Ammann, Bernd (2009) The smallest Dirac eigenvalue in a spin-conformal class and cmc immersions. COMMUNICATIONS IN ANALYSIS AND GEOMETRY, 17 (3). pp. 429-479. ISSN 1019-8385,
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Let us fix a conformal class [g(0)] and a spin structure sigma on a compact manifold M. For any g is an element of [g(0)], let lambda(+)(1) (g) be the smallest positive eigenvalue of the Dirac operator D on ( M, g, sigma). In a previous article we have shown that lambda(min)+(M, g(0), sigma) := lambda(+)(1) (g)is an element of[g(0)] (g) vol(M, g)(1/n) > 0. In the present article, we enlarge the conformal class by adding certain singular metrics. We will show that if lambda(+)(min)( M, g(0), sigma) < lambda(+)(min)(Sn), then the in. mum 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: D phi. = lambda vertical bar phi vertical bar(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 eigenvalues of the Dirac operator tend to +infinity and -infinity. Using the spinorial Weierstrass representation, the solution of this equation in dimension 2 shows the existence of many periodic constant mean curvature surfaces.
| Item Type: | Article |
|---|---|
| Uncontrolled Keywords: | QUATERNIONIC KAHLER-MANIFOLDS; 1ST EIGENVALUE; SCALAR-CURVATURE; YAMABE PROBLEM; OPERATOR; SURFACES; INVARIANT; BOUNDS; |
| Subjects: | 500 Science > 510 Mathematics |
| Divisions: | Mathematics > Prof. Dr. Bernd Ammann |
| Depositing User: | Dr. Gernot Deinzer |
| Date Deposited: | 14 Sep 2020 04:32 |
| Last Modified: | 14 Sep 2020 04:32 |
| URI: | https://pred.uni-regensburg.de/id/eprint/28733 |
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