On the nondegeneracy of constant mean curvature surfaces |
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| Authors: | N Korevaar R Kusner J Ratzkin |
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| Institution: | (1) Department of Mathematics, University of Utah, 155 South 1400 East, JWB 233, Salt Lake City, UT 84112-0090, USA;(2) GANG and Department of Mathematics & Statistics, University of Massachusetts, 710 North Pleasant Street, LGRT 1535, Amherst, MA 01003-9305, USA;(3) Department of Mathematics, University of Connecticut, 196 Auditorium Road, Storrs, CT 06269-3009, USA |
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| Abstract: | We prove that many complete, noncompact, constant mean curvature (CMC) surfaces
are nondegenerate; that is, the Jacobi operator Δf + | Af |2 has no L2 kernel. In fact, if ∑ has genus zero with k ends, and if f (∑) is embedded (or Alexandrov immersed) in a half-space, then we find an explicit upper bound for the dimension of the L2 kernel in terms of the number of non-cylindrical ends. Our main tool is a conjugation operation on Jacobi fields which linearizes
the conjugate cousin construction. Consequences include partial regularity for CMC moduli space, a larger class of CMC surfaces
to use in gluing constructions, and a surprising characterization of CMC surfaces via spinning spheres.
R.K. partially supported by NSF grants DMS-0076085 at GANG/UMass and DMS-9810361 at MSRI, and by a FUNCAP grant in Fortaleza,
Brazil. J.R. partially supported by an NSF VIGRE grant at Utah.
Received: January 2005; Accepted: June 2005 |
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| Keywords: | Constant mean curvature surfaces moduli space nondegeneracy |
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