On the Index of Willmore spheres

Abstract

We consider unbranched Willmore surfaces in the Euclidean space that arise as inverted complete minimal surfaces with embedded planar ends. Several statements are proven about upper and lower bounds on the Morse Index -the number of linearly independent variational directions that locally decrease the Willmore energy. We in particular compute the Index of a Willmore sphere in the three-space. This Index is m − d, where m is the number of ends of the corresponding complete minimal surface and d is the dimension of the span of the normals at the m-fold point. The dimension d is either two or three. For m = 4 we prove that d = 3. In general, we show that there is a strong connection of the Morse Index to the number of logarithmically growing Jacobi fields on the corresponding minimal surface.

Geometry of complete minimal surfaces at infinity and the Willmore index of their inversions

“…The authors of [9,8] computed numerically that there is a one-dimensional space of variations of the Morin surface decreasing the Willmore energy W to second order [9, Section 3]: that is, the Morin surface has Morse index one. This numerically indicated result was recently proven [12] by the first and third authors, who establish the following formula for the W-index of a Willmore sphere in R 3 :…”

“…In [12], the authors were able to draw conclusions for the Morse index of a Willmore sphere by studying area-Jacobi fields -functions in the kernel of the second order elliptic operator corresponding to the second variation of the area functional. Certain questions related to the geometry of complete minimal surfaces with embedded planar ends arose that are of independent interest.…”

“…Note that, in the case of four-ended surfaces, the whole moduli space of complete minimal spheres with embedded planar ends consists of the S 1 ×SO(3, C)orbit of the inverted Morin surface [19]. With this property in hand, we can simplify the proof [12] that each Willmore sphere with W = 16π has W-Morse index one. In Section 3, we also give a weaker condition than in [12] for a complete minimal surface with embedded planar ends of arbitrary genus to have W-Morse index at least one after inversion.…”

“…With this property in hand, we can simplify the proof [12] that each Willmore sphere with W = 16π has W-Morse index one. In Section 3, we also give a weaker condition than in [12] for a complete minimal surface with embedded planar ends of arbitrary genus to have W-Morse index at least one after inversion.…”

“…Finally, in Section 5, we study the p-equivariant W-Morse index -that is, the maximal dimension for a p-symmetry-invariant subspace of variations decreasing the W to second order -of the minimal flowers with 2p ends described in [15]. In [12], we already showed that the Morin surface (which is, after inversion, a minimal flower) has equivariant W-Morse index one, and thus the variational direction that decreases W to second order most rapidly must be the 2-symmetric variation. Here, we show that the p-equivariant Willmore Morse index of an inverted minimal flower is always 1 as well:…”