Pedro Ontaneda

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Pedro Ontaneda
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Alma materStony Brook University
EmployerBinghamton University
Known forKnown for his joint work with F.T. Farrell on the space of negatively curved metrics

Pedro Ontaneda is a Peruvian mathematician working in the United States where he is Professor of Mathematics at Binghamton University (State University of New York). Much of his research concerns the Riemannian geometry and topology of manifolds of negative curvature.

Ontaneda earned his Ph.D. at Stony Brook University in 1994, under the supervision of Lowell Edwin Jones. After a post-doctoral position at Vanderbilt University, he became a faculty member at Departamento de Matemática, Universidade Federal de Pernambuco, Brazil. He moved to the United States and to the Binghamton University Department of Mathematical Sciences in 2005.

In his doctoral dissertation he constructed piecewise linear exotic negatively curved Riemannian metrics on some manifolds, work that extended previous results by Farrell and Jones. Some of his research has dealt with a long-standing problem posed by S. T. Yau, namely: If f is a diffeomorphism between two compact negatively curved manifolds and h is the unique harmonic map homotopic to f, is h a homeomorphism? In joint work, Ontaneda, Farrell and Ragunathan showed that the answer is negative[1]. Building on this, Ontaneda and Farrell went on to show that even when a harmonic map can be approximated by a diffeomorphism, the harmonic map may not be a diffeomorphism. In summary, this work showed the (unexpected) limitations to the harmonic map technique.

Ontaneda is also known for his joint work with F.T. Farrell on the space of negatively curved metrics. They used the existence of exotic elements in topology (exotic spheres and the non-vanishing of certain homotopy groups of the space of stable pseudoisotopies of the circle) to show that the space of negatively curved metrics has complicated topology (if it is not empty). For instance they showed that the space of negatively curved metrics of any closed smooth manifold of dimension >9 is either empty or always has infinitely many path components.

Ontaneda’s most celebrated achievement concerns the existence of negatively curved Riemannian manifolds[2]. Prior to this work, every known example of a closed negatively curved Riemannian manifold of dimension greater than six was homeomorphic either to a hyperbolic manifold or a branched cover of such. The result for which Ontaneda is best known is his removal of singularities from a class of hyperbolized manifolds previously introduced by Charney and Davis, thereby making these manifolds negatively curved Riemannian (in fact with curvature arbitrarily close to −1). This gives an enormous array of examples. It was the culmination of a series of papers using ideas from analysis, topology and geometry.

In the media



  1. Farrell, F. T.; Ontaneda, P.; Raghunathan, M. S. (2000). "Non-Univalent Harmonic Maps Homotopic to Diffeomorphisms". Journal of Differential Geometry. 54 (2): 227–253. doi:10.4310/jdg/1214341646.
  2. "Riemannian Hyperbolization". Publications Mathmatiques de l'IHES. 121: 1-72. 2020.

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