Isoparameteric hypersurfaces in a Randers sphere of constant flag curvature

In this paper, I study the isoparametric hypersurfaces in a Randers sphere $(S^n,F)$ of constant flag curvature, with the navigation datum $(h,W)$. I prove that an isoparametric hypersurface $M$ for the standard round sphere $(S^n,h)$ which is tangent to $W$ remains isoparametric for $(S^n,F)$ after the navigation process. This observation provides a special class of isoparametric hypersurfaces in $(S^n,F)$, which can be equivalently described as the regular level sets of isoparametric functions $f$ satisfying $-f$ is transnormal. I provide a classification for these special isoparametric hypersurfaces $M$, together with their ambient metric $F$ on $S^n$, except the case that $M$ is of the OT-FKM type with the multiplicities $(m_1,m_2)=(8,7)$. I also give a complete classificatoin for all homogeneous hypersurfaces in $(S^n,F)$. They all belong to these special isoparametric hypersurfaces. Because of the extra $W$, the number of distinct principal curvature can only be 1,2 or 4, i.e. there are less homogeneous hypersurfaces for $(S^n,F)$ than those for $(S^n,h)$.Comment: The newest version adds a referenc

Examples of flag-wise positively curved spaces

A Finsler space $(M,F)$ is called flag-wise positively curved, if for any $x\in M$ and any tangent plane $\mathbf{P}\subset T_xM$, we can find a nonzero vector $y\in \mathbf{P}$, such that the flag curvature $K^F(x,y, \mathbf{P})>0$. Though compact positively curved spaces are very rare in both Riemannian and Finsler geometry, flag-wise positively curved metrics should be easy to be found. A generic Finslerian perturbation for a non-negatively curved homogeneous metric may have a big chance to produce flag-wise positively curved metrics. This observation leads our discovery of these metrics on many compact manifolds. First we prove any Lie group $G$ such that its Lie algebra $\mathfrak{g}$ is compact non-Abelian and $\dim\mathfrak{c}(\mathfrak{g})\leq 1$ admits flag-wise positively curved left invariant Finsler metrics. Similar techniques can be applied to our exploration for more general compact coset spaces. We will prove, whenever $G/H$ is a compact simply connected coset space, $G/H$ and $S^1\times G/H$ admit flag-wise positively curved Finsler metrics. This provides abundant examples for this type of metrics, which are not homogeneous in general.Comment: 9 pages. In the newest version, Theorem 1.3 is strenghened to provide many more example

The number of geometrically distinct reversible closed geodesics on a Finsler sphere with K≡1K\equiv 1

In this paper we study the Finsler sphere $(S^n,F)$ with $n>1$, which has constant flag curvature $K\equiv 1$ and only finite prime closed geodesics. In this case, the connected isometry group $I_0(S^n,F)$ must be a torus which dimension satisfies $0<\dim I(S^n,F) \leq[\frac{n+1}{2}]$. We will prove that the number of geometrically distinct reversible closed geodesics on $(S^n,F)$ is at least $\dim I(S^n,F)$. When $\dim I_0(S^n,F)=[\frac{n+1}{2}]$, the equality happens, and there are exactly $2[\frac{n+1}{2}]$ prime closed geodesics, which verifies Anosov conjecture in this special case

Homogeneous Finsler spaces with only one orbit of prime closed geodesics

When a closed Finsler manifold admits continuous isometric actions, estimating the number of orbits of prime closed geodesics seems a more reasonable substitution for estimating the number of prime closed geodesics. To generalize the works of H. Duan, Y. Long, H.B. Rademacher, W. Wang and others on the existence of two prime closed geodesics to the equivariant situation, we purpose the question if a closed Finsler manifold has only one orbit of prime closed geodesic if and only if it is a compact rank-one Riemannian symmetric space. In this paper, we study this problem in homogeneous Finsler geometry, and get a positive answer when the dimension is even or the metric is reversible. We guess the rank inequality and algebraic techniques in this paper may continue to play an important role for discussing our question in the non-homogeneous situation.Comment: 29 page

Geodesic orbit spheres and constant curvature in Finsler geometry

In this paper, we generalize the classification of geodesic orbit spheres from Riemannian geometry to Finsler geometry. Then we further prove if a geodesic orbit Finsler sphere has constant flag curvature, it must be Randers. It provides an alternative proof for the classification of invariant Finsler metrics with $K\equiv1$ on homogeneous spheres other than $Sp(n)/Sp(n-1)$.Comment: In version 2 of this paper, a reference has been added, and a minor mistake in the introduction section has been correcte

Maxwell's equal area law for Lovelock Thermodynamics

We present the construction of Maxwell's equal area law for the Guass-Bonnet AdS black holes in $d=5,6$ and third order Lovelock AdS black holes in $d=7,8$. The equal area law can be used to find the number and location of the points of intersection in the plots of Gibbs free energy, so that we can get the thermodynamically preferred solution which corresponds to the first order phase transition. We obtain the radius of the small and large black holes in the phase transition which share the same Gibbs free energy. The case with two critical points is explored in much more details. The latent heat is also studied.Comment: 16 pages, 10 figures. V2: minor corrections and new references. V3: more discussion about the relationship between latent heat and temperature added. Accepted by IJMP

Early thermalization of quark-gluon matter by elastic 3-to-3 scattering

The early thermalization is crucial to the quark-gluon plasma as a perfect liquid and results from many-body scattering. We calculate squared amplitudes for elastic parton-parton-parton scattering in perturbative QCD. Transport equations with the squared amplitudes are established and solved to obtain the thermalization time of initially produced quark-gluon matter and the initial temperature of quark-gluon plasma. We find that the thermalization times of quark matter and gluon matter are different.Comment: 5 pages, 1 figure, proceedings for Extreme QCD 201

Early Thermalization at RHIC

Triple-gluon elastic scatterings are briefly reviewed since the scatterings explain the early thermalization puzzle in Au-Au collisions at RHIC energies. A numerical solution of the transport equation with the triple-gluon elastic scatterings demonstrates gluon momentum isotropy achieved at a time of the order of 0.65 fm/c. Triple-gluon scatterings lead to a short thermalization time of gluon matter.Comment: LaTex, 8 pages and 4 figures, talk presented in the Weihai workshop on relativistic heavy ion collision

Origin of Temperature of Quark-Gluon Plasma in Heavy Ion Collisions

Initially produced quark-gluon matter at RHIC and LHC does not have a temperature. A quark-gluon plasma has a high temperature. From this quark-gluon matter to the quark-gluon plasma is the early thermalization or the rapid creation of temperature. Elastic three-parton scattering plays a key role in the process. The temperature originates from the two-parton scattering, the three-parton scattering, the four-parton scattering and so forth in quark-gluon matter.Comment: 6 pages, proceedings for the XXX-th International Workshop on High Energy Physic

On Bondage Numbers of Graphs -- a survey with some comments

The bondage number of a nonempty graph $G$ is the cardinality of a smallest edge set whose removal from $G$ results in a graph with domination number greater than the domination number of $G$. This lecture gives a survey on the bondage number, including the known results, problems and conjectures. We also summarize other types of bondage numbers.Comment: 80 page; 14 figures; 120 reference