Conformally invariant bending energy for hypersurfaces

Abstract

The most general conformally invariant bending energy of a closed four-dimensional surface, polynomial in the extrinsic curvature and its derivatives, is constructed. This invariance manifests itself as a set of constraints on the corresponding stress tensor. If the topology is fixed, there are three independent polynomial invariants: two of these are the straighforward quartic analogues of the quadratic Willmore energy for a two-dimensional surface; one is intrinsic (the Weyl invariant), the other extrinsic; the third invariant involves a sum of a quadratic in gradients of the extrinsic curvature -- which is not itself invariant -- and a quartic in the curvature. The four-dimensional energy quadratic in extrinsic curvature plays a central role in this construction.

16 pages

References in corpus (4)

• Geometric theory on the elasticity of bio-membranes
• Membrane geometry with auxiliary variables and quadratic constraints
• Exterior Differential Systems and Euler-Lagrange Partial Differential Equations
• Geometry of surface mediated interactions

Cited by in corpus (10)

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• On shape dependence of holographic entanglement entropy in AdS4/CFT3
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• Yang-Mills theory a la string
• Graham-Witten's conformal invariant for closed four dimensional submanifolds
• Conformal mechanics of space curves
• Residues of manifolds
• A duality theorem for a four dimensional Willmore energy