Spacelike Willmore surfaces in 4-dimensional Lorentzian space forms

Spacelike Willmore surfaces in 4-dimensional Lorentzian space forms, a topic in Lorentzian conformal geometry which parallels the theory of Willmore surfaces in $S^4$, are studied in this paper. We define two kinds of transforms for such a surface, which produce the so-called left/right polar surfaces and the adjoint surfaces. These new surfaces are again conformal Willmore surfaces. For them holds interesting duality theorem. As an application spacelike Willmore 2-spheres are classified. Finally we construct a family of homogeneous spacelike Willmore tori.Comment: 19 page

Complete stationary surfaces in R14\mathbb{R}^4_1 with total curvature −∫KdM=4π-\int K\mathrm{d}M=4\pi

Applying the general theory about complete spacelike stationary (i.e. zero mean curvature) surfaces in 4-dimensional Lorentz space $\mathbb{R}^4_1$, we classify those regular algebraic ones with total Gaussian curvature $-\int K\mathrm{d}M=4\pi$. Such surfaces must be oriented and be congruent to either the generalized catenoids or the generalized enneper surfaces. For non-orientable stationary surfaces, we consider the Weierstrass representation on the oriented double covering $\widetilde{M}$ (of genus $g$) and generalize Meeks and Oliveira's M\"obius bands. The total Gaussian curvature are shown to be at least $2\pi(g+3)$ when $\widetilde{M}\to\mathbb{R}^4_1$ is algebraic-type. We conjecture that there do not exist non-algebraic examples with $-\int K\mathrm{d}M=4\pi$.Comment: 22 page

Spacelike Willmore surfaces in 4-dimensional Lorentzian space forms

Spacelike Willmore surfaces in 4-dimensional Lorentzian space forms, a topic in Lorentzian conformal geometry which parallels the theory of Willmore surfaces in $S^4$, are studied in this paper. We define two kinds of transforms for such a surface, which produce the so-called left/right polar surfaces and the adjoint surfaces. These new surfaces are again conformal Willmore surfaces. For them holds interesting duality theorem. As an application spacelike Willmore 2-spheres are classified. Finally we construct a family of homogeneous spacelike Willmore tori.Comment: 19 page

Passive scheme with a photon-number-resolving detector for monitoring the untrusted source in a plug-and-play quantum-key-distribution system

A passive scheme with a beam splitter and a photon-number-resolving (PNR) detector is proposed to verify the photon statistics of an untrusted source in a plug-and-play quantum-key-distribution system by applying a three-intensity decoy-state protocol. The practical issues due to statistical fluctuation and detection noise are analyzed. The simulation results show that the scheme can work efficiently when the total number of optical pulses sent from Alice to Bob is above 10^8, and the dark count rate of the PNR detector is below 0.5 counts/pulse, which is realizable with current techniques. Furthermore, we propose a practical realization of the PNR detector with a variable optical attenuator combined with a threshold detector.Comment: 8 pages, 6 figure