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SUMMARY:CIMPA-ICTP Course: Recent Advances in the Spherical Cap Conjecture
DTSTART;VALUE=DATE-TIME:20260209T130000Z
DTEND;VALUE=DATE-TIME:20260212T150000Z
DTSTAMP;VALUE=DATE-TIME:20260412T094117Z
UID:indico-event-11277@ictp.it
DESCRIPTION:\n	Timetable and venue:\n	Monday 9 February: 14:00 - 16:00\n	T
 uesday 10 February: 14:00 - 16:00\n	Wednesday 11 February: 14:00 - 16:00\n
 	Thursday 12 February: 14:00 - 16:00Course Overview:\n	The Spherical Cap C
 onjecture posits that a compact embedded (non-zero) CMC surface in R3 boun
 ded by a circle is a spherical cap. This course will trace the conjecture
 ’s historical development\, present key results (e.g.\, Jellet’s and K
 oyso’s theorems)\, and explore recent advances\, including our work on s
 tarshaped surfaces [3]. The first two lectures ensure accessibility for Ma
 ster’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].\n\n	\n	1. Introduction to t
 he Spherical Cap Conjecture \n	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. Int
 roduction to the course structure and goals. Accessible to Master’s stud
 ents.\n	2. Preliminaries \n	Fundamentals of mean curvature\, tangency pri
 nciple\, mean curvature equation for graphs\, flux formula\, and height es
 timates. Emphasis on intuitive geometric interpretations and physical exam
 ples (e.g.\, soap bubbles). Introduction to Alexandrov’s reflection meth
 od and its applications to CMC surfaces. Discussion of Koyso’s theorem a
 nd the half-space property. Accessible to Master’s students.\n	3. The Di
 richlet Problem for the Mean Curvature Equation \n	Analysis of the Dirich
 let problem for radial versus vertical graphs. Techniques for studying 1 C
 MC surfaces via quasilinear PDEs\, with references to [2]. Advanced conten
 t for doctoral students.\n	4. Jellet’s Theorem and Its Extension to Surf
 aces with Boundary\n	The Convex Case Jellet’s theorem for closed starsha
 ped CMC surfaces and its extension to surfaces with a circular boundary un
 der local convexity\, as in [3]. Discussion of analytical methods and conv
 exity preservation. Recent efforts to remove the convexity hypothesis for 
 starshaped CMC surfaces\, using the continuity method and a priori estimat
 es. Open problems and future directions. Advanced content for doctoral stu
 dents.\n	References\n	[1] López\, R.\, Constant Mean Curvature Surfaces w
 ith Boundary\, Springer-Verlag\, Berlin Heidelberg\, 2013.\n	[2] Gilbarg\,
  D.\, Trudinger\, N.S.\, Elliptic Partial Differential Equations of Second
  Order\, Springer-Verlag\, Berlin\, 2001.\n	[3] Cruz\, F.\, An Extension o
 f Liebmann’s Theorem to Surfaces with Boundary\, Submitted\, arXiv:2412.
 03368.\n\n//indico.ictp.it/event/11277/
LOCATION:ICTP Leonardo Building - Luigi Stasi Seminar Room
URL://indico.ictp.it/event/11277/
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