Abstract
In this paper, we investigate the life-span of classical solutions to hyperbolic inverse mean curvature flow. Under the condition that the curve can be expressed in the form of a graph, we derive a hyperbolic Monge–Ampère equation which can be reduced to a quasilinear hyperbolic system in terms of Riemann invariants. By the theory on the local solution for the Cauchy problem of the quasilinear hyperbolic system, we discuss life-span of classical solutions to the Cauchy problem of hyperbolic inverse mean curvature.
Highlights
In this paper, we study hyperbolic inverse mean curvature flow (HIMCF): ⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨ z2c zt2 − k− → N + 1k− 1 → T, s ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
→ wher→e k denotes the mean curvature of the curve c and N and T are, respectively, the unit inner normal and tangent vectors of the curve c (·, t). c0 stands for a smooth strictly c→onvex closed curve, c1 denotes the initial velocity of c0, and N0 is the unit inner normal vector of c0
In [2], Huisken and Ilmanen proved a sharp lower bound of mean curvature, from which they proved that if the initial surface is the boundary of a strictly star-shaped domain and has nonnegative mean curvature, a smooth solution of the inverse mean flow will exist for all time and converge to a manifold
Summary
We study hyperbolic inverse mean curvature flow (HIMCF):. → wher→e k denotes the mean curvature of the curve c and N and T are, respectively, the unit inner normal and tangent vectors of the curve c (·, t). c0 stands for a smooth strictly c→onvex closed curve, c1 denotes the initial velocity of c0, and N0 is the unit inner normal vector of c0. In [3], Urbas proved that, for inverse mean curvature flow, the surfaces stay strictly convex and smooth for all time. In [2], Huisken and Ilmanen proved a sharp lower bound of mean curvature, from which they proved that if the initial surface is the boundary of a strictly star-shaped domain and has nonnegative mean curvature, a smooth solution of the inverse mean flow will exist for all time and converge to a manifold. The following theorem concerns the life-span of local (in space) smooth solutions of flow (1) that can be written as convex graphs over an interval R ⊂ R in the form c(t, z) (x, u(t, x)), for some u: [0, T) × R ⟶ R.
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