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**Constant mean curvature surfaces with Delaunay ends.**
*(English)*
Zbl 1005.53006

This paper presents a construction of complete surfaces in \(\mathbb{R}^3\) with finitely many ends and finite topology and with nonzero constant mean curvature, based on gluing perturbations of two different building blocks. This construction is parallel to the well-known original construction by Kapouleas. A topological difference between the two methods is that Kapouleas’ desingularization usually produces surfaces of high genus, while this article shows surfaces with the topology of the original minimal \(k\)-noid used as a building block.

Let \(\Sigma_0\subset \mathbb{R}^3\) be a \(k\)-noid, i.e., an Aleksandrov embedded complete minimal surface with finite total curvature, genus \(g\geq 0\) and \(k\geq 2\) ends, all of catenoid type, with the additional property of being nondegenerate. Then, there exist \(\varepsilon _0 >0\) and two distinct one-parameter families of CMC surfaces \(M_{\varepsilon }^+,M_{\varepsilon }^-\), \(\varepsilon \in (0,\varepsilon _0)\), of genus \(g\) and \(k\) Delaunay ends, with the following properties: (a) For any \(R>0\), the expansion \(\varepsilon ^{-1}M_{\varepsilon }^\pm\) restricted to the ball of radius \(R\) centered at the origin converges \(C^{\infty }\) as \(\varepsilon \) goes to zero to the restriction of \(\Sigma _0 \) to that ball. (b) If all the ends of \(\Sigma _0 \) have the same orientation, then all the ends of \(M_{\varepsilon }^+\) are asymptotic to Delaunay unduloids, while those of \(M_{\varepsilon }^-\) are asymptotic to Delaunay nodoids. If \(\Sigma _0\) has ends with different orientations, then each \(M_{\varepsilon }^+\) and each \(M_{\varepsilon }^-\) has ends of both Delaunay types. (c) \(M_{\varepsilon }^+,M_{\varepsilon }^-\) are nondegenerate.

The surfaces \(M_{\varepsilon }^\pm\) are constructed as follows: First the authors truncate the shrunk minimal surface \(\varepsilon \Sigma _0\) by removing suitable representatives of its ends, and parameterize the space of CMC surfaces near the truncated \(k\)-noid. They show that such CMC normal graphs are determined by their boundary values (graphs over \(\partial (\varepsilon \Sigma_0\))), for any \(\varepsilon >0\) small enough. Second, they choose appropriate half-Delaunay surfaces \(D_1,\dots ,D_k\), and also parameterize the space of CMC surfaces near each half-Delaunay \(D_i\) by their boundary values over \(\partial D_i\), \(i=1,\dots ,k\). Finally, they prove that the two boundary value problems can be simultaneously solved to give the desired globally defined CMC surface. This gluing step is done by matching both Dirichlet and Neumann boundary data along the common boundaries.

Let \(\Sigma_0\subset \mathbb{R}^3\) be a \(k\)-noid, i.e., an Aleksandrov embedded complete minimal surface with finite total curvature, genus \(g\geq 0\) and \(k\geq 2\) ends, all of catenoid type, with the additional property of being nondegenerate. Then, there exist \(\varepsilon _0 >0\) and two distinct one-parameter families of CMC surfaces \(M_{\varepsilon }^+,M_{\varepsilon }^-\), \(\varepsilon \in (0,\varepsilon _0)\), of genus \(g\) and \(k\) Delaunay ends, with the following properties: (a) For any \(R>0\), the expansion \(\varepsilon ^{-1}M_{\varepsilon }^\pm\) restricted to the ball of radius \(R\) centered at the origin converges \(C^{\infty }\) as \(\varepsilon \) goes to zero to the restriction of \(\Sigma _0 \) to that ball. (b) If all the ends of \(\Sigma _0 \) have the same orientation, then all the ends of \(M_{\varepsilon }^+\) are asymptotic to Delaunay unduloids, while those of \(M_{\varepsilon }^-\) are asymptotic to Delaunay nodoids. If \(\Sigma _0\) has ends with different orientations, then each \(M_{\varepsilon }^+\) and each \(M_{\varepsilon }^-\) has ends of both Delaunay types. (c) \(M_{\varepsilon }^+,M_{\varepsilon }^-\) are nondegenerate.

The surfaces \(M_{\varepsilon }^\pm\) are constructed as follows: First the authors truncate the shrunk minimal surface \(\varepsilon \Sigma _0\) by removing suitable representatives of its ends, and parameterize the space of CMC surfaces near the truncated \(k\)-noid. They show that such CMC normal graphs are determined by their boundary values (graphs over \(\partial (\varepsilon \Sigma_0\))), for any \(\varepsilon >0\) small enough. Second, they choose appropriate half-Delaunay surfaces \(D_1,\dots ,D_k\), and also parameterize the space of CMC surfaces near each half-Delaunay \(D_i\) by their boundary values over \(\partial D_i\), \(i=1,\dots ,k\). Finally, they prove that the two boundary value problems can be simultaneously solved to give the desired globally defined CMC surface. This gluing step is done by matching both Dirichlet and Neumann boundary data along the common boundaries.

Reviewer: Bang-yen Chen (East Lansing)

### MSC:

53A10 | Minimal surfaces in differential geometry, surfaces with prescribed mean curvature |

53C42 | Differential geometry of immersions (minimal, prescribed curvature, tight, etc.) |

58D27 | Moduli problems for differential geometric structures |

53C21 | Methods of global Riemannian geometry, including PDE methods; curvature restrictions |