Embedding Riemann surfaces with isolated punctures into the complex plane

We enlarge the class of open Riemann surfaces known to be holomorphically embeddable into the plane by allowing them to have additional isolated punctures compared to the known embedding results

Blow up for the critical gKdV equation II: minimal mass dynamics

We fully revisit the near soliton dynamics for the mass critical (gKdV) equation. In Part I, for a class of initial data close to the soliton, we prove that only three scenario can occur: (BLOW UP) the solution blows up in finite time $T$ in a universal regime with speed $1/(T-t)$; (SOLITON) the solution is global and converges to a soliton in large time; (EXIT) the solution leaves any small neighborhood of the modulated family of solitons in the scale invariant $L^2$ norm. Regimes (BLOW UP) and (EXIT) are proved to be stable. We also show in this class that any nonpositive energy initial data (except solitons) yields finite time blow up, thus obtaining the classification of the solitary wave at zero energy. In Part II, we classify minimal mass blow up by proving existence and uniqueness (up to invariances of the equation) of a minimal mass blow up solution $S(t)$. We also completely describe the blow up behavior of $S(t)$. Second, we prove that $S(t)$ is the universal attractor in the (EXIT) case, i.e. any solution as above in the (EXIT) case is close to $S$ (up to invariances) in $L^2$ at the exit time. In particular, assuming scattering for $S(t)$ (in large positive time), we obtain that any solution in the (EXIT) scenario also scatters, thus achieving the description of the near soliton dynamics

Blow up for the critical gKdV equation III: exotic regimes

We consider the blow up problem in the energy space for the critical (gKdV) equation in the continuation of part I and part II. We know from part I that the unique and stable blow up rate for solutions close to the solitons with strong decay on the right is $1/t$. In this paper, we construct non-generic blow up regimes in the energy space by considering initial data with explicit slow decay on the right in space. We obtain finite time blow up solutions with speed $t^{-\nu}$ where $\nu>11/13,$ as well as global in time growing up solutions with both exponential growth or power growth. These solutions can be taken with initial data arbitrarily close to the ground state solitary wave