Existence of 2D Skyrmions

In this paper, we consider the 2D Skyrme model $$E(u)=\frac{1}{2} \int\limits_{R^2}|du|^2dx+\frac{\lambda}{4} \int\limits_{R^2}|du\wedge du|^2dx+\frac{\mu}{16} \int\limits_{R^2}|u-{\bf{n}}|^4dx,$$ where ?? and??? > 0 are positive coupling constants and n = (0, 0, 1) is the north pole of S ². We derive a lower bound of 2D Skyrme model. Using this estimate, we prove the existence of 2D Skyrmion for any positive coupling constants ??, ??.     

The Blow-up Locus of Heat Flows for Harmonic Maps

Abstract Let M and N be two compact Riemannian manifolds. Let u _k (x, t) be a sequence of strong stationary weak heat flows from M×R ⁺ to N with bounded energies. Assume that u _k→u weakly in H ^{1, 2}(M×R ⁺, N) and that Σ^t is the blow-up set for a fixed t > 0. In this paper we first prove Σ^t is an H ^m−2-rectifiable set for almost all t∈R ⁺. And then we prove two blow-up formulas for the blow-up set and the limiting map. From the formulas, we can see that if the limiting map u is also a strong stationary weak heat flow, Σ^t is a distance solution of the (m− 2)-dimensional mean curvature flow [1]. If a smooth heat flow blows-up at a finite time, we derive a tangent map or a weakly quasi-harmonic sphere and a blow-up set ∪_t<0Σ^t× {t}. We prove the blow-up map is stationary if and only if the blow-up locus is a Brakke motion. This work is supported by NSF grant     

An analysis of the two-vortex case in the Chern-Simons Higgs model

Extending work of Caffarelli-Yang and Tarantello, we present a variational existence proof for two-vortex solutions of the periodic Chern-Simons Higgs model and analyze the asymptotic behavior of these solutions as the parameter coupling the gauge field with the scalar field tends to 0. Received September 24, 1997 / Accepted October 2, 1997     

Nonexistence of quasi-harmonic spheres with large energy

The absence of quasi-harmonic spheres is necessary for long time existence and convergence of harmonic map heat flows. Let (N, h) be a complete noncompact Riemannian manifold. Assume the universal covering of (N, h) admits a nonnegative strictly convex function with polynomial growth. Then there is no non-constant quasi-harmonic sphere ${u:\mathbb{R}^n\rightarrow N}$ such that $$\lim_{r \rightarrow \infty}r^ne^{-\frac{r^2}{4}}\int \limits_{|x|\leq r}e^{-\frac{|x|^2}{4}}|\nabla u|^2{\text {d}}x\,=\,0.$$ This generalizes a result of the first author and X. Zhu (Calc. Var., 2009). Our method is essentially the Moser iteration and thus comparatively elementary.     

Symplectic mean curvature flows in Kähler surfaces with positive holomorphic sectional curvatures

In this paper, we mainly study the mean curvature flow in Kähler surfaces with positive holomorphic sectional curvatures. We prove that if the ratio of the maximum and the minimum of the holomorphic sectional curvatures is less than $2$ , then there exists a positive constant $\delta $ depending on the ratio such that $\cos \alpha \ge \delta $ is preserved along the flow.     

Non existence of quasi-harmonic spheres

Let M and N be compact Riemannian manifolds. To prove the global existence and convergence of the heat flow for harmonic maps between M and N, it suffices to show the nonexistence of harmonic spheres and nonexistence of quasi-harmonic spheres. In this paper, we prove that, if the universal covering of N admits a nonnegative strictly convex function with polynomial growth, then there are no quasi-harmonic spheres nor harmonic spheres. This generalizes the famous Eells–Sampson’s theorem (Am J Math 86:109–169, [7]).     

Finite energy and totally geodesic maps from locally symmetric spaces of finite volume

LetM=G/K be a locally symmetric space of finite volume and rank 2. We show that any map fromM of weighted finite energy in the sense of Saper can be deformed into a finite energy map. As a consequence such maps can be deformed into totally geodesic ones, and a geometric generalization of Margulis' superrigidity theorem is obtained.     

Antimicrobial activity-guided identification of compounds from the deciduous leaves of Malus doumeri by HPLC-ESI-QTOF-MS/MS

This paper intends to identify the antimicrobial activity compounds from the deciduous leaves of Malus doumeri (Dong Li Tea) by HPLC-ESI-QTOF-MS/MS. The ethanol extracts of Malus doumeri were partitioned into petroleum ether, dichloromethane, ethyl acetate, n-butanol and water fraction, respectively. The antimicrobial screening experiments showed that ethyl acetate fraction has a certain antibacterial activity by inhibition zone method in vitro. And then we used the HPLC-ESI-QTOF-MS/MS method to verify the identities of bioactive compounds. Finally, 41 compounds were determined and 11 of which were firstly reported in this plant. Notably, compounds (32, 34, 38) are new dihydrochalcones, and three chlorogenic acid analogues (10, 13, 17) may be potential antimicrobial active ingredient. Which is of great significance to the isolation of novel compounds and the discovery of new natural preservative candidates from the deciduous leaves of Malus doumeri.