Tohoku University, Japan
Title: Harmonic maps and bi-harmonic maps on CR-manifolds and foliated Riemannian manifolds
Since G.Y. Jiang gave the first and second variation formulas for the bi-energy, the study of bi-harmonic maps has been developed by many geometers. B.Y. Chen proposed the conjecture that every bi-harmonic submanifold of the Euclidean space must be harmonic (minimal). The generalized B.Y. Chen¡¯s conjecture claims that every bi-harmonic submanifold of a Riemannian manifold of non-positive curvature must be harmonic (minimal). But, Y.L. Ou and L. Tang gave a counter example to this conjecture at 2012 in case of a submanifold in a negatively curved Riemannian manifold. On the other hand, we showed in joint works with N. Nakauchi that every bi-harmonic map from a complete manifold into a Riemannian manifold of non-positive curvature with finite energy and finite bi-energy must be harmonic. In our talk, that such rigidity of bi-harmonic maps still hold for CR-manifolds and foliated Riemannian manifolds will be shown.
Hajime Urakawa. He graduated at the undergraduate course at Tohoku University and at the master course at Osaka University, and has accomplished his doctoral degree of science at Nagoya University at 1977. He held an appointment at Nagoya Univ. at 1972 as an assistant professor, accepted an offer from Tohoku Univ. at 1978 as an associate professor, and became full professor at Tohoku Univ. since 1992, and professor emeritus and professor at Institute for Intern. Education, Tohoku Univ. since 2010. In 1979, he answered negatively to M. Berger¡¯s problem by giving a family of Riemannian metrics with a fixed volume whose first eigenvalues tend to infinity. In 1982, he answered to M. Kac¡¯s problem by giving two higher dimensional different shaped drums sounding the same tones. In 1988, he settled an equivariant Yang-Mills gauge theory in mathematical physics having an application producing a negative answer to the Atiyah-Jones conjecture. In 1993, he published ¡°Calculus of Variations and Harmonic Maps¡± (251pages) in the Amer. Math. Soc. He has published 99 mathematical journal papers cited in Math. Sci. Net., containing more than twenty papers in the recent five years.