zbMATH — the first resource for mathematics

Connected sums of constant mean curvature surfaces in Euclidean 3 space. (English) Zbl 0972.53010
We establish a general ‘gluing theorem’, which states roughly that if two nondegenerate constant mean curvature surfaces are juxtaposed, so that their tangent planes are parallel and very close to one another, but oppositely oriented, then there is a new constant mean curvature surface quite near to this configuration (in the Hausdorff topology), but which is a topological connected sum of the two surfaces. Here nondegeneracy refers to the invertibility of the Jacobi, or linearized mean curvature operator. This paper treats the simplest context for our result namely when the surfaces are compact with nonempty boundary, however the construction applies in the complete noncompact setting as well. The surfaces we produce here are nondegenerate for generic choices of the free parameters in the construction.

53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
Full Text: DOI arXiv
[1] J. Math. 6 (1841) pp 309–
[2] D. Gilbarg and N. Trudinger, Elliptic partial di erential equations of second order, Second edition, Springer Grundl. Math. Wiss. 224, Berlin-New York 1983. · Zbl 0562.35001
[3] N. Kapouleas, Complete constant mean curvature surfaces in Euclidean three space, Ann. Math. (2) 131 (1990), 239-330. · Zbl 0699.53007
[4] Geom. 33 pp 683– (1991)
[5] Invent. Math. 119 pp 443– (1995)
[6] Geom. 47 pp 95– (1997)
[7] Kusner R., Geom. Funct. Anal. 6 pp 120– (1996)
[8] R. Mazzeo and D. Pollack, Gluing and moduli for some noncompact geometric problems, in: Geometric Theory of Singular Phenomena in Partial Di erential Equations, Sympos. Math. XXXVIII, Cambridge Univ. Press (1998), 17-51. · Zbl 0976.53065
[9] Mazzeo R., Topol. Meth. Nonlin. Anal. 6 pp 2– (1995)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.