Generalizations of the Choe–Hoppe Helicoid and Clifford Cones in Euclidean Space

Abstract

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

Blackfolds, plane waves and minimal surfaces

“…Explicit coordinate embeddings for the Barbosa-Dajczer-Jorge helicoids into a subset R 2N +1 , where N is an integer, of R (D−1) or R (D−2) are given in [54]. These can be written in the form…”

New geometries for black hole horizons

“…, l along the flat x adirections. The resulting effective geometry is 31) and has topology R l+1 × S D−l−3 . Therefore we refer to these configurations as helicoidal black p-branes.…”

New construction techniques for minimal surfaces

“…An important example is the generalized helicoid constructed in [5] , which was noticed in [16] to be of the form (12), with u and v being specific twin harmonics, homogenous of degree 2.…”

“…Example 5.4 This example provides an irreducible homogeneous polynomial solution to (16) of arbitrary high degree. Let Z ∈ C m 2 denote the matrix with entries z ij , 1 ≤ i, j ≤ m. Then det Z is irreducible over C [15] and det Z ∈ T m 2 .…”