Higher-order geometric flow of hypersurfaces in a Riemannian manifold

Abstract

In this paper, we consider the high order geometric flows of a submanifolds M in a complete Riemannian manifold N with dim(N ) = dim(M ) + 1 = n + 1, which were introduced by Mantegazza in the case the ambient space is an Euclidean space, and extend some results due to Mantegazza to the present situation under some assumptions on N . Precisely, we show that if m ∈ N is strictly larger than the integer part of n/2 and ϕ(t) is a immersion for all t ∈ [0, T ) and if F m (ϕ 0 ) is bounded by a constant which relies on the injectivity radiusR > 0 and sectional curvatureK π (K π 1) of N , then T must be ∞.

Gradient Flows of Higher Order Yang–mills–higgs Functionals

“…Similar to the ones proposed by De Giorgi [8], Mantegazza [30] studied higher order generalizations of the mean curvature flow and proved that the flows do not hit singularities provided the order of the derivatives is sufficiently large. Very recently, Jia and Wang [20] extended some results due to Mantegazza to a more general ambient manifold. Actually, there have been many other important works on higher order flow such as Escher et al [12] and Wheeler [50] for surface diffusion flow, Kuwert and Schätzle [23,24] and Simonett [39] for Willmore flow of surfaces, Streets [42] for a certain flow of Riemannian curvatures, Bahuaud and Helliwell [3] and Kotschwar [22] for a certain flow of Riemannian metrics, Novaga and Okabe [33] for steepest descent flow and so on.…”

Gradient Flows of Higher Order Yang-Mills-Higgs Functionals

“…Similar to ones proposed by De Giorgi, Mantegazza [34] studied higher order generalisations of the mean curvature flow and proved that the flow do not hit singularities provided the order of the derivatives is sufficiently large. Just very recently, Jia-Wang [23] extended some results due to Mantegazza to a more general ambient manifold. Actually, there have been many other important works on higher order flow, such as Escher-Mayer-Simonett [12] and Wheeler [54] for surface diffusion flow, Kuwert-Schätzle [27,28] and Simonett [44] for Willmore flow of surfaces, Streets [46] for a certain flow of Riemannian curatures, Bahuaud-Helliwell [3] and Kotschwar [26] for a certain flow of Riemannian metrics, Novaga-Okabe [37] for steepest descent flow and so on.…”