In this paper, we give the asymptotic expansion of $n_{0,d}$ and $n_{1,d}$, where $(3d-1+g)!\, n_{g,d}$ counts the number of genus g curves in ${\mathbb {C}}P^2$ through $3d-1+g$ points in general position and can be identified with certain Gromov–Witten invariants.
In this paper, we study the isotropic–nematic phase transition for the nematic liquid crystal based on the Landau–de Gennes $\mathbf {Q}$-tensor theory. We justify the limit from the Landau–de Gennes flow to a sharp interface model: in the isotropic region, $\mathbf {Q}\equiv 0$; in the nematic region, the $\mathbf {Q}$-tensor is constrained on the manifolds $\mathcal {N}=\{s_+(\mathbf {n}\otimes \mathbf {n}-\frac{1}{3}\mathbf {I}), \mathbf {n}\in {\mathbb {S}^2}\}$ with $s_+$ a positive constant, and the evolution of alignment vector field $\mathbf {n}$ obeys the harmonic map heat flow, while the interface separating the isotropic and nematic regions evolves by the mean curvature flow. This problem can be viewed as a concrete but representative case of the Rubinstein–Sternberg–Keller problem introduced in Rubinstein et al. (SIAM J. Appl. Math. 49:116–133,
We derive the sharp Moser–Trudinger–Onofri inequalities on the standard n-sphere and CR $(2n+1)$-sphere as the limit of the sharp fractional Sobolev inequalities for all $n\ge 1$. On the 2-sphere and 4-sphere, this was established recently by Chang and Wang. Our proof uses an alternative and elementary argument.