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...

As a generalization of the classical duality between minimal graphs in E 3 and maximal graphs in L 3 , we construct the duality between graphs of constant mean curvature H in Bianchi-Cartan-Vranceanu space E 3 (κ, τ ) and spacelike graphs of constant mean curvature τ in Lorentzian Bianchi-Cartan-Vranceanu space L 3 (κ, H ).

We construct a twin correspondence between graphs with prescribed mean curvature in three-dimensional Riemannian Killing submersions and spacelike graphs with prescribed mean curvature in three-dimensional Lorentzian Killing submersions. Our duality extends the Calabi correspondence between minimal graphs in the Euclidean space R 3 and maximal graphs in the Lorentz-Minkowski spacetime L 3 , by allowing arbitrary prescribed mean curvature and bundle curvature. For instance, we transform the prescribed mean curvature equation in L 3 into the minimal surface equation in the generalized Heisenberg space with prescribed bundle curvature. We present several applications of the twin correspondence to the study of the moduli space of complete spacelike surfaces in certain Lorentzian spacetimes.2010 MSC: Primary 49Q05, 53A10; Secondary 53C50, 35B08.

Abstract. We extend Calabi's correspondence between minimal graphs in Euclidean space R 3 and maximal graphs in Lorentz-Minkowski space L 3 . We establish the twin correspondence between 2-dimensional minimal graphs in Euclidean space R n+2 carrying a positive area-angle function and 2-dimensional maximal graphs in pseudo-Euclidean space R n+2 n carrying the same positive area-angle function.We generalize Osserman's Lemma on degenerate Gauss maps of entire 2-dimensional minimal graphs in R n+2 and offer several Bernstein-Calabi type theorems. A simultaneous application of Harvey-Lawson Theorem on special Lagrangian equation and our extended Osserman's Lemma yields a geometric proof of Jörgens' Theorem on 2-variables unimodular Hessian equation.We introduce the correspondence from 2-dimensional minimal graphs in R n+2 to special Lagrangian graphs in C 2 , which induces an explicit correspondence from 2-variables symplectic Monge-Ampére equations to 2-variables unimodular Hessian equation.

In this paper, we survey known results on closed self-shrinkers for mean curvature flow and discuss techniques used in recent constructions of closed self-shrinkers with classical rotational symmetry. We also propose new existence and uniqueness problems for closed self-shrinkers with bi-rotational symmetry and provide numerical evidence for the existence of new examples.

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By sweeping out L indpendent Clifford cones in R 2N+2 via the multi-screw motion, we construct minimal submanifolds in R L(2N+2)+1 . Also, we sweep out the L-rays Clifford cone (introduced in Section 2.3) in R L(2N+2) to construct minimal submanifolds in R L(2N+2)+1 . Our minimal submanifolds unify various interesting examples: Choe-Hoppe's helicoid of codimension one, cone over Lawson's ruled minimal surfaces in S 3 , Barbosa-Dajczer-Jorge's ruled submanifolds, and Harvey-Lawson's volume-minimizing twisted normal cone over the Clifford torus 1 √ 2 This geometric observation gives an insight on generalizing classical helicoids into higher dimensions as in [4]. Choe and Hoppe [4, Theorem 2] gave an explicit construction of a minimal hypersurface in R 2N+1 foliated by Clifford hypercones in R 2N . The Choe-Hoppe helicoid in R 2N+1 is the hypersurface z = f x 1 , y 1 , · · · , x N , y N = arg (x 1 + iy 1 ) 2 + · · · + (x N + iy N ) 2 , up to homotheties. Recently, Del Pino, Musso, and Pacard [20] produced new solutions of the Allen-Cahn equation whose zero set is the