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Calc. Var. (2008) 33:133167 DOI 10.1007/s00526-008-0160-y Calculus of Variations

Summary: Calc. Var. (2008) 33:133­167
DOI 10.1007/s00526-008-0160-y Calculus of Variations
Generalized motion of level sets by functions
of their curvatures on Riemannian manifolds
D. Azagra · M. Jiménez-Sevilla · F. Maciŕ
Received: 18 July 2007 / Accepted: 21 January 2008 / Published online: 13 February 2008
© Springer-Verlag 2008
Abstract 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
ut + F(Du, D2u) = 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


Source: Azagra Rueda, Daniel - Facultad de Ciencias Matemáticas, Universidad Complutense de Madrid


Collections: Mathematics