The smallest Dirac eigenvalue in a spin-conformal class and cmc-immersions
by
Bernd Ammann
The smallest Dirac eigenvalue in a spin-conformal class and cmc-immersions
(.ps,.pdf)
Comm. Anal. Geom. 17 (2009), 429-479.
Let us fix a conformal class [g_0] and a spin structure σ on a compact manifold M.
For any g\in [g_0], let λ+1(g) be the
smallest positive eigenvalue of the Dirac operator D on (M,g,\si).
In a previous paper we have shown that
\lambda(M,g,\si):=\inf_{g\in [g_0]} λ+1(g) vol(M,g)^{1/n}>0.
In the present article, we enlarge the conformal class by certain singular metrics. We will show that if $\lambda(M,g,\si)<\lambda(S^n)$, then the infimum is attained on the enlarged conformal class. For proving this, we have to solve a system of semi-linear partial differential equations involving a nonlinearity with critical exponent: $D\phi= \la |\phi|^{2/(n-1)}\phi.$
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 spectrum of the Dirac operator is not bounded from below. The solution may have a nonempty zero set because a maximum principle is not available.
Using the Weierstraß representation,
the solution of this equation in dimension 2 provides a tool for the
construction of new constant mean curvature surfaces.
Typo in the published version
In the second last line at the end of section 4 the term k-1/3
should be replaced by k-1/2.
Warning for citations
Unfortuately the journal has changed the numbering of theorems, propositions, and equations in the final version in such a way that after equation (5.4) we have Theorem 5.2 and later Proposition 5.1. As I do not want to adapt the preprint version to this numbering, the numbering in the preprint version and the published version differ.
A further comment about regularity
(All numbers correspond to the preprint version linked above)
In Section 5 we followed the convention/definition that a weak solution is a solution in the sense of distributions.
In Theorem 5.1 the conclusion is that equation (5.2) holds weakly (=distributionally) on U, and Dφ is in Lq(U) for q, given by 1/p+1/q=1.
It is then natural to ask whether φ has "higher regularity". Let us emphasize that this higher regularity is neither claimed in Theorem 5.1, nor used in the article when applying Theorem 5.1.
In the subcritical case, i.e. in the case p<2n/(n-1), the question is easy to answer: a boot strap argument then implies φ∈C2,a for small a>0.
Our main case of consideration is the critical case p=2n/(n-1). Here it is interesting to know whether φ∈Lr for r>2n/(n-1). This Lr -condition then
implies φ∈C2,a for small a>0, see our Theorem 5.5 and Proposition 5.6, and away from the zeros of φ we get smoothness. This question was studied by other authors afterwards.
In the case n=2, the affirmative answer to the above question is published in Changyou Wang: A remark on nonlinear Dirac equations.
He showed that in this case solutions with the critical p=4=2n/(n-1) are already smooth.
In the higher dimensional case there is a strongly related preprint by Borrelli and Frank from which the affirmative answer follows on the round sphere.
It is likely that their proof extends to arbitrary Riemannian manifolds, and thus yields φ∈C2,a-regularity for critical p. However one does not get smoothness in the zeros of φ.
58J50, 53C27 (Primary) 58C40, 35P15, 35P30, 35B33 (Secondary)
Keywords
Dirac operator, eigenvalues, conformal geometry,
critical Sobolev exponents
Zurück zur Homepage
Bernd Ammann,
The Paper was written on 30.04.2003
Last update 15.5.2009