# José Fernando Escobar

### Quick Info

Born
20 December, 1954
Manizales, Colombia
Died
3 January, 2004
Cali, Colombia

Summary
José Escobar was a Colombian mathematician who made outstanding contributions to the Yamabe problem. He worked most of his life in the United States but made decisive contributions to developing and improving the mathematical level in Colombia.

### Biography

José Escobar was the son of Gustavo Escobar and Yadira Velásquez. José was widely known by the nickname 'Chepe', a common Spanish nickname for José. Gustavo and Yadira had three children, Maria Victoria Escobar (born 1951), Arturo Escobar (born 1952) and José Escobar, the subject of this biography. Maria Victoria became a psychologist. Arturo Escobar became a professor of anthropology and political ecology at the University of North Carolina, Chapel Hill, United States. Writing about his father, Gustavo, in [4], he explains:-
I want to remember my father, Gustavo, who died in 1990 still dreaming of his small hometown while trying (without great success in terms of conventional economic and development indicators) to make it in the big city so that his children could "get ahead" and become modern.
When José was very young, his parents moved from Manizales, about 120 km west of Bogotá in the mountainous coffee-growing region of western Colombia, to Cali, about 200 km south east of Manizales. Cali is famous for salsa dancing and, indeed, José would become an enthusiastic salsa dancer. José was brought up in Cali and he always considered that his hometown [11]:-
[Cali] became his home since early childhood and stamped characteristic features on his personality, like his joy for la salsa and la rumba, and his great love for sports. During his childhood and youth he was several times national champion of ornamental dives and in this specialty he won international titles for Colombia. He was extremely fond of soccer. He was almost as proud of his expertise in this matter as he was of his mathematical achievements, and he used to play it regularly during his entire life.
In fact he won the title of South American champion in diving and Central American and Caribbean champion in ornamental jumps.

Escobar was educated at the Colegio Lacordaire, a school in Cali which had been founded in 1956 by the Dominicans. In 1973 he graduated from the Colegio Lacordaire and entered the Universidad del Valle in Cali to study mathematics. He proved to be an outstanding student and graduated in 1977. In August of that year the Brazilian government awarded him a scholarship to allow him to undertake postgraduate studies at the Instituto Nacional de Matemática Pura e Aplicada in Río de Janeiro. His studies were supervised by the Uruguayan mathematician Ricardo Mañé Ramirez (1948-1995) who researched dynamical systems and ergodic theory. Escobar was awarded an M.Sc. in 1979 for his thesis Equações elíticas de 2ª ordem: soluções C* ao problema de Dirichlet . Being in Brazil for these years of postgraduate study meant that, in addition to his mathematical studies, he enjoyed learning about Brazilian music and dance. Of course, Brazil was a country where he could continue his passion for sport, especially soccer.

The exceptional quality of his work at the Instituto Nacional de Matemática Pura e Aplicada led to the award of a scholarship by the Brazilian National Council of Scientific and Technological Development to allow Escobar to undertake research at the University of California at Berkeley. His thesis advisor was Richard Melvin Schoen who was an expert on differential geometry and geometric analysis, making major contributions to the Yamabe problem. Escobar was, however, diagnosed with lymphatic cancer and spent two years being treated by Stanford's Medical Center. His exceptional fitness from his sporting activities together with his remarkable determination to overcome the disease, led to his eventual cure. He was helped in his recovery by María Elena Jiménez, a chemical laboratory specialist, whom he married in San Francisco on 22 October 1982. María Jimenez was aged 24 at the time of the marriage. After his recovery, Escobar made two research visits in 1985, one to the Mittag-Leffler Institute in Sweden which was holding a non-linear analysis year, and the other to the Institut des Hautes Études Scientifiques at Bures-sur-Yvette, south of Paris. He was awarded a Alfred P Sloan Doctoral Dissertation Fellowship for 1985-86.

Of course, the very serious health problem meant that Escobar did not complete his Ph.D. until 1986 when he was awarded the degree for his thesis Linear and Non-Linear Partial Differential Equations with Applications to Geometry. Fernando Codá Marques writes in [12] that a:-
... remarkable theorem was obtained in his Ph. D. thesis in collaboration with R Schoen. He completely determines the functions which can be realised as the scalar curvature of a conformal metric on nontrivial quotients of the three-sphere, extending the well-known Moser's result for $\mathbb{R}P^{2}$.
Let us note that the Moser referred to in this quote is Jürgen Kurt Moser. This result was published in a joint paper of Escobar and Schoen, Conformal metrics with prescribed scalar curvature which was published in Inventiones mathematicae in 1986. This was one of two papers Escobar published in 1986, the other being On the spectrum of the Laplacian on complete Riemannian manifolds.

The authors of [15] explain a little about Escobar's work on the Yamabe problem:-
A fundamental problem in differential geometry is the Yamabe problem, which asks whether every manifold can be mapped conformally to a manifold of constant curvature. When Chepe began his thesis work, the Yamabe problem had recently been solved affirmatively for manifolds without boundary by a group of mathematicians including Rich Schoen, his thesis advisor. Chepe's thesis, and much of his subsequent work, dealt with the Yamabe problem for manifolds with boundary, where there are additional difficulties to be overcome; for example, new ideas are required just to determine what conditions should hold for the boundary. Chepe was able to solve this problem in most cases, and to do so he had to introduce new methods in nonlinear partial differential equations.
Following the award of his doctorate, Escobar spent the rest of his career in the United States. He worked at the Courant Institute in New York (1986-1987) and then as a L E Dickson Instructor at the University of Chicago (1987-88). He was promoted to an Assistant Professor at Chicago in 1989. Peter May, who was Head of Mathematics at Chicago, wrote in 1989 (see, for example, [11]):-
... the assistant teachers of the University of Chicago form an extremely select group. Escobar is outstanding among the best, even within that group. The research in mathematics requires rare talents and there are very few people in the world with his capacity. People like him are the ones who are needed to fill in the best mathematics departments ...
Escobar then spent the years 1990 to 1994 as an associate professor at the University of Indiana at Bloomington before becoming a full professor at Cornell University in 1994.

Let us quote from [12] the overview given there of his research achievements:-
Escobar's powerful technical skills and his wide range of interests were certainly influenced by his Ph. D. studies under the supervision of Richard Schoen. During his mathematical career he touched several different and difficult problems lying in the domain of Geometric Analysis - the part of Differential Geometry concerned with the partial differential equations that naturally arise in geometrical questions. We can safely classify his results into two broad classes of problems.

The first class encompasses problems of prescribing the scalar curvature and the boundary mean curvature under conformal deformations of metrics. What is perhaps his greatest contribution falls in this class - the solution of the Yamabe Problem on manifolds with boundary in almost all cases. As a consequence of that achievement he proved a generalisation of the profound Riemann mapping theorem to higher dimensions, showing that, except possibly for dimensions 4 and 5, every smooth bounded open set of a Euclidean space admits a conformal scalar-flat metric with constant mean curvature on the boundary. ...

In the second class we can distinguish his work concerned with the spectral properties of the Laplacian on manifolds and isoperimetric inequalities. He was interested in the Laplace-Beltrami operator on complete and noncompact manifolds of nonnegative sectional curvature, together with similar questions on the differential form spectrum of the Hodge Laplacian. Motivated by his work on the Yamabe Problem he also became interested in the Steklov eigenvalue problem on manifolds, which somehow had been forgotten in the literature. He obtained various comparison results for the first nontrivial Steklov eigenvalue and studied its relation to isoperimetric constants. Very recently he also worked on isoperimetric inequalities in three dimensional PL-manifolds of nonpositive curvature.
Let us note that we [JJO'C and EFR] published a paper with Escobar in 1995 in the Obituario matemático 1992-1994 series of Lecturas Matemáticas, a joint publication of the Colombian Mathematical Society and the University of the Andes.

Although Escobar was employed in the United States, his influence on Colombian mathematics was enormous and decisive: for example after the award of his doctorate in 1986 he visited Colombia at least once a year, guided and worked with the Geometry and Topology Group of Universidad del Valle, collaborated actively with researchers from the National University in Bogotá and Medellín, and the Universidad de los Andes, organised Summer Schools in Differential Geometry, Partial Differential Equations and Numerical Analysis, as well as giving numerous lectures and workshops.

In 1992 Escobar was awarded a Presidential Faculty Fellowship. This programme was established in September 1991 at the request of President George Bush to [14]:-
... recognise, honour, and promote the integration of high-quality teaching and research in science and engineering fields; foster innovative and far-reaching developments in science and technology; create the next generation of academic leaders; and improve public understanding of the work of scientists and engineers.
Escobar visited the White House on 21 June 1992 to receive the award in person from President George Bush. He was the only mathematician to receive the award of \$100,000 per year for five years. Also in 1992 he was made an honorary member of the Colombian Mathematical Society and was awarded an honorary doctorate by the Universidad del Valle.

In the year 1992 he published another major contribution to extending the Yamabe Problem in his paper The Yamabe Problem on manifolds with boundary. In the same letter by Peter May from which we quoted above, he wrote about this paper:-
This article is an extraordinarily deep work, original and technically difficult, a really big contribution to Mathematics. Possibly no one without an exceptional ability could have proved the theorems existing in it. ... [It] is an article that any of the best mathematicians in the world would be proud to have written.
Escobar produced further outstanding results on the Yamabe Problem on compact manifolds with boundary in his 1996 paper Conformal deformation of a Riemannian metric to a constant scalar curvature metric with constant mean curvature on the boundary. In 2002 he published the book Topics in PDE's and differential geometry which was highly praised by Emmanuel Hebey in the review [6]:-
This is a very nice short book in which the author examines several topics on partial differential equations and their applications to Riemannian geometry. The book is very well written and contains very elegant mathematics. Chapter one discusses the Laplacian comparison theorem. Chapter two discusses the Steklov problem. Chapter three discusses conformal deformation of metrics. I strongly recommend this book, which I like very much.
We have looked at Escobar's outstanding research but we should note that he was also a charismatic teacher. Maria Antonia Garcés was a colleague of Escobar at Cornell University and wrote in an obituary which was published in the Cali newspaper El Pais:-
[In Cornell] Chepe knew how to create a special environment for Latin-American graduate students, mainly Colombians and Brazilians, who began to arrive at Cornell attracted by his brilliance and the things he published. As a teacher, Chepe's charisma, became proverbial in the Cornell Mathematics Department. Bringing down the barriers that separated great scientists from their students, Chepe built all environment of inspiration and camaraderie with his students, even the ones who were not doing their theses with him. He accomplished this by inviting them to his home and making them part of his family life.
The authors of [15] write about Escobar's political views:-
At the Universidad del Valle in Cali, Colombia, Escobar had met the mathematician María Mercedes Franco. His marriage to his first wife María Elena Jimenez ended in divorce and he married María Mercedes Franco. The book [5], which is written by her, contains the following biography:-
María Mercedes Franco Tello was born 1 October 1970 in Cali, Colombia , South America, the third child of Justo Pastor Franco Franco and Jeny Tello Mera. Her father died on 30 December 1976. Blessed with an extraordinary mother, wise twin brothers, a lively extended family (from her mother's side), the colourful people of el barrio San Juan Bosco in Cali, and the invaluable richness that only modest life can offer, Mercedes had a very happy and intense childhood and adolescence. Her interest in mathematics grew slowly and mainly because of her teaching experiences with high school classmates and students from other schools. She started her formal studies in Mathematics at the Universidad del Valle in Cali, Colombia, where she also worked as a computer lab assistant, mathematics tutor, and teaching assistant. Aside from the college life, she also worked as a mathematics high school teacher. She moved to the United States in 1994 to join her partner and future husband José Fernando Escobar Velásquez.
In early 2000, Escobar visited the Instituto de Matemática Pura e Aplicada in Rio de Janeiro where he was beginning a sabbatical year. While there, he was diagnosed with stomach cancer. He went to New York where he had a major operation. By August 2000 he was well enough to return to his teaching and research at Cornell. He published further major papers: Uniqueness and non-uniqueness of metrics with prescribed scalar and mean curvature on compact manifolds with boundary (2003); A new 3-dimensional curvature integral formula for PL-manifolds of non-positive curvature (2003); and Conformal metrics on the ball with zero scalar curvature and prescribed mean curvature on the boundary (2004). By the time this 2004 paper was published in the June part of the Journal of Functional Analysis, Escobar had already died. The paper, written jointly with Gonzalo García who had been a Ph.D. student of Escobar's graduating in 1999, has the following abstract:-
This paper considers the prescribed zero scalar curvature and mean curvature problem on the n-dimensional Euclidean ball for n ≥ 3. We consider the limits of solutions of the regularisation obtained by decreasing the critical exponent. We characterise those subcritical solutions which blow-up at the least possible energy level, determining the points at which they can concentrate, and their Morse indices. We show that when n = 3 this is the only blow-up which can occur for solutions. We use this in combination with the Morse inequalities for the subcritical problem to obtain a general existence theorem for the prescribed zero scalar curvature and mean curvature on the three-dimensional Euclidean ball. In the higher-dimensional case n ≥ 4, we give conditions on the function h to guarantee there is only one simple blow-up point.
It was in the second semester of 2003 that Escobar began to feel his health deteriorating again. He returned to Cali, Colombia, in November 2003, realising that he did not have long left and he wanted to be back in the home town he loved. His forty-ninth birthday was on 20 December 2003 and he died two weeks later.

The article [11] contains Jaime Lesmes' speech at the opening ceremony of the conference held in memory of Escobar. He said:-
I was bound up with Chepe by a friendship that goes back in time over 26 years, since I had the pleasure of being his teacher at the Instituto de Matematica Pura e Aplicada (IMPA) in Rio de Janeiro. This occasion allows me to express my great admiration and appreciation for his outstanding mathematical capacity, for his work as a scientist, as a teacher and guide, and for his honesty and remarkable human qualities. Admiration and appreciation which was always felt by these who knew Chepe. I would like to point out very specially the gratitude and recognition of the Colombian mathematical community for his untiring determination to be helpful, and his extensive work and decisive contributions to the development and rising of the mathematical level in Colombia. Chepe loved Colombia, his country and our country, and throughout his life he maintained close ties with it.
In 2005, Escobar's wife María Mercedes Franco published the book Existence and Regularity of Minimizers for a One-dimensional Elasticity Problem Without Convexity. She dedicated the work to Chepe:-
To my late husband José Fernando "Chepe" Escobar. For ten years we shared life, family, friends, joys, griefs, and dreams. We shared work and leisure times. We danced, drank wine, cooked, travelled, we were life partners and went through life events that ranged from absolutely joyous to extremely painful. Without a doubt, ours was an extraordinary journey and I will forever cherish the memories and value the lessons. Through him I got to meet and enjoyed many wonderful people, amongst them the members of the Escobar Velásquez family and mathematicians Alexander Freire, Jaime Lesmes, Sumio Yamada, Jia-ping Wang, Gonzalo García, Juan Carlos Uribe, Jean Carlos Cortissoz, Nelia Charalambous, Henrique Araújo, Fernando Codá Márques, Fernando Schwartz.

In 2017 the Colombian Mathematical Society instituted the "José Fernando Escobar" Prize for Research in Mathematics.

### References (show)

1. Escobar, José Fernando, in Who was who in America 16 (Marquis Who's Who, 2005), 80.
2. Escobar, José Fernando, in Who's Who in the Midwest (Marquis Who's Who, 1994), 218.
3. Escobar, José Fernando, in American Men & Women of Science (Thomson Gale, Detroit, 2005).
4. A Escobar, The making and unmaking of the Third World (Princeton University Press, 1995).
5. M M Franco, Existence and Regularity of Minimizers for a One-dimensional Elasticity Problem Without Convexity (Cornell University, 2005).
6. E Hebey, Review: Topics in PDE's and differential geometry, by José F Escobar, Mathematical Reviews MR1946561 (2003m:53054).
7. José Escobar "Chepe", The World of Mathematics in Colombia: Famous Mathematicians.
http://www.geocities.ws/jaynday529/
8. José Fernando "Chepe" Escobar, Ithaca Journal (31 January 2004).
9. José Fernando Escobar, prabook.
https://prabook.com/web/jose_fernando.escobar/3607817
10. José Fernando Escobar, El Tiempo (12 June 1992).
https://www.eltiempo.com/archivo/documento/MAM-135821
11. J Lesmes, Chepe Escobar, Matemáticas: Enseñanza Universitaria 15 (S1) (2007), 1-9.
12. F C Marques, On the mathematical work of José F Escobar, Matemática Contemporânea 29 (2005), 41-61.
13. Premio José Fernando Escobar a la Investigación en Matemáticas, Sociedad Colombiana de Matemáticas.
14. Presidential Faculty Fellows (PFF) program, National Science Foundation.
https://www.nsf.gov/pubs/2001/nsf01118/nsf01118_2.pdf
15. L Saloff-Coste, R Strichartz and L J Billera, José Fernando Escobar: December 20, 1954 - January 3, 2004, Memorial Statements of the Cornell University Faculty 2000-2009 8 (2010), 161-163.
https://ecommons.cornell.edu/bitstream/handle/1813/19086/Escobar_Jose_Fernando_2004.pdf?sequence=1&isAllowed=y
16. L Sun, Yamabe Problem on compact manifolds with boundary, Ph.D. Dissertation (Graduate School, New Brunswick Rutgers, The State University of New Jersey, 2017).