Generalized motion of level sets by functions of their curvatures on Riemannian manifolds

Details Generalized motion of level sets by functions of their curvatures on Riemannian manifolds Generalized motion of level sets by functions of their curvatures on Riemannian manifolds

2007-07-13

arxiv.org/abs/0707.2012v3

Mathematics

Analysis of PDEs

34 pages; third version (some misprints have been corrected and one important example added)

Scientific paper

We consider the generalized evolution of compact level sets by functions of their normal vectors and second fundamental forms on a Riemannian manifold M. The level sets of a function $u:M\to\mathbb{R}$ evolve in such a way whenever u solves an equation $u_{t}+F(Du, D^{2}u)=0$, for some real function F satisfying a geometric condition. We show existence and uniqueness of viscosity solutions to this equation under the assumptions that M has nonnegative curvature, F is continuous off Du=0, (degenerate) elliptic, and locally invariant by parallel translation. We then prove that this approach is geometrically consistent, hence it allows to define a generalized evolution of level sets by very general, singular functions of their curvatures. For instance, these assumptions on F are satisfied when F is given by the evolutions of level sets by their mean curvature (even in arbitrary codimension) or by their positive Gaussian curvature. We also prove that the generalized evolution is consistent with the classical motion by the corresponding function of the curvature, whenever the latter exists. When M is not of nonnegative curvature, the same results hold if one additionally requires that F is uniformly continuous with respect to D^2 u. Finally we give some counterexamples showing that several well known properties of the evolutions in $\mathbb{R}^{n}$ are no longer true when M has negative sectional curvature.

Affiliated with

Also associated with

No associations

Rating

Generalized motion of level sets by functions of their curvatures on Riemannian manifolds does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Generalized motion of level sets by functions of their curvatures on Riemannian manifolds, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Generalized motion of level sets by functions of their curvatures on Riemannian manifolds will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-60666