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.

Abstract

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.

Small Typo

In the second last line at the end of section 4 the term k-1/3 should be replaced by k-1/2.

Mathematics Subject Classification

58J50, 53C27 (Primary) 58C40, 35P15, 35P30, 35B33 (Secondary)

Keywords

Dirac operator, eigenvalues, conformal geometry, critical Sobolev exponents
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Bernd Ammann,
The Paper was written on 30.04.2003
Last update 15.5.2009