Construction of infinite time bubble tower solutions to critical wave maps equation

We construct infinite time bubble tower solutions to the critical wave maps equation taking values in the two-sphere. More precisely, for any integers $k\ge 3$ and $J\ge 1$, we construct a solution that is global in one time direction, has $k$-corotational symmetry, and asymptotically decomposes into $J$-many concentric bubbles of alternating signs with asymptotically vanishing radiation. The scales of each bubble are of order $t^{-{\alpha }_j}$ with ${\alpha }_j=(\frac{k}{k-2}{)}^{j-1}-1$. This shows the existence of multi-bubble solutions with an arbitrary number of bubbles in soliton resolution, provided that $k\ge 3$, global existence in one time direction, and alternating signs are considered. Our proof is based on modulation analysis with the method of backward construction. The key new ingredient is a Morawetz-type functional that provides suitable monotonicity estimates for solutions around multi-bubble configurations.