The First Dirac Eigenvalue in a Conformal Class.
November 01, 2005 02:30 PM to 03:30 PM
Speakers:
Ammann, Bernd
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Abstract: |
Let (M,g_0) be a compact n-diemnsional Riemannian manifold equiped with a fixed spin structure. Let [g_0] be the set of all metrics conformal to g_0 having volume 1. We study the first positive eigenvalue of the Dirac operator as a function on [g_0]. At first, we sketch the proof that the first positive Dirac eigenvalue is not bounded from above. Then we turn our attention to the infimum, denoted by \mu(M,[g_0]). We will show that \mu(M,[g_0]) is always positive. In order to discuss whether this infimum is attained, we reformulate the problem as a variational problem. The infimum is attained if \mu(M,[g_0])<\mu(\mathbb{S}^n) where \mathbb{S}^n denotes the round sphere. Roughly speaking, this inequality avoids concentration of minimizing sequences for our functional. We discuss the Euler-Lagrange equation of the variational problem. In dimension 2 the spinorial Weierstrass representation can be used to transform the Euler-Lagrange equation into an evenly branched conformal immersion into R^3 such that the image has constant mean curvature. The existence of certain periodic constant mean curvature surfaces is obtained as a corollary. In the remaining part we discuss several conditions implying the inequality \mu(M,[g_0])<\mu(\mathbb{S}^n). With an Aubin type construction of a test spinor, one sees that this inequality holds when M is not conformally flat and {\rm dim} M>6. Other conditions are known if M is conformally flat and for lower dimension. |
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