Immersed spheres of finite total curvature into manifolds

Abstract

Abstract. We prove that a sequence of possibly branched, weak immersions of the 2-sphere S 2 into an arbitrary compact Riemannian manifold .M m ; h/ with uniformly bounded area and uniformly bounded L 2 -norm of the second fundamental form either collapse to a point or weakly converges as current, modulo extraction of a subsequence, to a Lipschitz mapping of S 2 and whose image is made of a connected union of finitely many, possibly branched, weak immersions of S 2 with finite total curvature. We prove moreover that if the sequence belongs to a class of 2 .M m /, the limiting Lipschitz mapping of S 2 realizes this class as well.

Existence and Regularity of Spheres Minimising the Canham–Helfrich Energy

“…In this chapter we will prove compactness of sequences with uniformly bounded Willmore energy and area as well as lower semi-continuity of the Canham-Helfrich energy under this convergence, see Theorem 3.3. The proof of Theorem 3.3 will build on top of [MR14] and the next Lemma 3.1 which establishes the convergence of the constraints and the lower semi-continuity of the Willmore energy away from the branch points (Lemma 3.1 should be compared with [Riv16, Lemma 5.2]).…”

“…The next theorem establishes the weak closure of bubble trees, as well as the convergence of the constraints in the Helfrich problem and the lower semi-continuity of the Willmore energy. The proof builds on top of [MR14].…”

A Resolution of the Poisson Problem for Elastic Plates

“…Hence, if the topology of Σ is fixed, the difference between the functionals 4W and E is a null Lagrangian, consequently the Willmore and the total curvature energy are equivalent from the variational point of view for the treatment of the Poisson problem. The advantage of working with the total curvature energy is that it has a better coercivity property and that it controls the number of branch points 2 with their multiplicities (see for instance [50] or [59]), so the variational problem will be well-posed with E. We observe that there could be differences between minimizers of the two Lagrangians W (Σ) and E(Σ) in the case of the presence of interior branch points, since the identity (1.4) does not hold anymore.…”

Foliation by Area-constrained Willmore Spheres Near a Nondegenerate Critical Point of the Scalar Curvature

“…the existence of smooth immersed spheres minimizing quadratic curvature functionals in compact 3-dimensional Riemannian manifolds, was studied by the third author in collaboration with Kuwert and Schygulla in [15] (see also [30] for the non compact case). In collaboration with Rivière [28,29], the third author developed the necessary tools for the calculus of variations of the Willmore functional in Riemannian manifolds and proved the existence of areaconstrained Willmore spheres in homotopy classes (as well as the existence of Willmore spheres under various assumptions and constraints).…”