# Anthony J Tromba

Title | Distinguished Professor |

Division | Physical & Biological Sciences |

Department | Mathematics Department |

Phone | 831-459-2794 |

FAX | 831-459-4511 |

Office | McHenry Room #4151 |

Campus Mail Stop | Mathematics Department |

1156 High Street Santa Cruz, CA 95064 |

### Research Interests

Anthony Tromba's research interests are in the applications of global nonlinear analysis to various problems in partial differential equations. His main research during the last several years has been directed toward various questions concerning properties of minimal surfaces, in space and in Riemannian manifolds. In particular, he is interested in the question of isolatedness of solutions. To attack these problems, several new methods have had to be evolved, methods that require a synthesis of ideas from differential topology, differential geometry, and partial differential equations.Another of Tromba's interests is in the development of an effective Morse theory for problems in the calculus of variations that involves more than one independent variable.

He is also interested in a modern formulation of TeichmŸller space from the point of view of Riemannian geometry, and its applications to minimal surfaces and physics. This approach constructs TeichmŸller space directly as a differentiable manifold, and in so doing, completely bypasses the notions of quasi-conformal maps, the Beltrami equation, and nonstandard elliptic theory. As a consequence of this approach, several geometric descriptions of TeichmŸller space as a differentiable manifold can be given.

### Biography, Education and Training

B.S., Cornell UniversityM.A., Ph.D., Princeton University

### Selected Publications

- L. Andersson, V. Moncrief, and A. Tromba: On the global evolution problem in 2+1 gravit, J. Geometry and Physics 23 (1997), 191-205.
- A. Tromba: On a natural affine connector on the space of almost complex structures and the curvature of TeichmŸller space with respect to its Weil-Petersson metric. Manu-scripta Math 56 (1996), 475-497.
- F. Tomi and A. Tromba: The index theorem for minimal surfaces of higher genus, Mem. Amer. Math Soc 117, No. 560 (1995).
- F. Tomi and A. Tromba: Existence theorems for minimal surfaces of non-zero genus spanning a contour. Mem. Amer. Math. Soc. 71, no. 382 (1988).
- A. Tromba: A general approach to Morse theory. J. Differential Geometry 12:47Ð85. 1 (1977)