Solitons for the inverse mean curvature flow

Abstract

Abstract. We investigate self-similar solutions to the inverse mean curvature flow in Euclidean space. In the case of one dimensional planar solitons, we explicitly classify all homothetic solitons and translators. Generalizing Andrews' theorem that circles are the only compact homothetic planar solitons, we apply the Hsiung-Minkowski integral formula to prove the rigidity of the hypersphere in the class of compact expanders of codimension one. We also establish that the moduli space of compact expanding surfaces of codimension two is big. Finally, we update the list of Huisken-Ilmanen's rotational expanders by constructing new examples of complete expanders with rotational symmetry, including topological hypercylinders, called infinite bottles, that interpolate between two concentric round hypercylinders. Main resultsIn this paper, we study self-similar solutions to the inverse mean curvature flow in Euclidean space. After a brief introduction, we present an explicit classification of the one dimensional homothetic solitons (Theorem 5). Examples include circles, involutes of circles, and logarithmic spirals. Then, we prove that families of cycloids are the only translating solitons (Theorem 10), and we show how to construct translating surfaces via a tilted product of cyloids.Next, we consider the rigidity of homothetic solitons. In the class of closed homothetic solitons of codimension one, we prove that the round hyperspheres are rigid (Theorem 12). For the higher codimension case, we observe that any minimal submanifold of the standard hypersphere is an expander, so in light of Lawson's construction [18] of minimal surfaces in S 3 , there are compact embedded expanders for any genus in R 4 . We conclude with an investigation of homothetic solitons with rotational symmetry. First, we construct new examples of complete expanders with rotational symmetry, called infinite bottles, which are topological hypercylinders that interpolate between two concentric round hypercylinders (Theorem 15). Then, we show how the analysis in the proof of Theorem 15 can be used to construct other examples of complete expanders with rotational symmetry, including the examples from Huisken-Ilmanen [12]. Inverse mean curvature flow -history and applicationsRound hyperspheres in Euclidean space expand under the inverse mean curvature flow (IMCF) with an exponentially increasing radius. This behavior is typical for the flow. Gerhardt [10] and Urbas [21] showed that compact, star-shaped initial hypersurfaces with strictly positive mean curvature converge under IMCF, after suitable rescaling, to a round sphere.Strictly positive mean curvature is an essential condition. For the IMCF to be parabolic, the mean curvature must be strictly positive. Huisken and Ilmanen [15] proved that smoothness at later times is characterised by the mean curvature remaining bounded strictly away from zero (see also Smoczyk [22]). Within the class of strictly mean-convex surfaces, however, a solution to inverse mean curvature flow will, in general, become sin...

Strong spherical rigidity of ancient solutions of expansive curvature flows

“…As in [11,12], our result holds under very general hypotheses on F : besides positivity and monotonicity, no assumptions such as homogeneity, or convexity/concavity are needed. As a particular case, our theorem implies that the spheres are the only homothetically expanding compact solutions of these flows, a result which was first proved in [13] for the inverse mean curvature flow and in [10] for other 1-homogeneous flows.…”

Weighted Hsiung-Minkowski formulas and rigidity of umbilical hypersurfaces

“…Theorem 2 and 3 require no embeddedness assumption. Theorem 3 is proved in [10] for the inverse mean curvature flow.…”

Lagrangian Homothetic Solitons for the Inverse Mean Curvature Flow

“…The solutions of (2.1) when n = 1 and m = 2 are known as homothetic solitons for the inverse curve shortening flow. They are explicitly described in [DLW15] and include the classical logarithmic spirals and involutes of circles as expanders; the only closed ones are the circles centered at the origin ([And03]). When n ≥ 2, the simplest examples of expander hypersurfaces for the IMCF are (modulo homotheties) the standard sphere S n ⊂ R n+1 (a = 1/n) and the cylinders S k × R n−k ⊂ R n+1 (a = 1/k), 1 ≤ k ≤ n − 1.…”

“…The corresponding plane curves include the classical logarithmic spirals and involutes of circles as expanders and the only closed ones are the circles centered at the origin ( [And03]). When m = n + 1 and n ≥ 2, it is also proved in [DLW15] that the hyperspheres centered at the origin are exceptionally rigid since they are the only closed homothetic soliton hypersurfaces for the inverse mean curvature flow. However, in higher codimension, Drugan, Lee and Wheeler also observed in [DLW15] that any minimal submanifold of the standard hypersphere is an expander for the IMCF.…”

“…When n = 1 and m = 2, the solutions of (*) are explicitly described in [DLW15]. The corresponding plane curves include the classical logarithmic spirals and involutes of circles as expanders and the only closed ones are the circles centered at the origin ( [And03]).…”