Abstract

The investigation and construction of surfaces with special geometric properties has always been an important subject in differential geometry. Of particular interest are minimal surfaces and constant mean curvature (CMC) surfaces in R^3. Global properties surfaces were first considered by Hopf, showing that all CMC spheres are round. This result was generalized by Alexandrov [2] in the 1950s, who showed that the round spheres are the only embedded compact CMC surfaces in R^3, while there do not exist any compact minimal surfaces in R^3 due to the mean curvature comparison principle or by the property that the Gaussian curvature of minimal surfaces is non-positive.

One reason for the beauty and depth of minimal surface theory is that there are many different methods and tools with which to construct, study, and classify these surfaces.

In this course we will first derive basic properties of minimal, CMC and more generally, smooth surfaces in R^3 and introduce some general techniques. We mainly use differential geometric and complex analytical methods in this course but we will introduce analytic tools such as Brownian motion and stochastic tools including concrete example of universal super harmonic functions.

Prerequisite

It is necessary to be familiar with the basic concepts of linear algebra and calculus. In order to be able to follow the course throughout, it is beneficial to have some basic knowledge about differential geometry, point set topology, the topology of surfaces and to be familiar with some complex analysis. However, it is also possible to make up for this missing material within the course.

Syllabus

(a) The geometry of curves and surfaces in R^3. The Gauss or normal map N : M -> S^2 of an oriented surface M in R^3 and its derivative (called the second fundamental form of M).

(b) The norm of the second fundamental form of M in terms of its trace and determinate, which yield respectively, the mean curvature and Gaussian curvature functions of M.

(c) The definition of CMC surfaces M, as those surfaces with a constant value H for its mean curvature functional. A CMC surface with H = 0 is called a minimal surface.

(d) The construction of minimal surfaces in R^3 using the Schwartz reflection principle and the analyticWeierstrass representation. These examples include the plane, the catenoid, the helicoid, the Riemann minimal examples, Schwartz P and D surfaces, the gyroid, the Superman surface S, Scherk’s 1-periodic and 2-periodic minimal surfaces.

(e) Tools in minimal and CMC surface theory. These tools include the maximum principle for CMC surfaces, barrier constructions, the solution to Plateau, or equivalently described, general least-area problems.

(f) The notion of stable minimal and CMC stable surfaces and curvature estimates for complete stable minimal and CMC stable surfaces. These results depend on work of Fisher-Colbrie, Lopez, Ros, Rosenberg and Schoen.

(g) Applications of the previously mentioned tools and the Barrier Lemma (another tool) to prove the Strong Halfspace Theorem for minimal surfaces, the maximum principle at infinity and the existence of uniform regular neighborhoods for complete embedded minimal and CMC surfaces of bounded curvature.

(h) The notion of hitting or harmonic measure and the proof that almost all Brownian paths on a proper non-flat minimal surface in R^3 must intersect every plane in R^3 for an infinite number of divergent times.

(i) Classification results for complete minimal and CMC surfaces in R^3. These results include:

i. The classification of complete embedded minimal and CMC surfaces that are simply connected (topologically planes and spheres). These surfaces are flat planes, round spheres and the helicoid. Since complete proofs of these results are beyond the scope of this course, I will only give some of the ideas that go into their proofs.

ii. The classification of complete embedded minimal and CMC surfaces that are topologically equivalent to an annulus; these surfaces are all surfaces of revolution and so in the minimal case one only has the catenoid and in the CMC case one has the 1-periodic Delaunay (or nodoids) surfaces of revolution. Again, while a complete proof of these results is beyond the scope of this course, I will indicate some parts of these proofs.

iii. The classification of the asymptotic geometry of an annular end of a complete embedded CMC surfaceM in R^3; namely, each end ofM is asymptotic to the end of a flat plane, a helicoid or a catenoid in the minimal case and to the end of a Delaunay surface in the CMC case.

iv. Classification of properly embedded minimal surfaces of genus zero in R^3 and infinite topology; these surfaces are the Riemann minimal examples.

v. Throughout the course I will mention and discuss related unsolved problems, as they arise naturally in the course discussion.

Reference

[1] T. H. Colding andW. P. Minicozzi II. A course in minimal surfaces. Number 121 in Graduate Studies in Mathematics. American Mathematical Society, 2011. ISBN-10: 0-8218-5323-6, ISBN-13: 978-0-8218-5323-8.

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[3] M. do Carmo. Differential Geometry of Curves and Surfaces. Prentice Hall, New Jersey, 1976.

[4] C. Frohman and W. H. Meeks III. The ordering theorem for the ends of properly embedded minimal surfaces. Topology, 36(3):605–617, 1997. MR1422427, Zbl 878.53008.

[5] C. Frohman and W. H. Meeks III. The topological classification of minimal surfaces in R3. Ann. of Math., 167(3):681–700, 2008. MR2415385, Zbl 1168.49038.

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[12] W. H. Meeks III, J. P´erez, and A. Ros. Stable constant mean curvature surfaces. In Handbook of Geometrical Analysis, volume 1, pages 301–380. International Press, edited by Lizhen Ji, Peter Li, Richard Schoen and Leon Simon, ISBN: 978-1-57146-130-8, 2008. MR2483369, Zbl 1154.53009.

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[14] W. H. Meeks III, J. P´erez, and G. Tinaglia. Constant mean curvature surfaces. In Surveys in differential geometry 2016. Advances in geometry and mathematical physics, volume 21 of Surv. Differ. Geom., pages 179–287. Int. Press, Somerville, MA, 2016. MR3525098, Zbl 1360.53004.

[15] W. H. Meeks III and G. Tinaglia. The geometry of constant mean curvature surfaces in R3. To appear in JEMS, preprint at http://arxiv.org/pdf/1609.08032v1.pdf.

[16] W. H. Meeks III and G. Tinaglia. The dynamics theorem for CMC surfaces in R3. J. of Differential Geom., 85:141–173, 2010. MR2719411, Zbl 1203.53009.

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Lecturer Intro.

• Ph. D. from the University of California at Berkeley 1976.

• Primary research interests include classification results for minimal and CMC surfaces, curvature estimates for CMC disks, the Hoffman-Meeks conjecture, the embedded Calabi-Yau problem, the conformal type of proper minimal surfaces.

• Retired from University of Massachusetts at Amherst, 2017.

• Frequent research visitor at the University of Granada, Granada, Spain.

• Principal past research collaborators: Colin Adams, Pascal Colin, Charley Frohman, David Hoffman, Rob Kusner, Paco Martin, Joaquin Perez, Alvaro Ramos, Antonio Ros, Harold Rosenberg, Giuseppe Tinaglia, Brian White, S. T. Yau.