The smallest Dirac eigenvalue in a spin-conformal class and cmc immersions

Abstract

Let us fix a conformal class [g 0 ] and a spin structure σ on a compact manifold M . For any g ∈ [g 0 ], let λ + 1 (g) be the smallest positive eigenvalue of the Dirac operator D on (M, g, σ). In a previous article we have shown thatIn the present article, we enlarge the conformal class by adding certain singular metrics. We will show that if λ + min (M, g 0 , σ) < λ + min (S n ), then the infimum is attained on the enlarged conformal class. For proving this, we solve a system of semi-linear partial differential equations involving a nonlinearity with critical exponent: Dϕ = λ|ϕ| 2/(n−1) ϕ.The solution of this problem has many analogies to the solution of the Yamabe problem. However, our reasoning is more involved than in the Yamabe problem as the eigenvalues of the Dirac operator tend to +∞ and −∞. Using the spinorial Weierstraß representation, the solution of this equation in dimension 2 shows the existence of many periodic constant mean curvature surfaces.

A Sobolev-like inequality for the Dirac operator

“…The elliptic a-priori estimate gives that ϕ q ∈ H s 1 for all s > 1 and if we apply the Sobolev embedding theorem, one concludes that ϕ q ∈ C 0,α (M) for α ∈ (0, 1). Hence f |ϕ q | p−2 ϕ q ∈ C 0,α (M) as well, and the Schauder estimate (see [4]) gives ϕ q ∈ C 1,α (M). It is clear that one can carry on this argument on M \ ϕ −1 q (0) to obtain ϕ q ∈ C ∞ (M \ ϕ −1 q (0)).…”

“…The criterion obtained by Ammann in [4] is tightly related to the one involved in the Yamabe problem since he shows that if inequality (8) is strict then the spinor field solution of (7) is nontrivial (compare with (2)). …”

Existence and concentration of semi-classical solutions for a nonlinear Maxwell-Dirac system

“…Concerning the critical nonlinearity, the standard bootstrap argument does not yield regularity of finite action weak solutions (see [5,6]). Motivated by Takeshi Isobe [28], we give the following lemma Lemma 3.19.…”

A spinorial analogue of Aubin’s inequality

“…where A i jkαβγ ∈ R and where W ∈ ( 3 T V ), V ∈ (T V ), |W | ≤ C r 4 and |V| ≤ C r 2 (C and C being positive constants independent of ϕ).…”