Higher Order Non-Compact Curvature Flows

<p dir="ltr">In this thesis we are interested in trying to study higher order curvature flows of non-compact manifolds. Our particular aim is to prove that a unique solution exists to these flows and we want to understand their long time behaviour. Our first result is on the existence and uniqueness of solutions for curves under a general family of fourth order curvature flows. Remaining in the curve setting we introduce the entropy flow which is the first known fourth order curvature flow which preserves convexity. The main objectives for the entropy flow are to establish the existence and uniqueness of solutions to the entropy flow but to also determine the maximal time of existence and the global behaviour of solutions as we approach the maximal time of existence. We then move into the analysis of asymptotically Euclidean surfaces which serve as our model non-compact problem. We show that the expected energy estimates, lifespan theorem, interior estimates and blowup analysis hold for asymptotically Euclidean surfaces and show that the solution converges to the plane.</p>