We consider a priori estimates of Weyl’s embedding problem of (S^2,g) in general 3-dimensional Riemannian manifold (N^3,\bar g). We establish the mean curvature estimate under natural geometric assumption. Together with a recent work by Li-Wang, we obtain an isometric embedding of (S2,g) in Riemannian manifold. In addition, we reprove Weyl’s isometric embedding theorem in space form under the condition that g \in C^2 with D^2g Dini continuous. 

A vortex in a straining field is a canonical situation describing vortices in an irrotational flow. Exact solutions to this problem have been found in the form of vortex patches and hollow vortices, both of which can be viewed as desingularizations of point vortices. After a review of the history of point vortices, we discuss hollow vortices, which are steady vortex sheets. We then focus on the case of vortices in strain and examine hollow vortices and vortex patches, describing the bifurcation structure of the latter. Finally we consider Sadovskii vortices, which contain both interior vorticity and a vortex sheet on the boundary, and sketch the relations between the different solutions.

We discuss traveling front solutions u(t,x) = U(x-ct) of reaction-diffusion equations u_t = Lu + f(u) with ignition media f and diffusion operators L generated by symmetric Levy processes X_t. Existence and uniqueness of fronts are well-known in the case of classical diffusion (i.e., Lu = Laplacian(u)) and non-local diffusion (Lu = J*u - u). Our work extends these results to general Levy operators. In particular, we show that a strong diffusivity in the underlying process (in the sense that the first moment of X_1 is infinite) prevents formation of fronts, while a weak diffusivity gives rise to a unique (up to translation) front U and speed c>0.

Kumura showed that there are no eigenvalues embedded in the essential
spectrum of the Laplacian on $n$-dimensional noncompact
complete Riemannian manifold $(M_n, g)$, if the radial curvature $K_{\rm
rad}+1=o(r^{-1})$ as $r$ goes to infinity.

Given any finite/countable set of positive energies $\{\lambda_n\}$, we
can
construct a Riemannian manifold with the decay order
$K_{\rm rad}+1=O(r^{-1})$/$K_{\rm rad}+1=\frac{C(r)}{r}$, where $C(r)\geq
0$ and $C(r) $ goes to infinity arbitrarily slowly, such that the
eigenvalues $\{\frac{(n-1)^2}{4}+\lambda_n\}$ are embedded in the
essential
spectrum $\sigma_{{\rm ess}}(-\Delta_g)=\left[\frac{(n-1)^2}{4},\infty
\right)$.