Carathéodory conjecture
Encyclopedia
The Carathéodory conjecture is a mathematical conjecture
attributed to Constantin Carathéodory
by Hans Ludwig Hamburger in a session of the Berlin Mathematical Society in 1924, [1]. Other early references are the Invited Lecture [3] of Stefan Cohn-Vossen at the International Congress of Mathematicians
in Bologna
and the book [2] by Wilhelm Blaschke
. Carathéodory never committed the Conjecture into writing. In [1], John Edensor Littlewood
mentions the Conjecture and Hamburger's contribution [10] as an example of a mathematical claim that is easy to state but difficult to prove. Dirk Struik describes in [5] the formal analogy of the Conjecture with the Four Vertex Theorem for plane curves. Modern references for the Conjecture are the problem list of Shing-Tung Yau
in [6] and the book [7] of Marcel Berger
, as well as the books [18] and [19], [20], [21].
admits at least two umbilic points. In the sense of the Conjecture, the spheroid
with only two umbilic points and the sphere
, all points of which are umbilic, are examples of surfaces with minimal and maximal numbers of umbilics. For the conjecture to be well posed, or the umbilic points to be well-defined, the surface needs to be at least twice differentiable.
All the proofs mentioned above are based on a reduction of the Carathéodory conjecture to the following Loewner
conjecture: the index of every isolated umbilic point is never greater than one. Roughly speaking, the main difficulty lies in resolution of singularities generated by umbilical points. All the above-mentioned authors resolve the singularities by induction on 'degree of degeneracy' of the umbilical point, but none of them was able to present the induction process clearly.
In 2002, Vladimir Ivanov revisited the work of Hamburger on analytic surfaces with the following stated intent [17]:
First he follows the way passed by Gerrit Bol and Tilla Klotz, but later he proposes his own way for singularity resolution where crucial role belongs to complex analysis
(more precisely, to techniques involving analytic implicit function
s, Weierstrass preparation theorem
, Puiseux series
, and circular root system
s).
Conjecture
A conjecture is a proposition that is unproven but is thought to be true and has not been disproven. Karl Popper pioneered the use of the term "conjecture" in scientific philosophy. Conjecture is contrasted by hypothesis , which is a testable statement based on accepted grounds...
attributed to Constantin Carathéodory
Constantin Carathéodory
Constantin Carathéodory was a Greek mathematician. He made significant contributions to the theory of functions of a real variable, the calculus of variations, and measure theory...
by Hans Ludwig Hamburger in a session of the Berlin Mathematical Society in 1924, [1]. Other early references are the Invited Lecture [3] of Stefan Cohn-Vossen at the International Congress of Mathematicians
International Congress of Mathematicians
The International Congress of Mathematicians is the largest conference for the topic of mathematics. It meets once every four years, hosted by the International Mathematical Union ....
in Bologna
Bologna
Bologna is the capital city of Emilia-Romagna, in the Po Valley of Northern Italy. The city lies between the Po River and the Apennine Mountains, more specifically, between the Reno River and the Savena River. Bologna is a lively and cosmopolitan Italian college city, with spectacular history,...
and the book [2] by Wilhelm Blaschke
Wilhelm Blaschke
Wilhelm Johann Eugen Blaschke was an Austro-Hungarian differential and integral geometer.His students included Shiing-Shen Chern, Luis Santaló, and Emanuel Sperner....
. Carathéodory never committed the Conjecture into writing. In [1], John Edensor Littlewood
John Edensor Littlewood
John Edensor Littlewood was a British mathematician, best known for the results achieved in collaboration with G. H. Hardy.-Life:...
mentions the Conjecture and Hamburger's contribution [10] as an example of a mathematical claim that is easy to state but difficult to prove. Dirk Struik describes in [5] the formal analogy of the Conjecture with the Four Vertex Theorem for plane curves. Modern references for the Conjecture are the problem list of Shing-Tung Yau
Shing-Tung Yau
Shing-Tung Yau is a Chinese American mathematician working in differential geometry. He was born in Shantou, Guangdong Province, China into a family of scholars from Jiaoling, Guangdong Province....
in [6] and the book [7] of Marcel Berger
Marcel Berger
Marcel Berger is a French mathematician, doyen of French differential geometry, and a former director of the Institut des Hautes Études Scientifiques , France...
, as well as the books [18] and [19], [20], [21].
Mathematical content
The Conjecture claims that any convex, closed and sufficiently smooth surface in three dimensional Euclidean spaceEuclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
admits at least two umbilic points. In the sense of the Conjecture, the spheroid
Spheroid
A spheroid, or ellipsoid of revolution is a quadric surface obtained by rotating an ellipse about one of its principal axes; in other words, an ellipsoid with two equal semi-diameters....
with only two umbilic points and the sphere
Sphere
A sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point...
, all points of which are umbilic, are examples of surfaces with minimal and maximal numbers of umbilics. For the conjecture to be well posed, or the umbilic points to be well-defined, the surface needs to be at least twice differentiable.
Mathematical research on an approach by a local index estimate
For analytic surfaces, an affirmative answer to this conjecture was given in 1940 by Hans Ludwig Hamburger in a long paper published in three parts [10]. The approach of Hamburger was via a local index estimate for isolated umbilics, which he showed to imply the Conjecture in his earlier work [8], [9]. In 1943, a shorter proof was proposed by Gerrit Bol [11], see also [23], but, in 1959, Tilla Klotz found and corrected a gap in Bol's proof in [10]. Her proof, in turn, was announced to be incomplete in Hanspeter Scherbel's dissertation [13] (no results of that dissertation related to the Carathéodory conjecture were published for decades, at least nothing was published up to June 2009). Among other publications we refer to papers [14]—[16].All the proofs mentioned above are based on a reduction of the Carathéodory conjecture to the following Loewner
Charles Loewner
Charles Loewner was an American mathematician. His name was Karel Löwner in Czech and Karl Löwner in German.Loewner received his Ph.D...
conjecture: the index of every isolated umbilic point is never greater than one. Roughly speaking, the main difficulty lies in resolution of singularities generated by umbilical points. All the above-mentioned authors resolve the singularities by induction on 'degree of degeneracy' of the umbilical point, but none of them was able to present the induction process clearly.
In 2002, Vladimir Ivanov revisited the work of Hamburger on analytic surfaces with the following stated intent [17]:
- "First, considering analytic surfaces, we assert with full responsibility that Carathéodory was right. Second, we know how this can be proved rigorously. Third, we intend to exhibit here a proof which, in our opinion, will convince every reader who is really ready to undertake a long and tiring journey with us."
First he follows the way passed by Gerrit Bol and Tilla Klotz, but later he proposes his own way for singularity resolution where crucial role belongs to complex analysis
Complex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics,...
(more precisely, to techniques involving analytic implicit function
Implicit function
The implicit function theorem provides a link between implicit and explicit functions. It states that if the equation R = 0 satisfies some mild conditions on its partial derivatives, then one can in principle solve this equation for y, at least over some small interval...
s, Weierstrass preparation theorem
Weierstrass preparation theorem
In mathematics, the Weierstrass preparation theorem is a tool for dealing with analytic functions of several complex variables, at a given point P...
, Puiseux series
Puiseux series
In mathematics, Puiseux series are a generalization of formal power series, first introduced by Isaac Newton in 1676 and rediscovered by Victor Puiseux in 1850, that allows for negative and fractional exponents of the indeterminate...
, and circular root system
Root system
In mathematics, a root system is a configuration of vectors in a Euclidean space satisfying certain geometrical properties. The concept is fundamental in the theory of Lie groups and Lie algebras...
s).
See also
- Differential geometry of surfacesDifferential geometry of surfacesIn mathematics, the differential geometry of surfaces deals with smooth surfaces with various additional structures, most often, a Riemannian metric....
- Second fundamental form
- Principal curvaturePrincipal curvatureIn differential geometry, the two principal curvatures at a given point of a surface are the eigenvalues of the shape operator at the point. They measure how the surface bends by different amounts in different directions at that point.-Discussion:...
- Umbilical pointUmbilical pointIn the differential geometry of surfaces in three dimensions, umbilics or umbilical points are points that are locally spherical. At such points both principal curvatures are equal, and every tangent vector is a principal direction....
External links
- http://www.w-volk.de/BMG/ Berliner Mathematische Gesellschaft
- :de:Wilhelm Blaschke Wilhelm Blaschke
- :de:Gerrit Bol Gerrit Bol
- :de:Carathéodory Constantin Carathéodory
- :de:Stefan Cohn-Vossen Stefan Cohn-Vossen
- :de:Hans Hamburger Hans Hamburger
- http://owpdb.mfo.de/person_detail?id=5641 Tilla Klotz (1934 - 2002), photos