Self-Similar Solutions to the Inverse Mean Curvature Flow in $$\mathbb {R}^2$$ R 2
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Self-Similar Solutions to the Inverse Mean Curvature Flow in R2 Jui-En Chang Abstract. In this paper, we obtain an explicit list of self-similar solutions of inverse mean curvature flow in R2 . Mathematics Subject Classification. 53A04. Keywords. Inverse mean curvature flow, Self-similar solutions, Solitons, Complete solutions.
1. Introduction The inverse mean curvature flow is given by dx = H −1 N, (1.1) dt where x is the position vector on the hypersurface, N is a unit normal vector to the hypersurface, and the mean curvature H is defined as the trace of ∇N . Urbas [18] proved that for a convex hypersurface, the inverse mean curvature flow can be defined at all positive time and the hypersurface will converge to a round sphere. Later, this was generalized to star-shaped hypersurfaces by Urbas [17] and Gerhardt [6]. In other cases, singularities may occur in finite time. In the 2-dimensional case, Smoczyk [15] showed that singularities can only occur when the mean curvature H vanishes somewhere. The inverse mean curvature flow has several important applications in geometry. Huisken and Ilmanen [11], [12] used the inverse mean curvature flow to establish the The author is supported by Ministry of Science and Technology, Taiwan (Grant No. MOST107-2115-M-002-015-MY3). 0123456789().: V,-vol
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Riemannian Penrose inequality. Bray and Neves [2] also used inverse mean curvature flow to establish the Poincare conjecture for 3-manifolds with Yamabe constant greater than that of RP3 . We are interested in the solutions which move by a self-similar motion under the inverse mean curvature flow. In previous works, two types of self-similar motions were considered extensively: The homothety and the translation. If a solution moves by homothety, it means that there exists a constant d = 0 such that the hypersurface Σ moves by Σt = edt Σ. The constant d is regarded as the expanding rate. If d > 0, the solution is called a self-expander. If d < 0, the solution is called a self-shrinker. In R2 , Andrews [1] studied the curves moving by a power of the curvature and established that, for the inverse mean curvature flow, the only compact self-similar solution is a circle. Its expanding rate is 1. For all other expanding rates d, Drugan et al. [5] gave a classification of the one-dimensional homothetic solitons. In higher dimension, an important expander is Sn (exp( nt )) ⊂ Rn+1 . This solution has expanding rate n1 . Drugan et al. [5], Gerhardt [6], and Urbas [17] established the rigidity of solutions with expanding rate n1 . A compact expander must be a round sphere centered at the origin. The other homothetic solitons must either be open or with singularities. Two topological types of solutions with rotational symmetry are considered: hyperplanes and hypercylinders. Huisken and Ilmanen [10] used a phase-plane analysis to establish homothetic solitons which are topological hyperplanes. This was also estab1 . For the solution lished by Hsu [7] by a different method for n1 < d
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