In
differential geometry
Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of Calculus, single variable calculus, vector calculus, lin ...
, the Carathéodory conjecture is a mathematical
conjecture
In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis or Fermat's conjecture (now a theorem, proven in 1995 by Andrew Wiles), ha ...
attributed to
Constantin Carathéodory
Constantin Carathéodory (; 13 September 1873 – 2 February 1950) was a Greeks, Greek mathematician who spent most of his professional career in Germany. He made significant contributions to real and complex analysis, the calculus of variations, ...
by
Hans Ludwig Hamburger in a session of the Berlin Mathematical Society in 1924.
[''Sitzungsberichte der Berliner Mathematischen Gesellschaft'', 210. Sitzung am 26. März 1924, Dieterichsche Universitätsbuchdruckerei, Göttingen 1924] Carathéodory never committed the conjecture into writing, but did publish a paper on a related subject. In
John Edensor Littlewood
John Edensor Littlewood (9 June 1885 – 6 September 1977) was a British mathematician. He worked on topics relating to analysis, number theory, and differential equations and had lengthy collaborations with G. H. Hardy, Srinivasa Ramanu ...
mentions the conjecture and Hamburger's contribution
[ H. Hamburger, ''Beweis einer Caratheodoryschen Vermutung. I'', Ann. Math. (2) 41, 63—86 (1940); ''Beweis einer Caratheodoryschen Vermutung. II'', Acta Math. 73, 175—228 (1941), and ''Beweis einer Caratheodoryschen Vermutung. III'', Acta Math. 73, 229—332 (1941)] as an example of a mathematical claim that is easy to state but difficult to prove.
Dirk Struik describes in the formal analogy of the conjecture with the
four-vertex theorem
In geometry, the four-vertex theorem states that the curvature along a simple, closed, smooth plane curve has at least four local extrema (specifically, at least two local maxima and at least two local minima). The name of the theorem derives ...
for
plane curve
In mathematics, a plane curve is a curve in a plane that may be a Euclidean plane, an affine plane or a projective plane. The most frequently studied cases are smooth plane curves (including piecewise smooth plane curves), and algebraic plane c ...
s. Modern references to the conjecture are the problem list of
Shing-Tung Yau
Shing-Tung Yau (; ; born April 4, 1949) is a Chinese-American mathematician. He is the director of the Yau Mathematical Sciences Center at Tsinghua University and professor emeritus at Harvard University. Until 2022, Yau was the William Caspar ...
, the books of
Marcel Berger, as well as the books.
[I. Nikolaev, ''Foliations on Surfaces'', Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge A, Series of Modern Surveys in Mathematics, Springer 2001 ]
The local real analytic version of the conjecture has had a troubled history with published proofs
which contained gaps.
The proof for surfaces of
Hölder smoothness by Brendan Guilfoyle and Wilhelm Klingenberg, first announced in 2008,
was published in three parts
concluding in 2024. Their proof involves techniques spanning a number of areas of mathematics, including neutral
Kähler geometry,
parabolic PDEs, and
Sard-Smale theory.
Statement of the conjecture
The conjecture claims that any convex, closed and sufficiently smooth surface in three dimensional
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...
needs to admit at least two
umbilic points. In the sense of the conjecture, the
spheroid
A spheroid, also known as an ellipsoid of revolution or rotational ellipsoid, is a quadric surface (mathematics), surface obtained by Surface of revolution, rotating an ellipse about one of its principal axes; in other words, an ellipsoid with t ...
with only two umbilic points and the
sphere
A sphere (from Ancient Greek, Greek , ) is a surface (mathematics), surface analogous to the circle, a curve. In solid geometry, a sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...
, all points of which are umbilic, are examples of surfaces with minimal and maximal numbers of the umbilicus. For the conjecture to be well posed, or the umbilic points to be well-defined, the surface needs to be at least twice differentiable. The proof of Guilfoyle and Klingenberg requires that the surface have
Hölder third-order derivatives, a reflection of their use of second order parabolic methods in the 1-jet of the surface.
The case of real analytic surfaces
The invited address of
Stefan Cohn-Vossen to the
International Congress of Mathematicians
The International Congress of Mathematicians (ICM) is the largest conference for the topic of mathematics. It meets once every four years, hosted by the International Mathematical Union (IMU).
The Fields Medals, the IMU Abacus Medal (known before ...
of 1928 in
Bologna
Bologna ( , , ; ; ) is the capital and largest city of the Emilia-Romagna region in northern Italy. It is the List of cities in Italy, seventh most populous city in Italy, with about 400,000 inhabitants and 150 different nationalities. Its M ...
was on the subject and in the 1929 edition of
Wilhelm Blaschke
Wilhelm Johann Eugen Blaschke (13 September 1885 – 17 March 1962) was an Austrian mathematician working in the fields of differential and integral geometry.
Education and career
Blaschke was the son of mathematician Josef Blaschke, who taugh ...
's third volume on Differential Geometry he states:
While this book goes into print, Mr. Cohn-Vossen has succeeded in proving that closed real-analytic surfaces do not have umbilic points of index > 2 (invited talk at the ICM in Bologna 1928). This proves the conjecture of Carathéodory for such surfaces, namely that they need to have at least two umbilics.
Here Blaschke's index is twice the usual definition for an index of an umbilic point, and the global conjecture follows by the
Poincaré–Hopf index theorem. No paper was submitted by Cohn-Vossen to the proceedings of the International Congress, while in later editions of Blaschke's book the above comments were removed. It is, therefore, reasonable to assume that this work was inconclusive.
For analytic surfaces, an affirmative answer to this conjecture was given in 1940 by
Hans Hamburger in a long paper published in three parts.
The approach of Hamburger was also via a local index estimate for isolated umbilics, which he had shown to imply the conjecture in his earlier work.
In 1943, a shorter proof was proposed by
Gerrit Bol,
see also, but, in 1959,
Tilla Klotz found and corrected a gap in Bol's proof.
Her proof, in turn, was announced to be incomplete in Hanspeter Scherbel's dissertation
(no results of that dissertation related to the Carathéodory conjecture were published for decades). Among other publications we refer to the following papers.
All the proofs mentioned above are based on Hamburger's reduction of the Carathéodory conjecture to the following conjecture: the index of every isolated umbilic point is never greater than one.
Roughly speaking, the main difficulty lies in the 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:
"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, 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 helpful in many branches of mathematics, including algebraic ...
(more precisely, to techniques involving analytic
implicit function
In mathematics, an implicit equation is a relation of the form R(x_1, \dots, x_n) = 0, where is a function of several variables (often a polynomial). For example, the implicit equation of the unit circle is x^2 + y^2 - 1 = 0.
An implicit func ...
s,
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''. It states that such a function is, up to multiplication by a function not zero at ''P'', a poly ...
,
Puiseux series
In mathematics, Puiseux series are a generalization of power series that allow for negative and fractional exponents of the indeterminate. For example, the series
: \begin
x^ &+ 2x^ + x^ + 2x^ + x^ + x^5 + \cdots\\
&=x^+ 2x^ + x^ + 2x^ + x^ + ...
, and circular
root system
In mathematics, a root system is a configuration of vector space, vectors in a Euclidean space satisfying certain geometrical properties. The concept is fundamental in the theory of Lie groups and Lie algebras, especially the classification and ...
s).
Application of the analytic index bound
Hamburger’s umbilic index bound for analytic surfaces leads to restrictions on the position of the roots of certain types of holomorphic polynomials. In particular, a holomorphic polynomial is said to be ''self-inversive'' if the set of roots is invariant under reflection in the unit circle. It can be shown that for a polynomial with self-inversive second derivative, none of whose roots lie on the unit circle, the number of roots (counted with multiplicity) inside the unit circle is less than or equal to ⌊N/2⌋ + 1. The proof takes any holomorphic polynomial with the stipulated properties and constructs a real analytic surface with an isolated umbilic point. The index is determined by the number of zeros of the polynomial that lie inside the unit circle, and then Hamburger’s bound yields the stated result.
The general smooth case
In 2008, Brendan Guilfoyle and Wilhelm Klingenberg announced
a proof of the global conjecture for surfaces of smoothness
. The proof was published in three parts.
Their method uses neutral
Kähler geometry of the
Klein quadric to define an associated Riemann-Hilbert boundary value problem, and then applies
mean curvature flow
In the field of differential geometry in mathematics, mean curvature flow is an example of a geometric flow of hypersurfaces in a Riemannian manifold (for example, smooth surfaces in 3-dimensional Euclidean space). Intuitively, a family of sur ...
and the Sard–Smale Theorem on regular values of Fredholm operators to prove a contradiction for a surface with a single umbilic point.
In particular, the boundary value problem seeks to find a
holomorphic curve with boundary lying on the Lagrangian surface in the
Klein quadric determined by the normal lines to the surface in Euclidean 3-space. Previously it was proven that the number of isolated umbilic points contained on the surface in
determines the Keller-Maslov class of the boundary curve
and therefore, when the problem is Fredholm regular, determines the dimension of the space of holomorphic disks.
All of the geometric quantities referred to are defined with respect to the canonical ''neutral'' Kähler structure, for which surfaces can be both holomorphic and Lagrangian.
In addressing the global conjecture, the question is “''what would be so special about a smooth closed convex surface in
with a single umbilic point?''” This is answered by Guilfoyle and Klingenberg:
the associated Riemann-Hilbert boundary value problem would be Fredholm regular. The existence of an isometry group of sufficient size to fix a point has been proven to be enough to ensure this, thus identifying the size of the Euclidean isometry group of
as the underlying reason why the Carathéodory conjecture is true. This is reinforced by a more recent result in which ambient smooth metrics (without symmetries) that are different but arbitrarily close to the Euclidean metric on
, are constructed that admit smooth convex surfaces violating both the local and the global conjectures.
By Fredholm regularity, for a generic convex surface close to a putative counter-example of the global Carathéodory Conjecture, the associated Riemann-Hilbert problem would have no solutions. The second step of the proof is to show that such solutions always exist, thus concluding the non-existence of a counter-example. This is done using co-dimension 2 mean curvature flow with boundary. The required interior estimates for higher codimensional mean curvature flow in an indefinite geometry appear in.
The final part is the establishment of sufficient boundary control under mean curvature flow to ensure weak convergence. This is carried out while also proving a related conjecture of
Toponogov regarding umbilic points on complete planes for which the same methods work.
In 2012 the proof was announced of a weaker version of the local index conjecture for smooth surfaces, namely that an isolated umbilic must have index less than or equal to 3/2. The proof follows that of the global conjecture, but also uses more topological methods, in particular, replacing hyperbolic umbilic points by totally real cross-caps in the boundary of the associated Riemann-Hilbert problem. It leaves open the possibility of a smooth (non-real analytic by Hamburger
) convex surface with an isolated umbilic of index 3/2.
In 2012, Mohammad Ghomi and Ralph Howard showed, using a
Möbius transformation
In geometry and complex analysis, a Möbius transformation of the complex plane is a rational function of the form
f(z) = \frac
of one complex number, complex variable ; here the coefficients , , , are complex numbers satisfying .
Geometrically ...
, that the global conjecture for surfaces of smoothness
can be reformulated in terms of the number of umbilic points on graphs subject to certain asymptotics of the gradient.
See also
*
Differential geometry of surfaces
In mathematics, the differential geometry of surfaces deals with the differential geometry of smooth manifold, smooth Surface (topology), surfaces with various additional structures, most often, a Riemannian metric.
Surfaces have been extensiv ...
*
Second fundamental form
*
Principal curvature
In differential geometry, the two principal curvatures at a given point of a surface (mathematics), surface are the maximum and minimum values of the curvature as expressed by the eigenvalues of the shape operator at that point. They measure how ...
*
Umbilical point
In the differential geometry of surfaces in three dimensions, umbilics or umbilical points are points on a surface that are locally spherical. At such points the normal curvatures in all directions are equal, hence, both principal curvatures are e ...
References
{{DEFAULTSORT:Caratheodory conjecture
Conjectures
Unsolved problems in geometry
Differential geometry of surfaces