Embedded area-constrained Willmore tori of small area in Riemannian three-manifolds, II: Morse Theory
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EMBEDDED AREA-CONSTRAINED WILLMORE TORI OF SMALL AREA IN RIEMANNIAN THREE-MANIFOLDS, II: MORSE THEORY By NORIHISA IKOMA, ANDREA MALCHIODI, and ANDREA MONDINO Abstract. This is the second part of a series of two papers where we construct embedded Willmore tori with small area constraint in Riemannian three-manifolds. In both papers the construction relies on a Lyapunov-Schmidt reduction, the difficulty being the Möbius degeneration of the tori. In the first paper the construction was performed via minimization, here by Morse Theory. To this aim we establish new geometric expansions of the derivative of the Willmore functional on small Clifford tori (in geodesic normal coordinates) which degenerate to small geodesic spheres with a small handle under the action of the Möbius group. By using these sharp asymptotics we give sufficient conditions, in terms of the ambient curvature tensors and Morse inequalities, for having existence/multiplicity of embedded tori which are stationary for the Willmore functional under the constraint of prescribed (sufficiently small) area. 1. Introduction. This is the second part of a series of two papers where embedded area-constrained Willmore tori in Riemannian 3-manifolds are constructed. Here the construction is performed via Morse theory, whereas in the previous paper [12] it was achieved via minimization/maximization. Let us start by recalling the basic definitions and properties of the Willmore functional. Given an immersion i : Σ → (M,g) of a closed (compact without boundary) 2-dimensional surface Σ into a Riemannian 3-manifold (M,g), the Willmore functional is defined by W(i) :=  Σ H2 dσ where dσ is the area form induced by the immersion and H is the mean curvature (we adopt the convention that H is the sum of the principal curvatures or, in other words, H is the trace of the second fundamental form Aij with respect to the induced metric ḡij, i.e., H := ḡijAij). An immersion i is called Willmore surface (or Willmore immersion) if it is a critical point of the Willmore functional with respect to normal perturbations or, equivalently, if it satisfies the associated Euler-Lagrange equation ΔḡH +H|Å|2 +H Ric(n,n) = 0. (1) Manuscript received April 3, 2015; revised September 14, 2016. Research of the first author supported by JSPS Research Fellowships 24-2259; research of the second author supported by the SNS Project Geometric Variational Problems and by MIUR Bando PRIN 2015 2015KB9WPT 001: the second author is also a member of G.N.A.M.P.A., which is part of INdAM; research of the third author supported by the ETH fellowship. American Journal of Mathematics 139 (2017), 1315–1378. c  2017 by Johns Hopkins University Press. 1315 1316 N. IKOMA, A. MALCHIODI, AND A. MONDINO Here Δḡ is the Laplace-Beltrami operator corresponding to the induced metric ḡ, (Å)ij := Aij − 1 2Hḡij is the trace-free second fundamental form, n is a normal unit vector to i, and Ric is the Ricci tensor of the ambient manifold (M,g). Since of course a minimal immersion (i.e., an immersion with vanishing mean curvature) satisfies the Willmore equation, Willmore surfaces are a natural higher order generalization of minimal surfaces. Analogously, area-constrained Willmore surfaces satisfy the equation ΔḡH +H|Å|2 +H Ric(n,n) = λH, for some λ ∈ R playing the role of Lagrange multiplier. These immersions are naturally linked to the Hawking mass mH(i) :=  Area(i) 64π3/2 (16π −W(i)), a quantity introduced in general relativity to measure the mass of a portion of space by means of the bending effect on light rays. Clearly, by the latter formula, the critical points of the Hawking mass under area constraint are exactly the areaconstrained Willmore immersions (see [18] and the references therein for more material about the Hawking mass). In case the ambient manifold is the Euclidean three-dimensional space, the Willmore functional is invariant under the action of the Möbius group (i.e., under composition of the immersion with isometries, homotheties and inversions with respect to spheres), so the theory of Willmore surfaces can be seen as a natural merging between conformal invariance and minimal surface theory. This was indeed the motivation of Blaschke...