RT Journal Article
T1 Generalized motion of level sets by functions of their curvatures on Riemannian manifolds
A1 Azagra Rueda, Daniel
A1 Jiménez Sevilla, María del Mar
A1 Macia Lang, Fabricio
AB 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 -> 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 R(n) are no longer true when M has negative sectional curvature.
PB Springer
SN 0944-2669
YR 2008
FD 2008
LK https://hdl.handle.net/20.500.14352/49814
UL https://hdl.handle.net/20.500.14352/49814
LA eng
NO MTM
NO UCM-CAM
NO Ministerio de Educacion y Ciencia
NO Juan de la Cierva
NO UCM-BSCH
DS Docta Complutense
RD 8 may 2024