Timetable and venue:
Monday 9 February: 14:00 - 16:00
Tuesday 10 February: 14:00 - 16:00
Wednesday 11 February: 14:00 - 16:00
Thursday 12 February: 14:00 - 16:00

Course Overview:
The Spherical Cap Conjecture posits that a compact embedded (non-zero) CMC surface in R3 bounded by a circle is a spherical cap. This course will trace the conjecture’s historical development, present key results (e.g., Jellet’s and Koyso’s theorems), and explore recent advances, including our work on starshaped surfaces [3]. The first two lectures ensure accessibility for Master’s students by introducing CMC surfaces, their physical relevance (e.g., soap bubbles), and fundamental geometric and analytical tools. The latter two lectures introduce advanced PDE techniques and open problems, suitable for doctoral students. The course is prepared with great care to serve as a model for students globally, with clear explanations, visual aids, and references to resources like [1].

1. Introduction to the Spherical Cap Conjecture 
Motivation, historical aspects, current state of the art, and future perspectives. Overview of the conjecture: a compact, embedded CMC surface bounded by a circle is a spherical cap. Introduction to the course structure and goals. Accessible to Master’s students.

2. Preliminaries 
Fundamentals of mean curvature, tangency principle, mean curvature equation for graphs, flux formula, and height estimates. Emphasis on intuitive geometric interpretations and physical examples (e.g., soap bubbles). Introduction to Alexandrov’s reflection method and its applications to CMC surfaces. Discussion of Koyso’s theorem and the half-space property. Accessible to Master’s students.

3. The Dirichlet Problem for the Mean Curvature Equation 
Analysis of the Dirichlet problem for radial versus vertical graphs. Techniques for studying 1 CMC surfaces via quasilinear PDEs, with references to [2]. Advanced content for doctoral students.

4. Jellet’s Theorem and Its Extension to Surfaces with Boundary
The Convex Case Jellet’s theorem for closed starshaped CMC surfaces and its extension to surfaces with a circular boundary under local convexity, as in [3]. Discussion of analytical methods and convexity preservation. Recent efforts to remove the convexity hypothesis for starshaped CMC surfaces, using the continuity method and a priori estimates. Open problems and future directions. Advanced content for doctoral students.

References
[1] López, R., Constant Mean Curvature Surfaces with Boundary, Springer-Verlag, Berlin Heidelberg, 2013.
[2] Gilbarg, D., Trudinger, N.S., Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 2001.
[3] Cruz, F., An Extension of Liebmann’s Theorem to Surfaces with Boundary, Submitted, arXiv:2412.03368.