I propose to give a 3 week mini-course on the subject of minimal and constant mean curvature surfaces at IMPA during January 2015. In the first week of the course I plan to cover the standard material in the classical theory of minimal surfaces in R^{3}; for the most part this is the more standard material covered in the first half of my joint book [4] with Joaquin Perez. During the second week of the course I will focus my attention on the classical theory of constant mean curvature H > 0 surfaces in R^{3}; this is in contrast to the material in the first week of the course where the mean curvature of the surfaces being considered is assumed H = 0. The material of second week is again the standard material in this subject with an emphasis on the special case of embedded (H > 0)-surfaces in R3 as well possibly some more recent work on curvature estimates for (H > 0)-disks by Meeks and Tinaglia. In each of these first 2 weeks of the course I will include some nontrivial classical global results such as the characterization of ends of H-surfaces with H > 0 as being asymptotic to the ends of Delaunay surfaces of revolution. In the third week of the course I will focus on the study of embedded H-surfaces in Riemannian 3 manifolds N with an emphasis on the case where *N* is homogeneous; see my joint article [3] with Joaquin Pérez for some of this last material and also the closely related articles [1, 2]. I hope to have some preliminary lecture notes and slides of lectures available to course participants before the course begins.

[1] W. H. Meeks III, P. Mira, J. Pérez, and A. Ros. Constant mean curvature spheres in homogeneous three-spheres. Preprint at http://arxiv.org/abs/1308.2612.

[2] W. H. Meeks III, P. Mira, J. Pérez, and A. Ros. Isoperimetric domains of large volume in homoge-neous three-manifolds. Preprint at http://arxiv.org/pdf/1303.4222.pdf.

[3] W. H. Meeks III and J. Pérez. Constant mean curvature surfaces in metric Lie groups. In Geometric Analysis, volume 570, pages 25{110. Contemporary Mathematics, edited by J. Galvez, J. Pérez, 2012.

[4] W. H. Meeks III and J. Pérez. A survey on classical minimal surface theory, volume 60 of University Lecture Series. AMS, 2012. ISBN: 978-0-82; Preprint available at

http://www.ugr.es/local/jperez/papers/papers.htm

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