Curvature Theorem Research Articles

The Spacelike-Characteristic Cauchy Problem of General Relativity in Low Regularity

In this paper we study the spacelike-characteristic Cauchy problem for the Einstein vacuum equations. Given initial data on a maximal spacelike hypersurface $$\Sigma \simeq \overline{B_1} \subset {{\mathbb {R}}}^3$$ and the outgoing null hypersurface $${{\mathcal {H}}}$$ emanating from $${\partial }\Sigma $$ , we prove a priori estimates for the resulting future development in terms of low-regularity bounds on the initial data at the level of curvature in $$L^2$$ . The proof uses the bounded $$L^2$$ curvature theorem [22], the extension procedure for the constraint equations [12], Cheeger-Gromov theory in low regularity [13], the canonical foliation on null hypersurfaces in low regularity [15] and global elliptic estimates for spacelike maximal hypersurfaces.

The Canonical Foliation On Null Hypersurfaces in Low Regularity

Let \({{\mathcal {H}}}\) denote the future outgoing null hypersurface emanating from a spacelike 2-sphere S in a vacuum spacetime \(({{\mathcal {M}}},\textbf{g})\). In this paper we study the so-called canonical foliation on \({{\mathcal {H}}}\) introduced in [13, 22] and show that the corresponding geometry is controlled locally only in terms of the initial geometry on S and the \(L^2\) curvature flux through \({{\mathcal {H}}}\). In particular, we show that the ingoing and outgoing null expansions \({\textrm{tr}}\chi \) and \({\textrm{tr}}{{{\underline{\chi }}}}\) are both locally uniformly bounded. The proof of our estimates relies on a generalisation of the methods of [15,16,17] and [1, 2, 26, 32] where the geodesic foliation on null hypersurfaces \({{\mathcal {H}}}\) is studied. The results of this paper, while of independent interest, are essential for the proof of the spacelike-characteristic bounded \(L^2\) curvature theorem [12].

The principal curvature theorem and its applications to constant mean curvature hypersurfaces in Euclidean space

The principal curvature theorem and its applications to constant mean curvature hypersurfaces in Euclidean space

A short note on a class of Weingarten hypersurfaces in $$\mathbb {R}^{n + 1}$$

We provide sharp bounds for the squared norm of the second fundamental form of a wide class of Weingarten hypersurfaces in Euclidean space satisfying $$H_r = aH + b$$ , for constants $$a, b \in \mathbb {R}$$ , where $$H_r$$ stands for the rth mean curvature and H the mean curvature of the hypersurface. Besides we are able to characterize those hypersurfaces for which these bounds are attained by showing that they must be a cylinder of the type $$\mathbb {R}\times \mathbb {S}^{n - 1}(\rho )$$ . Moreover, with the additional assumption that the Gauss–Kronnecker curvature of the hypersurface does not change sign, we prove that it is the cylinder $$\mathbb {R}\times \mathbb {S}^{n - 1}(\rho )$$ . In order to proof our results, we will use the method introduced in Alias and Garca-Martinez (Geom Dedic 156:31–47, 2012), Alias and Melendez (Geom Dedic 182:117–131, 2016), which is based on the so called principal curvature theorem due to Smyth and Xavier (Invent Math 90:443–450, 1987).

Analytic torsion for surfaces with cusps II. Regularity, asymptotics and curvature theorem

We study the Quillen metric on the determinant line bundle associated with a family of complex singular curves with hyperbolic cusp singularities.More precisely, we fix a family of complex curves, which admit at most double-point singularities. We endow the fibers of this family with Kähler metrics, which are defined away from a finite set of points, a divisor on the total space of the family, in the vicinity of which the metric has Poincaré-type singularities. We fix a holomorphic vector bundle over the total space of the family and endow it with a Hermitian metric with at most logarithmic singularities on the divisor, coming from the power of the relative canonical line bundle twisted by the divisor. An important example is the family of stable surfaces with marked points endowed with constant scalar curvature metric.The associated determinant line bundle is naturally endowed with the Quillen metric defined using the analytic torsion from the first paper of this series, generalizing the classical definition of Ray-Singer. We study the regularity of this Quillen metric near the locus of singular curves. The singularities turn out to be reasonable enough, so that the curvature of the Chern connection of the determinant line bundle endowed with the Quillen norm is well-defined as a current. We derive the explicit formula for this current, which gives a refinement of Riemann-Roch-Grothendieck theorem at the level of currents. This generalizes the curvature theorems of Takhtajan-Zograf and Bismut-Bost.

Analytic Torsion for Surfaces with Cusps I: Compact Perturbation Theorem and Anomaly Formula

Let $\overline{M}$ be a compact Riemann surface and let $g^{TM}$ be a metric over $\overline{M} \setminus D_M$, where $D_M \subset \overline{M}$ is a finite set of points. We suppose that $g^{TM}$ is equal to the Poincar\'e metric over a punctured disks around the points of $D_M$. The metric $g^{TM}$ endows the twisted canonical line bundle $\omega_M(D)$ with the induced Hermitian norm $\|\cdot\|_M$ over $\overline{M} \setminus D_M$. Let $(\xi, h^{\xi})$ be a holomorphic Hermitian vector bundle over $\overline{M}$. In this article we define the analytic torsion $T(g^{TM}, h^{\xi} \otimes \|\cdot\|_M^{2n})$ associated with $(M, g^{TM})$ and $(\xi \otimes \omega_M(D)^n, h^{\xi} \otimes \|\cdot\|_M^{2n})$ for $n \leq 0$. We prove that $T(g^{TM}, h^{\xi} \otimes \|\cdot\|_M^{2n})$ is related to the analytic torsion of non-cusped surfaces. Then we prove the anomaly formula for the associated Quillen norm. The results of this paper will be used in the sequel to study the regularity of the Quillen norm and its asymptotics in a degenerating family of Riemann surfaces with cusps and to prove the curvature theorem. We also prove that our definition of the analytic torsion for hyperbolic surfaces is compatible with the one obtained through Selberg trace formula by Takhtajan-Zograf.

Numerical simulation and experimental investigations on a three-roller setting round process for thin-walled pipes

In order to reveal the process principle of three-roller setting round process and optimize the setting round strategy, the three-roller setting round process is simulated and experimented. The results show that there are three positive bending regions and three reverse bending regions in the cross section of pipe in the setting round process. The absolute value of equivalent stress and equivalent strain not only decreases from both ends to the geometric neutral layer along the thickness direction of pipe but also decreases from the center of each positive and negative bending region to both sides along the circumferential direction of pipe. The distribution of maximum stress and minimum stress conforms to the characteristics of pure bending deformation in the setting round process. The direction of tangential stress is the direction of main stress, and the direction of tangential strain is the direction of main strain. The geometrical dimensions of pipe do not change in axial and radial directions. The residual ovality of pipes with different initial ovality is basically the same, which proves that the uniform curvature theorem of reciprocating bending is correct. The residual ovality of pipes decreases with the increase of the reduction. With the increase of the relative thickness of pipes, the optimum reduction of pipes decreases gradually. Comparing experimental results with simulation results, the residual ovality of pipes can be less than 0.2%.

The Localised Bounded $$L^2$$-Curvature Theorem

In this paper, we prove a localised version of the bounded $L^2$-curvature theorem of Klainerman-Rodnianski-Szeftel. More precisely, we consider initial data for the Einstein vacuum equations posed on a compact spacelike hypersurface $\Sigma$ with boundary, and show that the time of existence of a classical solution depends only on an $L^2$-bound on the Ricci curvature, an $L^4$-bound on the second fundamental form of $\partial \Sigma \subset \Sigma$, an $H^1$-bound on the second fundamental form, and a lower bound on the volume radius at scale $1$ of $\Sigma$. Our localisation is achieved by first proving a localised bounded $L^2$-curvature theorem for small data posed on $B(0,1)$, and then using the scaling of the Einstein equations and a low regularity covering argument on $\Sigma$ to reduce from large data on $\Sigma$ to small data on $B(0,1)$. The proof uses the author's previous work, and the bounded $L^2$-curvature theorem as black boxes.

Trajectory curvature guidance for Mars landings in hazardous terrains

Focusing on the trajectory curvature, this paper presents an innovative and analytical guidance law for the construction of geometrically convex trajectories. Moving along such trajectories, the lander can increase the probability of a safe landing in hazardous terrains. Initially, the curvature theorems of the powered descent trajectories are developed. In these theorems, the inner relationship between trajectory curvature and lander states is revealed, and the state constraints for a geometrically convex trajectory are derived. Next, the trajectory curvature guidance is developed in an analytical formulation by satisfying the state constraints for a convex trajectory, and the selection of the key guidance parameters is investigated. Finally, the performance of the trajectory curvature guidance is analyzed in detail, illustrating its superior hazard avoidance and the camera’s field of view.

Remarks on hypersurfaces with constant higher order mean curvature in Euclidean space

In this paper we consider complete oriented hypersurfaces of Euclidean space with constant higher order mean curvature and having two principal curvatures, one of them simple. As an application of the so called principal curvature theorem, a purely geometric result on the principal curvatures of the hypersurface given by Smyth and Xavier (Invent Math 90:443–450, 1987), we characterize those hypersurfaces for which the Gauss–Kronecker curvature does not change sign, extending to the general n-dimensional case a previous result for surfaces due to Klotz and Osserman.

PT-symmetry, indefinite metric, and nonlinear quantum mechanics

If a Hamiltonian of a quantum system is symmetric under space-time reflection, then the associated eigenvalues can be real. A conjugation operation for quantum states can then be defined in terms of space-time reflection, but the resulting Hilbert space inner product is not positive definite and gives rise to an interpretational difficulty. One way of resolving this difficulty is to introduce a superselection rule that excludes quantum states having negative norms. It is shown here that a quantum theory arising in this way gives an example of Kibble’s nonlinear quantum mechanics, with the property that the state space has a constant negative curvature. It then follows from the positive curvature theorem that the resulting quantum theory is not physically viable. This conclusion also has implications to other quantum theories obtained from the imposition of analogous superselection rules.

On complete hypersurfaces with constant mean and scalar curvatures in Euclidean spaces

Generalizing a theorem of Huang, Cheng and Wan classified the complete hypersurfaces of $\mathbb R^4$ with non-zero constant mean curvature and constant scalar curvature. In our work, we obtain results of this nature in higher dimensions. In particular, we prove that if a complete hypersurface of $\mathbb R^5$ has constant mean curvature $H\neq 0$ and constant scalar curvature $R\geq\frac{2}{3}H^2$, then $R=H^2$, $R=\frac{8}{9}H^2$ or $R=\frac{2}{3}H^2$. Moreover, we characterize the hypersurface in the cases $R=H^2$ and $R=\frac{8}{9}H^2$, and provide an example in the case $R=\frac{2}{3}H^2$. The proofs are based on the principal curvature theorem of Smyth-Xavier and a well known formula for the Laplacian of the squared norm of the second fundamental form of a hypersurface in a space form.

Warped products admitting a curvature bound

This paper completes a fundamental construction in Alexandrov geometry. Previously we gave a new construction of metric spaces with curvature bounds either above or below, namely warped products with intrinsic metric space base and fiber, and with possibly vanishing warping functions – thereby extending the classical cone and suspension constructions from interval base to arbitrary base, and furthermore encompassing gluing constructions. This paper proves the converse, namely, all conditions of the theorems are necessary. Note that in the cone construction, both the construction and its converse are widely used. We also show that our theorems for curvature bounded above and below, respectively, are dual. We give the first systematic development of basic properties of warped products of metric spaces with possibly vanishing warping functions, including new properties.

Uniform Curvature Theorem by Reciprocating Bending and Its Experimental Verification

Uniform Curvature Theorem by Reciprocating Bending and Its Experimental Verification

Hypersurfaces with constant higher order mean curvature in Euclidean space

In this paper we derive sharp estimates for the infimum and for the supremum of the squared norm of the second fundamental form of complete oriented hypersurfaces of Euclidean space with constant higher order mean curvature and having two principal curvatures, one of them simple. Besides, we characterize those hypersurfaces for which any of these bounds is attained. Our results will be an application of a purely geometric result on the principal curvatures of the hypersurface, the so called principal curvature theorem, given by Smyth and Xavier (Invent Math 90:443–450, 1987).

A discrete Gauss-Bonnet type theorem

We discuss a curvature theorem for subgraphs of the flat triangular tessellation of the plane. These graphs play the analogue of domains in two dimensional Euclidean space. We show that with the curvature K(p) = 2|S1(p)|−|S2(p)| the discrete Hopf Umlaufsatz ∑ p∈δG K(p) = 12χ(G) holds, where χ(G) is the Euler characteristic of the graph, δG is the boundary of G and where |Sr| is the arc length of the sphere of radius r in G. Dedicated to Ernst Specker to his 90th birthday.

Characterization of Holomorphic Bisectional Curvature ofGCR-Lightlike Submanifolds

We obtain the expressions for sectional curvature, holomorphic sectional curvature, and holomorphic bisectional curvature of aGCR-lightlike submanifold of an indefinite Kaehler manifold. We discuss the boundedness of holomorphic sectional curvature ofGCR-lightlike submanifolds of an indefinite complex space form. We establish a condition for aGCR-lightlike submanifold of an indefinite complex space form to be null holomorphically flat. We also obtain some characterization theorems for holomorphic sectional and holomorphic bisectional curvature.

Movement consensus of delayed multi‐agent systems with directed weighted networks

PurposeThe purpose of this paper is to study the moving consensus of multi‐agent dynamical systems with time delays and directed weighted networks.Design/methodology/approachThe approach used in the study, the topologies of multi‐agent dynamical systems with directed weighted networks is graph theories. The frequency domain is applied to research the movement characteristics of multi‐agent systems with time delays. The generalized Nyquist criterion and curvature theorem are utilized to analyze the consensus algorithm with heterogeneous input delays and heterogeneous communication delays.FindingsIt was discovered that the consensus for the delayed multi‐agent systems with asymmetric coupling weights can be achieved with the hypothesis of directed weighted network composed of n agents with a globally reachable node. The convergence condition is a decentralized consensus condition which uses only local information of each agent.Originality/valueThe novelty associated with this work is to present a new approach to study the consensus of delayed multi‐agent dynamical systems with directed weighted networks. The consensus condition obtained in the paper is less conservative than the consensus condition given in references.

Load Balancing by Network Curvature Control

The traditional heavy-tailed interpretation of congestion is challenged in this paper. A counter example shows that a network with uniform degree can have significant traffic congestion when the degree is larger than 6. A profound understanding of what causes congestion is reestablished, based on the network curvature theorem. A load balancing algorithm based on curvature control is presented with network applications.

Comparison geometry for the Bakry-Emery Ricci tensor

For Riemannian manifolds with a measure (M, g, edvolg) we prove mean curvature and volume comparison results when the ∞-Bakry-Emery Ricci tensor is bounded from below and f is bounded or ∂rf is bounded from below, generalizing the classical ones (i.e. when f is constant). This leads to extensions of many theorems for Ricci curvature bounded below to the Bakry-Emery Ricci tensor. In particular, we give extensions of all of the major comparison theorems when f is bounded. Simple examples show the bound on f is necessary for these results.