# Biographical sketch

A.V. Pogorelov was born on March 3, 1919, in Korocha town near Belgorod (Russia). On his farther's "farm" there was just one cow and one horse. During the collectivization they were taken from him. Once his father came to the collective-farm stable and found his horse exhausted, dying from thirst, while the groom was drank. Vasily Stepanovich hit the groom, a former pauper. This incident was reported as if a kulak has beaten a peasant, and Vasily Stepanovich was forced to escape the town, with wife Ekaterina Ivanovna, without even taking the children. A week later Ekaterina Ivanovna has secretly returned for the children. This is how A. V. Pogorelov came to Kharkov, where his father became a construction worker on the building of the tractor factory. A.V.Pogorelov told me the story of how his parents have suffered during the collectivization I have heard from him only in 2000. In my opinion, these events had a strong influence on his life and on the way of his public behavior. He was always very cautious in expressions and liked to quote his mother who kept saying: silence is gold. However, he never did the things contradicting his political views. Several times, he successfully escaped becoming a member of the Communist Party (which was almost compulsory for a person of his scale in the USSR). As far as I know, he never signed any letters of condemnation of dissidents, but, again, any letters in their support, as well. Several times he was elected to the Supreme Soviet of Ukraine (although, as he said later, against his will).

The mathematical abilities of A.V.Pogorelov became apparent already at school. His school nickname was Pascal. He became the winner of one of the first school mathematical competitions organized by the Kharkov University, and then of several All-Ukrainian Mathematical Olympiads. Another talent of A.V.Pogorelov was the painting. The parents did not know, which profession to choose for him. His mother asked the son's mathematics teacher for advice. He had a look at the paintings and said that the boy has brilliant abilities, but in the time of industrialization the painting will not give the resource for life. This advice determined their choice. In 1937, Alexey Vasilyevich became a student of the Department of Mathematics at the Faculty of Physics and Mathematics of the Kharkov University.

His passion to mathematics immediately drew the attention of the teachers. Professor P.A.Solovjev gave him the book by T.Bonnezen and V.Fenchel "Theory of convex bodies". From that moment and for the rest of his life, geometry became the main interest of Alexey Vasilyevich. His study was interrupted by the War. He was conscripted and sent to study at the Air Force Zhukovski Academy. But he still thinks about geometry. In August 1943, in a letter to Professor Ya.P.Blank he says: "Very much I regret, that I left in Kharkov the abstracts of Bonnezen and Fenchel on the convex bodies. There are many interesting problems in geometry "in the large"... Do you have any interesting problem of geometry "in the large" or of geometry in general in mind?"

The first audition lasted for ten minutes. Sitting on a backpack, A.D.Aleksandrov asked Alexey Vasilyevich the following question: is it true, that on a closed convex surface of the Gauss curvature $$K \leq 1$$, any geodesic segment of lengths at most $$\pi$$ is minimizing? It took A.V. a year to answer this question (in affirmative) and to publish the result in 1946. The multidimensional generalization of his theorem is a well-known theorem of Riemannian geometry, which was proved in 1959 by W.Klingenberg: on a complete simply connected Riemannian manifold $$M^{2n}$$ of sectional curvature satisfying $$0 < K_{\sigma}\leq\lambda$$, a geodesic of the length $$\leq \pi / \sqrt{\lambda}$$ is minimizing. In the odd-dimensional case, one needs a two-sided bound for the curvature to obtain the same result, namely $$0 < \frac{1}{4} \lambda \leq K_{\sigma} \leq \lambda$$ (and the inequality cannot be improved).

Few years ago, I asked Alexey Vasilyevich, why the Soviet mathematicians at that time showed not much interest to the global Riemannian geometry. He answered: "We had enough interesting problems to think about". However, as V.A.Toponogov told me later, the first person who appreciated his comparison theorem for triangles in a Riemannian space was A.V.Pogorelov (in my opinion, it would be more correct to call this theorem the Aleksandrov-Toponogov theorem, since A.D.Aleksandrov discovered and proved it for general convex surfaces in the three-dimensional Euclidean space).

Alexey Vasilyevich became a postgraduate-in-correspondence at Moscow State University under the supervision of professor N.V.Efimov. Having read the manuscript of the A.D.Aleksandrov's book "Intrinsic geometry of convex surfaces", he starts his work in the geometry of general convex surfaces.

One of the main roles of a supervisor, in the opinion of N.V.Efimov, was to inspire a post-graduate student to solving difficult and challenging problems. I gave numerous talks both at the N.V.Efimov's and the A.V.Pogorelov's seminars. They were very different by style. The N.V.Efimov's seminar was long gathered, then the talk lasted for two hours or more, and the talk was always praised very warmly, so it was almost impossible to understand the real value of the result. A.V. always started on time, very punctually. The report lasted for at most an hour. A.V. did not like to go through the details of the proof (probably because in many cases, after the theorem was stated, he could prove it immediately).

In the estimation of the results he was strict and even severe. For example, in 1968, three applicants for the Doctor degree presented their theses at the Pogorelov’s seminar in Kharkov. He supported only one of them, V.A.Toponogov, and rejected the other two, who went to Novosibirsk to A.D.Aleksandrov. All three theses were later successfully defended.

A.V. praised rarely, but when he did – that meant that the result was really good. He had a very fast thinking, an enormous geometric intuition, and grasped the essence of the result very fast. Many seminar participants were afraid to ask questions not to look foolish.

In 1947, A.V.Pogorelov defended his Candidate thesis. The main result of his thesis was the following theorem: every general closed convex surface possesses three closed quasi-geodesics. This theorem generalizes the Lusternik - Shnirelman theorem on the existence of three closed geodesic on a closed regular convex surface (a quasi-geodesic is a generalization of a geodesics; both the left and the right "turns" of a quasi-geodesic are nonnegative; for instance, the union of two generatrices of a round cone dividing the cone angle in two halves is a quasi-geodesic).

After defending his Candidate thesis, A.V. discharges from the military service and moves to Kharkov (probably, this was not an easy thing to do at that time: he was discharges by the same Order of the Defence Minister, as the son of M.M.Litvinov, the former Soviet Minister for Foreign Affairs). In one year, he defends his Doctor Thesis on the unique determination of a convex surface of bounded relative curvature. Soon after that, he proves the theorem on the unique determination in the most general settings.

Until 1970, A.V.Pogorelov lectured at Kharkov University. Based on this lecture notes, he published a series of brilliant textbooks on analytic and differential geometry and the foundation of geometry. Sometimes, during routine lectures, he was thinking about his research. Anecdote says that on one of such lectures reflecting on something completely different he started improvising and became lost. Then he opened the textbook with the words: "What does the author say on the topic? Oh, yes, it is obvious . . . ". In contrast, when lecturing on a topic interesting to him, A.V.Pogorelov was very enthusiastic and inspired (I remember one of his topology courses for the 4th year students). But perhaps the best of his lecturing brilliance was seen when he was presenting his own results. His talks were real fine art performances. In his opinion, one of the most valuable qualities of a mathematical result is its beauty and naturalness. That is why he usually omitted technicalities, and for the sake of simplicity and beauty was ready to sacrifice the generality.

A.V.Pogorelov was the author of one of the most popular school textbooks in geometry. This began as follows. He was a member of the commission on the school education whose head was A.N.Kolmogorov. A.V. disagreed with the textbook written by A.N.Kolmogorov and his coauthors and wrote his own manual for teachers on elementary geometry, in which he built the whole school geometry course starting with a set of natural and intuitive axioms. The manual was published in 1969 and formed a basis for his school textbook. A.V. used to say: "My textbook is the Kiselyov's improved textbook" ("Elementary geometry" by A.P.Kiselyov is probably the most well-known Russian-language school geometry textbook; it was first published in 1892, with the last edition in 2002; many generations of students studied the Kiselyov's "Geometry"). The first version of the A.V.Pogorelov's textbook sparked sharp criticism from A.D.Aleksandrov whom Pogorelov deeply respected. This criticism was based on implementing the axiomatic approach as early as in year six at school: "What is the point to prove 'obvious' statements (from the student's point of view)?". After reworking of the textbook, these disagreements were resolved, and they remained in strong friendship till the last days of A.D.Aleksandrov.

Alexey Vasilyevich was a person of the highest decency. When a five year contract with the "Prosvescheniye" Publisher was coming to an end, another publisher offered a very tempting contract to him. He refused on the unique ground that it will be unfair to the editor of the textbook. It should be noted that the money for the school textbook republishing were the main source of his living in the middle of the 90-th.

A.V.Pogorelov told me that I.G.Petrovsky invited him to the Moscow University, I.M.Vinogradov invited to Moscow Mathematical Institute, A.D.Aleksandrov invited to Leningrad several times. He even spent one year (1955-1956) in Leningrad, but then returned to Kharkov. He preferred to stay in Kharkov, far from the fuss and noise of the capitals. In Kharkov he proved his theorems, and to Moscow and Leningrad he went to shine.

Alexey Vasilyevich Pogorelov was a person blessed by an incredible natural talent combined with a constant tireless labor.

# Scientific interests

By the beginning of the 20th century, the methods for solving of local problems related to regular surfaces were developed. By the thirties, there were developed the methods for solving the problems in geometry "in the large". These methods were related mainly to the theory of partial differential equations. Mathematicians were helpless when the surfaces were irregular (had conic points, ribbed points) and when the intrinsic geometry was given not by a regular positive definite quadratic form, but simply by a metric space of a fairly general form. A breakthrough in the study of irregular metrics and irregular surfaces was made by an outstanding geometer A.D. Aleksandrov. He constructed the theory of metric spaces of nonnegative curvature, later named by Aleksandrov's spaces. As a special case the theory covered the intrinsic geometry of general convex surfaces (the boundary of an arbitrary convex body). A.D. Aleksandrov began to study the connections between the intrinsic and extrinsic geometries of irregular convex surfaces. He proved that any metric of non-negative curvature given on a two-dimensional sphere (including an irregular metric defined as a metric space with inner metric) can be isometrically immersed into a three-dimensional Euclidean space in a form of closed convex surface, but the answers to the following fundamental questions were unknown:

1. Whether the immersion is unique up to motion?
2. If the metric given on the sphere is a regular one of positive Gaussian curvature, is then the surface with this metric regular?
3. G. Minkowski proved the existence theorem for a closed convex surface with the Gaussian curvature given as function of the unit normal under some natural condition for this function. But there was an open problem: if the function is regular on a sphere, is the surface is regular itself?

After solving these problems, the theory created by Aleksandrov would have received "full citizenship" in mathematics and could be applied also in the classical regular case. Each of these 3 questions were answered positively by A.V. Pogorelov. Using the synthetic geometric methods, he developed geometric methods to obtain the a priori estimates for solutions of the Monge-Ampère equations. On the one hand, he used these equations to solve geometric problems; on the other hand, based on geometric reasons, he constructed a generalized solution of the Monge-Ampère equation, and then proved its regularity for regular right-hand side. In fact, in these pioneering works A.V. Pogorelov laid the foundation of geometrical analysis. On this way, he received the following fundamental results:

1. Let $$F^1$$ and $$F^2$$ be two closed convex isometric surfaces in a three-dimensional Euclidean space or in a spherical space. Then the surfaces coincide up to the motion in space.
2. A closed convex surface in a space of constant curvature is rigid outside the flat domains on the surface. This means that it admits only trivial infinitesimal bendings.
3. If the metric of a convex surface is regular in the class $$C^k$$ $$(k \geq 2)$$ in a space of constant curvature c and the Gaussian curvature of the surface $$k > c$$, then the surface is regular of class $$C^{k-1,\alpha}$$.

For the domains on convex surfaces, the assertions 1) and 2) are false. The local and global properties of surfaces are significantly different. By proof of the assertion 1) A.V. Pogorelov completed the solution of the problem open for more than a century. The first result in this direction was obtained by Cauchy for closed convex polyhedra in 1804. We recall that two surfaces are said to be isometric if there exists a mapping of one surface onto another which preserves the lengths of the corresponding curves.

The theorems proved by Pogorelov formed the basis for his nonlinear theory of thin shells. In this theory, there are considered those elastic states of the shell, which differ significantly from the original form. Under such deformations, the middle surface of the thin shell undergoes bending with preservation of the metric. This makes it possible, by using theorems proved by Pogorelov for convex surfaces, to investigate the stability loss and the over critical elastic state of convex shells under the action of a given load. Such shells are the most common elements of modern designs.

The results 1) and 2) were generalized for regular surfaces in a the Riemannian space. In addition, the Weyl problem for Riemannian space was solved: it was proved that a regular metric of Gaussian curvature greater than the constant $$c$$ on a two-dimensional sphere can be isometrically immersed into complete three-dimensional Riemannian space of curvature $$< c$$ in a form of a regular surface. Studying the methods developed in the proof of this result, the Abel Prize laureate M. Gromov introduced the concept of pseudoholomorphic curves, which are the main tool in modern symplectic geometry.

A closed convex hypersurface is uniquely defined not only by the metric, but also by the Gaussian curvature as a function of a unit normal. Moreover, the hypersurface is uniquely determined up to a parallel transport. This was proved by G. Minkowski. But whether the hypersurface is regular under the condition that the Gaussian curvature $$K(n)$$ is a regular function of a unit normal? Pogorelov proved that if the positive function $$K(n)$$ belongs to the class $$C^k$$ $$(k \geq 3)$$, then the support function will be of regularity class $$C^{k+1, \nu}$$ $$(0 < \nu < 1)$$.

The hardest part of the proof of the theorem was to obtain a priori estimates for the derivatives of the support function of a hypersurface up to third order inclusively. Pogorelov's method of a priori estimates was used by S.-T. Yau to obtain the a priori estimates of the solutions of the complex Monge-Ampere equation. This was the main stage in the proof of the existence of the Calabi-Yau manifolds, which play an important role in theoretical physics. The Monge-Ampère equation has the form $$|Z_{ij}|=f(x_1,...,x_n,Z,Z_1,...,Z_n)$$

A priori estimates in the Minkowski problem are a priori for the solution of the Monge-Ampère equation with the function $$f=\frac{1}{K(1+x_1^2+...+x_n^2)^{n/2+1}}$$

At that time there was no approach to studying this completely nonlinear equation. A. V. Pogorelov has created the theory of the Monge-Ampère equation by using the geometric methods. First, going from polyhedra, he proved the existence of generalized solutions under natural conditions on the right-hand side. After that he has found the a priori estimates for the derivatives up to the third order inclusively for the regular solutions. Using the a priori estimates, he has proved the regularity of strictly convex solutions, the existence of solutions of the Dirichlet problem and their regularity. The Monge-Ampère equation is an essential component of the Monge-Kantorovich transport problem; it is used in conformal, affine, Kähler geometries, in meteorology and in financial mathematics. Somehow A.V. Pogorelov said about the Monge-Ampère equation: "This is a great equation, which I had the honor to deal with".

One of the most conceptual works of A.V. Pogorelov refers to the cycle of works about smooth surfaces of bounded extrinsic curvature. A.D. Aleksandrov created a theory of general metric manifolds that naturally generalize the Riemannian manifolds. In particular, he introduced the class of two-dimensional manifolds of bounded curvature. They exhaust the class of all metrized two-dimensional manifolds that admit, in a neighborhood of each point, a uniform approximation by Riemannian metrics with absolute integral curvature (i.e., the integral of the module of Gaussian curvature) bounded in aggregate.

Naturally, the question arose about the class of surfaces in three-dimensional Euclidean space carrying such a metric with preservation of connections between the metric and the extrinsic geometry of the surface. Partially answering this question, A.V. Pogorelov introduced the class of $$C^1$$-smooth surfaces with the requirement on the area of a spherical image to be bounded, taking into account the multiplicity of the covering in some neighborhood of each point of the surface. Such surfaces are called surfaces of bounded curvature.

For such surfaces there is also a very close connection between the intrinsic geometry of the surface and its external shape: a complete surface with a bounded extrinsic curvature and a nonnegative intrinsic curvature (not equal to zero) is either a closed convex surface or an infinite convex surface; a complete surface with zero intrinsic curvature and bounded extrinsic curvature is a cylinder.

The first work of A.V. Pogorelov on surfaces of bounded extrinsic curvature was published in 1953. In 1954, J. Nash published a paper on $$C^1$$-isometric immersions, which was improved by N. Kuiper in 1955. It follows from these studies that the Riemannian metric defined on a two-dimensional manifold, under very general assumptions, admits a realization on a $$C^1$$-smooth surface in a three-dimensional Euclidean space. Moreover, this realization is carried out as freely as a topological immersion into the space of the manifold on which the metric is given. Hence it is clear that for $$C^1$$- surfaces, even with a good intrinsic metric, it is impossible to preserve the connections between the intrinsic and extrinsic curvatures. Even in case if $$C^1$$-surface carries a regular metric of positive Gaussian curvature, then this does not imply the local convexity of the surface. This emphasizes the naturalness of the class of surfaces of bounded extrinsic curvature introduced by A.V Pogorelov.

A. V. Pogorelov solved the fourth problem of Hilbert, set by D. Hilbert at the II International Congress of Mathematicians in Paris in 1900. He found all, up to isomorphism, realizations of the systems of axioms of classical geometries (Euclid, Lobachevsky and elliptic) if one omit the congruence axioms containing the concept of angle and supplement these systems with the axiom of "triangle inequality".

# Selected publications

• Die eindentige Bestimmung allgemeiner konvexer Flachen. - Berlin: Akad. Verl., 1956.- 79 s.
• Die Verbiegung konvexer Flachen. - Berlin: Verl., 1957.-135 s.
• Einige Untersuchungen zur Riemannschen Geometrie “im Grossen” – Berlin: VEB Deutsch. Verl. Wiss., 1960. – 71s.
• Topics in the theory of surfaces in elliptic space – New York: Gordon and Breach, 1961. – 130 p.
• Monge – Ampere equations of elliptic type. - Groningen: P. Noordhoff, 1964.- 114 p.
• Some results on surface theory in the large. – Advances math. – 1964. – 1, №2. – P. 191-264.
• Extrinsic geometry of convex surfaces. – Providence, R. I.: AMS, 1973. - 665 p.
• The Minkowski multidimensional problem. - Washington: Scripta, 1978. - 106 p.
• Hilbert’s fourth problem. - Washington: Scripta, 1979. – 97 p.
• Bending of surfaces and stability of shells. - Providence, R. I., AMS, 1989. – 77 p.
• Multidimensional Monge-Ampere equation / Harwood Academic Publishers // Rev. in Math. And Math. Phys. – 1995. – 10. - 103 p. Cambridge Scientific Publishers // Rev. in Math. And Math. Phys., 2009. – 110 p.
• Busemann regular G-spaces. Harwood Academic Publishers // Rev. in Math. And Math. Phys. – 1998. – 10. – Part 4. - 102 p.
• Differential geometry. - Groningen: P. Noordhoff, 1957. - 172 p.: 2-nd ed. 1967.
• Lectures on foundations of geometry. - Groningen: P. Noordhoff, 1966. – 137 p.
• Geometry (manual for higher school). Mir Publishers, Moscow, 1987. – 312 с.
• A. V. Pogorelov "Analytical Geometry" Mir Publishers, Moscow, 1980