This work considers problems pertaining to the regularity theory and the analysis of singularities of geometric partial differential equations that stem from the theory of isometric immersions and geometric flows. In the first of two largely independent parts, we employ the Uhlenbeck–Rivière theory of Coulomb gauges to prove that a Pfaffian system with coefficients in the critical space $L^2_\mathrm{loc}$ on a simply connected open subset of $\mathbb{R}^2$ has a non-trivial solution in the Sobolev space $W^{1,2}_\mathrm{loc}$ if the coefficients are antisymmetric and satisfy a compatibility condition. As an application of this result, we show that the fundamental theorem of surface theory holds for prescribed first and second fundamental forms of optimal regularity in the classes $W^{1,2}_\mathrm{loc}$ and $L^2_\mathrm{loc}$, respectively, that satisfy a compatibility condition equivalent to the Gauss–Codazzi–Mainardi equations. Finally, we give a weak compactness theorem for surface immersions in the class $W^{2,2}_\mathrm{loc}$. The second part of this work is concerned with the analysis of singularities of the curve shortening and mean curvature flows. In particular, we show a cylindrical estimate for the mean curvature flow of $k$-convex hypersurfaces, extending estimates that had previously been introduced in the context of Huisken–Sinestrari’s surgery procedure for $2$-convex flows. Furthermore, we consider curve shortening flow of arbitrary codimension in an Euclidean background. For type-II singularities, we prove the existence of a sequence of space-time points along which the curvature tends to infinity such that a rescaling of the solution along it converges to the Grim Reaper solution, paralleling Altschuler’s work in the case of space curves. Finally, we demonstrate that the curve shortening flow of initial curves with an entropy bound converges to a round point in finite time.

Type

Publication

PhD thesis