Complete self-shrinkers with bounded the second fundamental form in Rn+1\mathbb{R}^{n+1}

Let $X:M^n\to \mathbb{R}^{n+1}$ be a complete properly immersed self-shrinker. In this paper, we prove that if the squared norm of the second fundamental form $S$ satisfies $1\leq S< C$ for some constant $C$, then $S=1$. Further we classify the $n$-dimensional complete proper self-shrinkers with constant squared norm of the second fundamental form in $\mathbb{R}^{n+1}$, which solve the conjecture proposed by Q.M. Cheng and G. Wei when the self-shrinker is proper.Comment: 17 pages, 0 figure, All comments are welcom

A Liouville theorem for the fractional Ginzburg-Landau equation

In this paper, we are concerned with a Liouville-type result of the nonlinear integral equation \begin{equation*} u(x)=\int_{\mathbb{R}^{n}}\frac{u(1-|u|^{2})}{|x-y|^{n-\alpha}}dy, \end{equation*} where $u: \mathbb{R}^{n} \to \mathbb{R}^{k}$ with $k \geq 1$ and $1<\alpha<n/2$. We prove that $u \in L^2(\mathbb{R}^n) \Rightarrow u \equiv 0$ on $\mathbb{R}^n$, as long as $u$ is a bounded and differentiable solution.Comment: 7 page

Exposure to various abscission-promoting treatments suggests substantial ERF subfamily transcription factors involvement in the regulation of cassava leaf abscission

AP2/ERF genes that exhibited the same expression patterns during ethylene and water-deficit stress treatments. (XLS 19 kb

Spectroscopic study of light scattering in linear alkylbenzene for liquid scintillator neutrino detectors

We has set up a light scattering spectrometer to study the depolarization of light scattering in linear alkylbenzene. From the scattering spectra it can be unambiguously shown that the depolarized part of light scattering belongs to Rayleigh scattering. The additional depolarized Rayleigh scattering can make the effective transparency of linear alkylbenzene much better than it was expected. Therefore sufficient scintillation photons can transmit through the large liquid scintillator detector of JUNO. Our study is crucial to achieving the unprecedented energy resolution 3\%/$\sqrt{E\mathrm{(MeV)}}$ for JUNO experiment to determine the neutrino mass hierarchy. The spectroscopic method can also be used to judge the attribution of the depolarization of other organic solvents used in neutrino experiments.Comment: 6 pages, 5 figure