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

By Bernd Ammann. 2003

84 citations recorded by the source corpus.

Let us x a conformal class [g0] and a spin structure on a compact manifold M. For any g 2 [g0], let + (g) be the smallest positive eigenvalue of the Dirac operator D on (M; g; ). In a previous paper we have shown that + (M; g0; ) := inf g2[g0] + (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 + (M; g0; ) < + (S n ), then the inm um is attained on the enlarged conformal class. For proving this, we have to solve a system of semi-linear partial dieren tial equations involving a nonlinearity with critical exponent: D’ = j’j 2=(n 1) ’: 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 +1 and 1 . Using the Weierstra representation, the solution of this equation in dimension 2 provides a tool for constructing new periodic constant mean curvature surfaces.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/95941f49c6c59c5d8e328d445004e02c4e560559

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In Applied Mathematics

1. E. Humbert
2. Bernd Ammann
3. Haizhong Li
4. Linfen Cao

In Mathematical Physics

1. E. Humbert
2. Bernd Ammann

Citations in the corpus

• The second Yamabe invariant (2005, result): “Then, proceeding in a way analogous to [1,2], one shows that”
• A spinorial analogue of Aubin’s inequality (2003, background): “the Removal of singularity theorem [ 4, Theorem 5.1 ]), and one sees that � is an eigenspinor to the eigenvalue n/2 on the round sphere S n . The equality”