In this article we consider nonlinear Schrödinger (NLS) equations in ${\mathbb{R}}^d$ for $d=1$, $2$, and $3$. We consider nonlinearities satisfying a flatness condition at zero and such that solitary waves are stable. Let $R_k(t,x)$ be $K$ solitary wave solutions of the equation with different speeds $v_1,v_2,\dots ,v_K$. Provided that the relative speeds of the solitary waves $v_k-v_{k-1}$ are large enough and that no interaction of two solitary waves takes place for positive time, we prove that the sum of the $R_k(t)$ is stable for $t\ge 0$ in some suitable sense in $H^1$. To prove this result, we use an energy method and a new monotonicity property on quantities related to momentum for solutions of the nonlinear Schrödinger equation. This property is similar to the $L^2$ monotonicity property that has been proved by Martel and Merle for the generalized Korteweg–de Vries (gKdV) equations (see [12, Lem. 16, proof of Prop. 6]) and that was used to prove the stability of the sum of $K$ solitons of the gKdV equations by the authors of the present article (see [15, Th. 1(i)]).

We prove that the closure of a complete embedded minimal surface $M$ in a Riemannian three-manifold $N$ has the structure of a minimal lamination when $M$ has positive injectivity radius. When $N$ is ${\mathbb{R}}^3$, we prove that such a surface $M$ is properly embedded. Since a complete embedded minimal surface of finite topology in ${\mathbb{R}}^3$ has positive injectivity radius, the previous theorem implies a recent theorem of Colding and Minicozzi in [5, Corollary 0.7]; a complete embedded minimal surface of finite topology in ${\mathbb{R}}^3$ is proper. More generally, we prove that if $M$ is a complete embedded minimal surface of finite topology and $N$ has nonpositive sectional curvature (or is the Riemannian product of a Riemannian surface with $\mathbb{R}$), then the closure of $M$ has the structure of a minimal lamination

We consider $u(x,t)$, a solution of $u_t=\Delta u+\left|u\right|{}^{p-1}u$ which blows up at some time $T>0$, where $u:{\mathbb{R}}^N\times [0,T)\to \mathbb{R}$, $p>1$, and $(N-2)p<N+2$. Under a nondegeneracy condition, we show that the mere hypothesis that the blow-up set $S$ is continuous and $(N-1)$-dimensional implies that it is $C^2$. In particular, we compute the $N-1$ principal curvatures and directions of $S$. Moreover, a much more refined blow-up behavior is derived for the solution in terms of the newly exhibited geometric objects. Refined regularity for $S$ and refined singular behavior of $u$ near $S$ are linked through a new mechanism of algebraic cancellations that we explain in detail

Let $M$ be a closed connected Riemannian manifold, and let $\alpha$ be a homotopy class of free loops in $M$. Then, for every compactly supported time-dependent Hamiltonian on the open unit disk cotangent bundle which is sufficiently large over the zero section, we prove the existence of a $1$-periodic orbit whose projection to $M$ represents $\alpha$. The proof shows that the Biran-Polterovich-Salamon capacity of the open unit disk cotangent bundle relative to the zero section is finite. If $M$ is not simply connected, this leads to an existence result for noncontractible periodic orbits on level hypersurfaces corresponding to a dense set of values of any proper Hamiltonian on $T^{*}M$ bounded from below, whenever the levels enclose $M$. This implies a version of the Weinstein conjecture including multiplicities; we prove existence of closed characteristics—one associated to each nontrivial $\alpha$—on every contact-type hypersurface in $T^{*}M$ enclosing $M$

This article gives a uniform construction of infinitely many primitive Teichmüller curves $V\subset {\mathcal{M}}_g$ for $g=2$, $3$, and $4$

Let $X\subset {\mathbb{R}}^n$ be a set that is definable in an o-minimal structure over $\mathbb{R}$. This article shows that in a suitable sense, there are very few rational points of $X$ which do not lie on some connected semialgebraic subset of $X$ of positive dimension