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Generalized motion of level sets by functions of their curvatures on Riemannian manifolds. (English) Zbl 1148.53048

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 of the form \(u_{t} + F (Du, D^{2} u) = 0\), 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 transportation. 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.

MSC:

53C44 Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010)
58J05 Elliptic equations on manifolds, general theory
35D05 Existence of generalized solutions of PDE (MSC2000)
35J70 Degenerate elliptic equations
47J35 Nonlinear evolution equations
35G25 Initial value problems for nonlinear higher-order PDEs
35J60 Nonlinear elliptic equations
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