| dc.description.abstract | 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
L2
loc on a simply connected open subset of R2 has a non-trivial solution in the Sobolev
space W
1,2
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
loc and L2
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
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. | en_US |
