Existence of minimal surfaces of arbitrary large Morse index

We show that in a closed 3-manifold with a generic metric of positive Ricci curvature, there are minimal surfaces of arbitrary large Morse index, which partially confirms a conjecture by F. Marques and A. Neves. We prove this by analyzing the lamination structure of the limit of minimal surfaces with bounded Morse index.Comment: One figure, final versio

A maximum principle for free boundary minimal varieties of arbitrary codimension

We establish a boundary maximum principle for free boundary minimal submanifolds in a Riemannian manifold with boundary, in any dimension and codimension. Our result holds more generally in the context of varifolds.Comment: revised version, to appear in Comm. Anal. Geo

Ghost free massive gravity with singular reference metrics

An auxiliary metric (reference metric) is inevitable in massive gravity theory. In the scenario of gauge/gravity duality, a singular reference metric corresponds to momentum dissipations, which describes the electric and heat conductivity for normal conductors. We demonstrate in detail that the massive gravity with singular reference metric is ghost-free.Comment: 6pages, no figur

Absolutely split metacyclic groups and weak metacirculants

Let $m,n,r$ be positive integers, and let $G=\langle a\rangle: \langle b\rangle \cong \mathbb{Z}_n: \mathbb{Z}_m$ be a split metacyclic group such that $b^{-1}ab=a^r$. We say that $G$ is {\em absolutely split with respect to $\langle a\rangle$} provided that for any $x\in G$, if $\langle x\rangle\cap\langle a\rangle=1$, then there exists $y\in G$ such that $x\in\langle y\rangle$ and $G=\langle a\rangle: \langle y\rangle$. In this paper, we give a sufficient and necessary condition for the group $G$ being absolutely split. This generalizes a result of Sanming Zhou and the second author in [arXiv: 1611.06264v1]. We also use this result to investigate the relationship between metacirculants and weak metacirculants. Metacirculants were introduced by Alspach and Parsons in $1982$ and have been a rich source of various topics since then. As a generalization of this classes of graphs, Maru\v si\v c and \v Sparl in 2008 posed the so called weak metacirculants. A graph is called a {\em weak metacirculant} if it has a vertex-transitive metacyclic automorphism group. In this paper, it is proved that a weak metacirculant of $2$-power order is a metacirculant if and only if it has a vertex-transitive split metacyclic automorphism group. This provides a partial answer to an open question in the literature

Quasinormal Modes and Late-Time Tails of Canonical Acoustic Black Holes

In this paper, we investigate the evolution of classical wave propagation in the canonical acoustic black hole by numerical method and discuss the details of tail phenomenon. The oscillating frequency and damping time scale both increase with the angular momentum $l$. For the lower $l$, numerical results show the lowest WKB approximation gives the most reliable result. We also find that time scale of the interim region from ringing to tail is not affected obviously by changing $l$.Comment: 5 pages, 6 figure

Anisotropy of incommensurate spin and charge fluctuations in detwinned YBa2_2Cu3_3O6+δ_{6+\delta}

Motivated by a recent neutron scattering experiment on detwinned YBa$_2$Cu$_3$O$_{6+\delta}$ superconductor, we examine the frequency and doping dependence of the anisotropy in the spin and charge fluctuation arising from the coupling between the plane and the chain. Starting from the two-dimensional $t$-$t^{'}$-$J$ model and using the random-phase approximation (RPA), we find a pronounced anisotropy of the incommensurate (IC) peaks in the spin channel, namely the peak intensity at the $(\pi\pm\delta,\pi)$ direction is stronger than that at the $(\pi,\pi\pm\delta)$ direction in a wide frequency range from $\omega=0.2J$ to the resonance frequency $\omega_r=0.35J$. Above the resonance frequency, the IC peaks reemerge. Their intensities shift to the diagonal direction and no anisotropy exists along the two diagonal directions. We find that this anisotropy is robust with respect to the possible variation of the RPA correction factor and to the dopings. The charge fluctuation is also found to be incommensurate for all energies considered and peak at $(0,\delta)$ and $(\delta,0)$. An anisotropy in its IC peak intensity along the $k_x$ and the $k_y$ direction exists, but in sharp contrast to the spin channel, the maximum intensity of the IC peak is along the $k_y$ direction. Moreover, the IC peak in the charge channel exhibits an upward dispersion, in contrast to the downward dispersion below the spin resonance frequency for the spin IC peak. We explain these features based on the effect of the plane-chain coupling on the topology of the Fermi surface.Comment: 8 page

De Sitter ground state of scalar-tensor gravity and its fluctuation with dust

An exact de Sitter solution of scalar-tensor gravity is found, in which the non-minimal coupling scalar is rolling along a non-constant potential. Based on this solution, a dust-filled FRW universe is explored in frame of scalar-tensor gravity. The effective dark energy induced by the sole non-minimal scalar can be quintessence-like, phantom-like, and more significantly, can cross the phantom divide. The rich and varied properties of scalar-tensor gravity even with only one scalar is shown in this article.Comment: 9 pages, 9fig

Global monopoles in the Brans-Dicke theory

A gravitating global monopole produces a repulsive grativational field outside the core in addition to a solid angular deficit in the Brans-Dicke theory. As a new feature, the angular deficit is dependent on the values of \phi_{\infty} and \omega, where \phi_{\infty} is asymptotic value of scalar field in space-like infinity and \omega is the Brans-Dicke parameter.Comment: Latex 6 pages, To be published in Phys. Rev.

Black hole solutions in de Rham-Gabadadze-Tolley massive gravity

We present a detailed study of the static spherically symmetric solutions in de Rham-Gabadadze-Tolley (dRGT) theory. Since the diffeomorphism invariance can be restored by introducing the St\"{u}ckelberg fields $\phi^a$, there is new invariant $I^{ab}=g^{\mu\nu}\partial_{\mu}\phi^a\partial_\nu\phi^b$ in the massive gravity, which adds to the ones usually encountered in general relativity (GR). In the unitary gauge $\phi^a=x^\mu\delta_\mu^a$, any inverse metric $g^{\mu\nu}$ that has divergence including the coordinate singularity in GR would exhibit a singularity in the invariant $I^{ab}$. Therefore, there is no conventional Schwarzschild metric if we choose unitary gauge. In this paper, we obtain a self-consistent static spherically symmetric ansatz in the nonunitary gauge. Under this ansatz, we find that there are seven solutions including the Schwarzschild solution, Reissner-Nordstr\"{o}m solution and five other solutions. These solutions may possess an event horizon depending upon the physical parameters (Schwarzschild radius $r_s$, scalar charge $S$ and/or electric charge $Q$). If these solutions possess an event horizon, we show that the singularity of $I^{ab}$ is absent at the horizon. Therefore, these solutions may become candidates for black holes in dRGT.Comment: 12pages, no figur

Analytical expression for a class of spherically symmetric solutions in Lorentz breaking massive gravity

We present a detailed study of the spherically symmetric solutions in Lorentz breaking massive gravity. There is an undetermined function $\mathcal{F}(X, w_1, w_2, w_3)$ in the action of St\"{u}ckelberg fields $S_{\phi}=\Lambda^4\int{d^4x\sqrt{-g}\mathcal{F}}$, which should be resolved through physical means. In the general relativity, the spherically symmetric solution to the Einstein equation is a benchmark and its massive deformation also play a crucial role in Lorentz breaking massive gravity. $\mathcal{F}$ will satisfy the constraint equation $T_0^1=0$ from the spherically symmetric Einstein tensor $G_0^1=0$, if we maintain that any reasonable physical theory should possess the spherically symmetric solutions. The St\"{u}ckelberg field $\phi^i$ is taken as a 'hedgehog' configuration $\phi^i=\phi(r)x^i/r$, whose stability is guaranteed by the topological one. Under this ans\"{a}tz, $T_0^1=0$ is reduced to $d\mathcal{F}=0$. The functions $\mathcal{F}$ for $d\mathcal{F}=0$ form a commutative ring $R^{\mathcal{F}}$. We obtain a general expression of solution to the functional differential equation with spherically symmetry if $\mathcal{F}\in R^{\mathcal{F}}$. If $\mathcal{F}\in R^{\mathcal{F}}$ and $\partial\mathcal{F}/\partial X=0$, the functions $\mathcal{F}$ form a subring $S^{\mathcal{F}}\subset R^{\mathcal{F}}$. We show that the metric is Schwarzschild, AdS or dS if $\mathcal{F}\in S^{\mathcal{F}}$. When $\mathcal{F}\in R^{\mathcal{F}}$ but $\mathcal{F}\notin S^{\mathcal{F}}$, we will obtain some new metric solutions. Using the general formula and the basic property of function ring $R^{\mathcal{F}}$, we give some analytical examples and their phenomenological applications. Furthermore, we also discuss the stability of gravitational field by the analysis of Komar integral and the results of QNMs.Comment: A typo is corrected. Some sentences are polished. 26pages, 1 figure. arXiv:1503.08952 [gr-qc