# Self-Translating Solitons

Specialeforsvar: Christopher Junker Pugh

Titel: Self-Translating Solitons

Abstract: In this thesis we study self-translating solitons for mean curvature flow. First, we introduce the first and the second variation formulas and the notion of stability for minimal surfaces. Then we show that 2-dimensional immersed surfaces in R
3 with small curvature are locally graphical and we give a compactness result for minimal surfaces. Next we introduce translators and the notion of stability for translators. In particular, we show that translators are minimal surfaces with respect to a conformal metric and we prove that complete mean convex 2-dimensional translators are stable. Furthermore, in the 2-dimensional setting we discuss inequalities for complete mean convex translators and give a mean value inequality for proper embedded surfaces in R 3 with bounded mean curvature. Moreover, we consider geometric and topological assumptions for
translators under which we obtain curvature bounds. Finally, we provide new results of independent interest for parabolic surfaces and prove that complete 2-dimensional mean convex translators are convex.

Vejleder: Niels Martin Møller
Censor:   Andrew Swann, Aarhus Universitet