A Global Invariant of Conformal Mappings in Space

Abstract

Abstract.This paper shows that the total integral of the square of the mean curvature for a compact orientable surface in E3 is an invariant of a conformai space mapping. This result is then used to answer a problem raised by T. Willmore and B.-Y. Chen concerning embeddings of compact orientable surfaces, and in particular tori, for which this integral is a minimum.A conformai mapping on Euclidean three-space has the property that it carries spheres into spheres. Such a mapping can be decomposed into a product of similarity transformations and inversions. Let x:M2^-E3 be a C3 regular immersion of a compact orientable surface into Euclidean three-space, and let T=(XI2 H2 dA, where H is the mean curvature of the immersed surface and dA is the area element. This note proves the result that T is invariant under a conformai space mapping and applies this result to the problem mentioned by T. J. Willmore That ris invariant under similarity transformations (Euclidean motions and homotheties) is obvious. What is not so apparent is that Tis invariant under inversions. (We stipulate here that the center of inversion does not lie on the immersed surface.) It was observed locally by Blaschke [1] that the quantity (H2-K)dA is an inversion invariant, where K is the Gauss curvature.To prove this fact let the center of inversion be taken as the origin. Then, if c is the radius of inversion, the position vector x of a point on the inverse surface, corresponding to the point x on the original surface, has the direction of x and the magnitude c2/|x| where |x| denotes the length of x. Therefore, we may write x=c2x:/|x|2. If N denotes the usual surfaceReceived by the editors June 15, 1972. AMS (MOS) subject classifications (1970). Primary 53A05; Secondary 53A30, 53C45.

On shape dependence of holographic entanglement entropy in AdS4/CFT3

“…Given an oriented, smooth and closed two dimensional surface Σ g with genus g embedded in R 3 , the Willmore energy functional evaluated on Σ g is defined as [77][78][79][80][81] A is also a critical point of the functional (2.25) [82]. Among the large number of papers in the mathematical literature about the Willmore functional, let us mention [102][103][104][105][106][107][108][109][110].…”

Boundary value problems for variational integrals involving surface curvatures

“…3The combination (H2 -K) dA is invariant under conformal transformations of i?3 ; see [8,74]. The same is true for the differential equation (25); see [9], On the phenomenological level, there are suggestions of such integrands; see, e.g., [3,17,37,38].…”

“…If there is no volume constraint, the term fiQ(x) is missing in (70), but this fact does not alter the conclusion (74).…”

Local Palais–Smale sequences for the Willmore functional

“…Its importance is largely due to its invariance under conformal transformations [9,34]. As the second author of the present paper showed in [23], the Euler-Lagrange equation arising from a two-dimensional conformally invariant Lagrangian with quadratic growth can be written in divergence form.…”