# A fully nonlinear flow of three-convex hypersurfaces

We will discuss a fully nonlinear geometric flow of three-convex hypersurfaces, where the normal speed at each point of the solution is given by a concave function of the second fundamental form. Three-convexity means that at each point, the sum of the smallest three principal curvatures is positive. The flow smoothly deforms any compact three-convex initial hypersurface, preserving three-convexity, until its curvature becomes unbounded. Our main result is a convexity estimate, which says that high-curvature regions are approximately convex. Such an estimate is known to hold for mean-convex mean curvature flow, and for a large class of fully nonlinear flows where the speed function is convex. For concave speeds, previous results of this kind assume two-convexity

### About Pure mathematics seminars

We present regular seminars on a range of pure mathematics interests. Students, staff and visitors to UQ are welcome to attend, and to suggest speakers and topics.

Seminars are usually held on Tuesdays from 3pm to 3.50pm.

Talks comprise 45 minutes of speaking time plus five minutes for questions and discussion.

#### Information for speakers

Researchers in all pure mathematics fields attend our seminars, so please aim your presentation at a general mathematical audience.

#### Contact us

To volunteer to talk or to suggest a speaker, email Ole Warnaar or Travis Scrimshaw.