GEOMETRY OF (HYPER)SURFACES

Recent advances on minimal surfaces, CMC surfaces and beyond

Every session includes an introduction to the problem and a research seminar. Students are welcome and encouraged to participate.

SPEAKERS

SCHEDULE & ABSTRACTS

• 10:00 - 11:00: J. Pérez, Some recent advances on classical minimal surface theory (I) - Introduction

ABSTRACT: In this survey talk we will start with an introduction to classical minimal surface theory; by this we mean complete, embedded minimal surfaces in Euclidean three-space. We will review some recent results about structure, classification and open problems. A key tool in tackling these problems consists on understanding the different ways of taking limits of a sequence of embedded minimal surfaces: from the classical theory assuming uniform local bounds of area and curvature, passing through minimal laminations, to the most recent Colding-Minicozzi theory.

• 11:15 - 12:15: J. Pérez, Some recent advances on classical minimal surface theory (II) - Applications

ABSTRACT: We will devote the second talk to explain one of the most recent applications of the tools described in the first talk; namely, the existence of a bound for the number of ends of an embedded minimal surface in R^3 with finite total curvature solely in terms of its genus (a partial answer to the Hoffman-Meeks conjecture). This is joint work with Bill Meeks and Antonio Ros.

• 12:15 - 14:15: Lunch

• 14:15 - 15:45: B. Nelli, Hypersurfaces with constant higher order mean curvature (Mini Intro + Research Seminar)

ABSTRACT: We give an overview of some old and new results about the shape of hypersurfaces whose one of the symmetric functions of the principal curvature is constant.

• 15:45 - 16:15: Break

• 16:15 - 17:45: L. Nicolodi, Bonnet Mates and Isothermic Surfaces (Mini Intro + Research Seminar)

ABSTRACT: The Bonnet Problem asks whether an immersion x : M → R^3 of a surface M admits a Bonnet mate, which is another noncongruent immersion x̃ : M → R^3 with the same induced metric, dx̃ ·dx̃ = dx·dx, and the same mean curvature function H ̃ = H. This talk discusses some local and global aspects of the Bonnet Problem in relation with the property of an immersion of being isothermic. If x possesses a Bonnet mate, it is called a Bonnet immersion. If it possesses more than one distinct mate, it is called a proper Bonnet immersion. Bonnet (1867) proved that umbilic free proper Bonnet immersions are isothermic. Graustein (1924) proved that if x is isothermic and Bonnet, then it is proper Bonnet. Jensen, Musso, and Nicolodi (2016) proved that if x is totally nonisothermic (to be made precise) then it has a unique Bonnet mate. According to Bianchi (1903) and Kamberov, Pedit, and Pinkall (1998), Bonnet pairs can be (locally) built from isothermic surfaces. This is known as the KPP construction. As for the global geometry of Bonnet surfaces, Lawson and Tribuzy (1981) proved that there are no proper Bonnet immersions of a compact surface M. In particular, Roussos and Hernàndez (1990) proved that x : M → R^3 has no Bonnet mate if M is compact and x is a surface of revolution with nonconstant mean curvature. Since then, the question whether there are any compact Bonnet pairs has been open. A necessary condition that x be Bonnet is that its set of umbilics is a discrete subset of M. Sabitov (2012) gives a sufficient condition for nonexistence of a Bonnet mate for a compact immersion. Jensen, Musso, and Nicolodi (2018) proved the following result, which generalizes that of Roussos–Hernandez and provides a geometrical clarification of the Sabitov result. Let x : M → R^3 be an immersion of a compact, connected surface, all of whose umbilics are isolated. (1) If x is isothermic, then it has no Bonnet mate. (2) If x is totally nonisothermic, then it has no Bonnet mate. This is based on joint work with Gary R. Jensen and Emilio Musso.

Local Organizing Commitee: Paolo Mastrolia (paolo.mastrolia@unimi.it), Marco Rigoli (marco.rigoli@unimi.it).

Participation is free, but we strongly encourage the registration in order to simplify the organization. Thank you!