The conformal Willmore functional: a perturbative approach. (English) Zbl 1276.53068

Mondino, Andrea

J. Geom. Anal. 23, No. 2, 764-811 (2013).

The conformal Willmore functional of a compact orientable surface \(S\) embedded in a three-dimensional Riemannian manifold \((M,g)\) is defined to be the integral \[ I(S) = \int_S \left( {H^2 \over 4} - D \right) d\Sigma, \] where \(H\) is the mean curvature, \(D\) is the product of the principal curvatures, and \(d\Sigma\) is the area form of \(S\) induced from the embedding of \(S\) in \(M\). This functional is conformally invariant which means that if \(\Psi : M \to M\) is a conformal diffeomorphism, then \(I(\Psi(S)) = I(S)\).
The critical points of \(I\) are called conformal Willmore surfaces. In the paper under review, the author investigates existence and nonexistence of conformal Willmore surfaces using a perturbative method called Lyapunov-Schmidt reduction. It is noteworthy that, unlike previous studies of conformal Willmore surfaces, the ambient manifolds considered here do not have constant curvature.
In the case when \((M,g)\) is \(({\mathbb R}^3, g_\varepsilon)\), where \(g_\varepsilon\) is a certain metric sufficiently close to and asymptotic to the flat Euclidean metric on \({\mathbb R}^3\), the author shows that there exist perturbed spheres which are conformal Willmore surfaces. On the other hand, for a general three-dimensional Riemannian manifold \((M,g)\), if the traceless Ricci tensor of \(M\) does not vanish at the point \(p \in M\), then no sufficiently small perturbed sphere about \(p\) can be a Willmore surface.

Reviewer: James Hebda (St. Louis)

MSC:

53C42	Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
53C21	Methods of global Riemannian geometry, including PDE methods; curvature restrictions
53C20	Global Riemannian geometry, including pinching
58E99	Variational problems in infinite-dimensional spaces
58J60	Relations of PDEs with special manifold structures (Riemannian, Finsler, etc.)
35J60	Nonlinear elliptic equations

Keywords:

Willmore functional; conformal geometry; perturbative method; mean curvature; nonlinear elliptic PDE

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