integral norm of curvature sufficiently pinched relative to the inverse of its Sobolev constant, then the Ricci flow starting from this manifold exists for all positive times and converges to flat Euclidean space. In particular our result implies that the initial manifold must have been diffeomorphic to $\mathbb{R}^n$.

Generalized Oppenheimer–Snyder gravitational collapse into regular black holes

We shall study the formation of a particular class of regular black holes (BHs) from the gravitational collapse of a massive star. The inside geometry is described by spatially flat Friedmann–Robertson–Walker (FRW) metric and the stellar matter is distributed uniformly without any pre-assumption about its equation of state. Our model is a generalization of Oppenheimer–Snyder collapse for regular BHs. We have obtained the density and pressure of star by applying the condition of smooth joining of metrics at the freely falling surface of star. Specifying the regular BHs to Hayward and Bardeen cases, we see that the stellar matter is described by a polytropic equation of state and moreover, for the radius smaller than a certain value, the strong energy condition becomes invalid. Then for both BHs, the interior apparent and event horizons and also the stellar surface are obtained as functions of the proper time of star. At the end, we have constructed a new two parametric family of regular BHs jointed smoothly to the flat FRW interior metric of a polytropic star with an arbitrary index.

On the Cerbo Conjecture on Lie Groups with the Left-Invariant Lorentzian Metric

Manifolds with constraints on tensor fields include Einstein manifolds, Einstein-like manifolds, conformally flat manifolds, and a number of other important classes of manifolds. The work of many mathematicians is devoted to the study of such manifolds, which is reflected in the monographs of A. Besse, M. Berger, M.-D. Cao, M. Wang.
 Ricci solitons are one of the natural generalizations of Einstein's metrics. If a Riemannian manifold is a Lie group, one speaks of invariant Ricci solitons.
 Invariant Ricci solitons were studied in most detail in the case of unimodular Lie groups with left-invariant Riemannian metrics and the case of low dimension. Thus, L. Cerbo proved that all invariant Ricci solitons are trivial on unimodular Lie groups with left-invariant Riemannian metric and Levi-Civita connection.A similar result up to dimension four was obtained by P.N. Klepikov and D.N. Oskorbin for the non-unimodular case.
 We study invariant Ricci solitons on three-dimensional unimodular Lie groups with the Lorentzian metric.The study results show that unimodular Lie groups with left-invariant Lorentzian metric admit invariant Ricci solitons other than trivial ones. In this paper, a complete classification of invariant Ricci solitons on three-dimensional unimodular Lie groups with leftinvariant Lorentzian metric is obtained.

Hyperbolic geometry of shapes of convex bodies

We use the intrinsic area to define a distance on the space of homothety classes of convex bodies in the $n$-dimensional Euclidean space, which makes it isometric to a convex subset of the infinite dimensional hyperbolic space. The ambient Lorentzian structure is an extension of the intrinsic area form of convex bodies, and Alexandrov–Fenchel inequality is interpreted as the Lorentzian reversed Cauchy–Schwarz inequality. We deduce that the space of similarity classes of convex bodies has a proper geodesic distance with curvature bounded from below by $-1$ (in the sense of Alexandrov). In dimension 3, this space is homeomorphic to the space of distances with non-negative curvature on the 2-sphere, and this latter space contains the space of flat metrics on the 2-sphere considered by W. P. Thurston. Both Thurston’s and the area distances rely on the area form. So the latter may be considered as a generalization of the “real part” of Thurston’s construction.

Isoperimetry for asymptotically flat 3-manifolds with positive ADM mass

Let $$(M^3, g)$$ be an asymptotically flat 3-manifold with positive ADM mass. In this paper, we show that each leaf of the canonical foliation consisting of stable constant mean curvature spheres is the unique isoperimetric surface for the volume it encloses. Our proof is based on “fill-in” argument and sharp isoperimetric inequality on asymptotically flat 3-manifold with non-negative scalar curvature.

New Characterizations of the Whitney Spheres and the Contact Whitney Spheres

In this paper, based on the classical Yano’s formula, we first establish an optimal integral inequality for compact Lagrangian submanifolds in the complex space forms, which involves the Ricci curvature in the direction \(J\mathbf {H}\) and the norm of the covariant differentiation of the second fundamental form h, where J is the almost complex structure and \(\mathbf {H}\) is the mean curvature vector field. Second and analogously, for compact Legendrian submanifolds in the Sasakian space forms with Sasakian structure \((\varphi ,\xi ,\eta ,g)\), we also establish an optimal integral inequality involving the Ricci curvature in the direction \(\varphi \mathbf {H}\) and the norm of the modified covariant differentiation of the second fundamental form. The integral inequality is optimal in the sense that all submanifolds attaining the equality are completely classified. As direct consequences, we obtain new and global characterizations for the Whitney spheres in complex space forms as well as the contact Whitney spheres in Sasakian space forms. Finally, we show that, just as the Whitney spheres in complex space forms, the contact Whitney spheres in Sasakian space forms are locally conformally flat manifolds with sectional curvatures non-constant.

∗-Ricci Tensor onα-Cosymplectic Manifolds

In this paper, we study α-cosymplectic manifoldMadmitting∗-Ricci tensor. First, it is shown that a∗-Ricci semisymmetric manifoldMis∗-Ricci flat and aϕ-conformally flat manifoldMis anη-Einstein manifold. Furthermore, the∗-Weyl curvature tensorW∗onMhas been considered. Particularly, we show that a manifoldMwith vanishing∗-Weyl curvature tensor is a weakϕ-Einstein and a manifoldMfulfilling the conditionRE1,E2⋅W∗=0isη-Einstein manifold. Finally, we give a characterization for α-cosymplectic manifoldMadmitting∗-Ricci soliton given as to be nearly quasi-Einstein. Also, some consequences for three-dimensional cosymplectic manifolds admitting∗-Ricci soliton and almost∗-Ricci soliton are drawn.

On a constant curvature statistical manifold

We will show that a statistical manifold $$(M, g, \nabla )$$ has a constant curvature if and only if it is a projectively flat conjugate symmetric manifold, that is, the affine connection $$\nabla $$ is projectively flat and the curvatures satisfies $$R=R^*$$ , where $$R^*$$ is the curvature of the dual connection $$\nabla ^*$$ . Moreover, we will show that properly convex structures on a projectively flat compact manifold induces constant curvature $$-1$$ statistical structures and vice versa.

Properties of the combinatorial Hantzsche-Wendt groups

The combinatorial Hantzsche-Wendt group Gn=〈x1,...,xn|xi−1xj2xixj2,∀i≠j〉 was defined by W. Craig and P.A. Linnell in [4]. For n=2 it is a fundamental group of 3-dimensional oriented flat manifold with non cyclic holonomy group. We calculate the Hilbert-Poincaré series of Gn,n≥1 with Q and F2 coefficients. Moreover, we prove that the cohomological dimension of Gn is equal to n+1. Some other properties of this group are also considered.

Bavard's systolically extremal Klein bottles and three dimensional applications

A compact manifold is called Bieberbach if it carries a flat Riemannian metric. Bieberbach manifolds satisfy an isosystolic inequality by a general and fundamental result of M. Gromov. In dimension 3, there exist four classes of non-orientable Bieberbach manifolds up to an affine diffeomorphism. In this paper, we prove the existence on each diffeomorphism class of non-orientable Bieberbach 3-manifolds of a two-parameter family of singular Riemannian metrics that are systolically extremal in their conformal class. The proof uses a one-parameter family of singular Riemannian metrics on the Klein bottle discovered by C. Bavard ([3]): each one of these metrics is extremal in its conformal class.

Global uniqueness of large stable CMC spheres in asymptotically flat Riemannian 3-manifolds

Let (M,g) be a complete Riemannian 3-manifold that is asymptotic to Schwarzschild with positive mass and whose scalar curvature vanishes. We unconditionally characterize the large, embedded stable constant mean curvature (CMC) spheres in (M,g).

Towers of Bubbles for Yamabe-Type Equations and for the Brézis–Nirenberg Problem in Dimensions $$n \ge 7$$

Let (M, g) be a closed locally conformally flat Riemannian manifold of dimension $$n \ge 7$$ and of positive Yamabe type. If $$h \in C^1(M)$$ and $$\xi _0$$ is a non-degenerate critical point of the mass function we prove the existence, for any $$ k \ge 1$$ of a positive blowing-up solution $$u_\varepsilon $$ of $$\begin{aligned} \triangle _g u_\varepsilon +\big ( c_n S_g +\varepsilon h\big ) u_\varepsilon = u_\varepsilon ^{2^*-1} \end{aligned}$$ that blows up, as $$\varepsilon \rightarrow 0$$ , like the superposition of k positive bubbles concentrating at different speeds at $$\xi _0$$ . The method of proof combines a finite-dimensional reduction with the sharp pointwise analysis of solutions of a linear problem. As another application of this method of proof we construct sign-changing blowing-up solutions $$u_\varepsilon $$ for the Brézis–Nirenberg problem $$\begin{aligned} \triangle _{\xi } u_\varepsilon - \varepsilon u_\varepsilon = |u_\varepsilon |^{\frac{4}{n-2}} u_\varepsilon \text { in } \Omega , \quad u_\varepsilon = 0 \text { on } \partial \varOmega \end{aligned}$$ on a smooth bounded open set $$\varOmega \subset {\mathbb {R}}^n$$ , $$n \ge 7$$ , that look like the superposition of k positive bubbles of alternating sign as $$\varepsilon \rightarrow 0$$ .

Geodesic and Conformally Reeb Vector Fields on Flat 3-Manifolds

A unit vector field on a Riemannian manifold $M$ is called geodesic if all of its integral curves are geodesics. We show, in the case of $M$ being a flat 3-manifold not equal to $\mathbb{E}^3$, that every such vector field is tangent to a 2-dimensional totally geodesic foliation. Furthermore, it is shown that a geodesic vector field $X$ on a closed flat 3-manifold is (up to rescaling) the Reeb vector field of a contact form if and only if there is a contact structure transverse to $X$ that is given as the orthogonal complement of some other geodesic vector field. An explicit description of the lifted contact structures (up to diffeomorphism) on the 3-torus is given in terms of the volume of $X$. Finally, similar results for non-closed flat 3-manifolds are discussed.

A Spherically Symmetric Gravitational Solution of Nearly Conformally Flat Metric Measure Space

A Spherically Symmetric Gravitational Solution of Nearly Conformally Flat Metric Measure Space

Locally Conformally Flat Doubly Twisted Product Complex Finsler Manifolds

Let be a doubly twisted product manifold of two strongly pseudoconvex complex Finsler manifolds (M1, F1) and (M2, F2). In this study, we give a characterization of locally conformally flat doubly twisted product complex Finsler manifold. We also obtain a necessary and sufficient condition for a doubly twisted product of two locally conformally flat complex Finsler manifolds to be locally conformally flat.

Riemannian concircular structure manifolds

In this manuscript, we give the definition of Riemannian concircular structure manifolds. Some basic properties and integrability condition of such manifolds are established. It is proved that a Riemannian concircular structure manifold is semisymmetric if and only if it is concircularly flat. We also prove that the Riemannian metric of a semisymmetric Riemannian concircular structure manifold is a generalized soliton. In this sequel, we show that a conformally flat Riemannian concircular structure manifold is a quasi-Einstein manifold and its scalar curvature satisfies the partial differential equation ?r = ?2r/?t2 + ?(n?1)?r/?t. To validate the existence of Riemannian concircular structure manifolds, we present some non-trivial examples. In this series, we show that a quasi-Einstein manifold with a divergence free concircular curvature tensor is a Riemannian concircular structure manifold.

Gradient Ricci–Yamabe Soliton on Twisted Product Manifolds

In this paper, we study the twisted product manifolds with gradient Ricci–Yamabe solitons. Then, we classify and characterize the warped product and twisted product spaces with gradient Ricci–Yamabe solitons. We also study the construction of the model space of gradient Ricci–Yamabe solitons in the Riemannian product manifolds and the warped product manifolds. Finally, we study the geometric characterization of the conformally flat twisted product manifolds with gradient Ricci–Yamabe solitons.

Some almost-Schur type inequalities and applications on sub-static manifolds

<abstract><p>In this paper, we establish some almost-Schur type inequalities on sub-static manifolds, naturally arising in General Relativity. In particular, our results generalize those in <sup>[<xref ref-type="bibr" rid="b1">1</xref>]</sup> of Li-Xia for $ r $-th mean curvatures of closed sub-static hypersurfaces in space forms and $ k $-scalar curvatures for closed locally conformally flat sub-static manifolds. Moreover, our results also generalize those of Cheng <sup>[<xref ref-type="bibr" rid="b2">2</xref>]</sup>.</p></abstract>