Umbilic hypersurfaces of constant sigma-k curvature in the Heisenberg group

We study immersed, connected, umbilic hypersurfaces in the Heisenberg group $H_{n}$ with $n$ $\geq$ $2.$ We show that such a hypersurface, if closed, must be rotationally invariant up to a Heisenberg translation. Moreover, we prove that, among others, Pansu spheres are the only such spheres with positive constant sigma-k curvature up to Heisenberg translations.Comment: 28 pages, 6 figure

Umbilicity and characterization of Pansu spheres in the Heisenberg group

For $n\geq 2$ we define a notion of umbilicity for hypersurfaces in the Heisenberg group $H_{n}$. We classify umbilic hypersurfaces in some cases, and prove that Pansu spheres are the only umbilic spheres with positive constant $p$(or horizontal)-mean curvature in $H_{n}$ up to Heisenberg translations.Comment: 32 pages, 2 figures; in Crelle's journal, 201

A Codazzi-like equation and the singular set for C1C^{1} smooth surfaces in the Heisenberg group

In this paper, we study the structure of the singular set for a $C^{1}$ smooth surface in the $3$-dimensional Heisenberg group $\boldsymbol{H}_{1}$. We discover a Codazzi-like equation for the $p$-area element along the characteristic curves on the surface. Information obtained from this ordinary differential equation helps us to analyze the local configuration of the singular set and the characteristic curves. In particular, we can estimate the size and obtain the regularity of the singular set. We understand the global structure of the singular set through a Hopf-type index theorem. We also justify that Codazzi-like equation by proving a fundamental theorem for local surfaces in $\boldsymbol{H}_{1}$.Comment: 64 pages, 17 figure

Bidirectional outflows as evidence of magnetic reconnection leading to a solar microflare

Magnetic reconnection is a rapid energy release process that is believed to be responsible for flares on the Sun and stars. Nevertheless, such flare-related reconnection is mostly detected to occur in the corona, while there have been few studies concerning the reconnection in the chromosphere or photosphere. Here we present both spectroscopic and imaging observations of magnetic reconnection in the chromosphere leading to a microflare. During the flare peak time, chromospheric line profiles show significant blueshifted/redshifted components on the two sides of the flaring site, corresponding to upflows and downflows with velocities of $\pm$(70--80) km s$^{-1}$, comparable with the local Alfv\'{e}n speed as expected by the reconnection in the chromosphere. The three-dimensional nonlinear force-free field configuration further discloses twisted field lines (a flux rope) at a low altitude, cospatial with the dark threads in He I 10830 \r{A} images. The instability of the flux rope may initiate the flare-related reconnection. These observations provide clear evidence of magnetic reconnection in the chromosphere and show the similar mechanisms of a microflare to those of major flares.Comment: 16 pages, 5 figures, accepted for publication in ApJ

Synthesizing and characterization of hole doped nickel based layer superconductor (La1−x_{1-x}Srx_{x})ONiAs

We report the synthesizing and characterization of the hole doped Ni-based superconductor ($La_{1-x}Sr_{x})ONiAs$. By substituting La with Sr, the superconducting transition temperature $T_c$ is increased from 2.75 K of the parent phase $LaONiAs$ to 3.7 K at the doping levels x= 0.1 - 0.2. The curve $T_c$ versus hole concentration shows a symmetric behavior as the electron doped samples $La(O_{1-x}F_{x})NiAs$. The normal state resistivity in Ni-based samples shows a good metallic behavior and reveals the absence of an anomaly which appears in the Fe-based system at about 150 K, suggesting that this anomaly is not a common feature for all systems. Hall effect measurements indicate that the electron conduction in the parent phase $LaONiAs$ is dominated by electron-like charge carriers, while with more Sr doping, a hole-like band will emerge and finally prevail over the conduction, and accordingly the superconducting transition temperature $T_c$ increases.Comment: 4 pages, 5 figure