| Grant number: | 06/02023-4 |
| Support Opportunities: | Regular Research Grants |
| Start date: | September 01, 2006 |
| End date: | August 31, 2008 |
| Field of knowledge: | Physical Sciences and Mathematics - Mathematics - Analysis |
| Principal Investigator: | Alexandre Casassola Gonçalves |
| Grantee: | Alexandre Casassola Gonçalves |
| Host Institution: | Faculdade de Filosofia, Ciências e Letras de Ribeirão Preto (FFCLRP). Universidade de São Paulo (USP). Ribeirão Preto , SP, Brazil |
Abstract
Partial Differential Equations on manifolds areusually treated with classical analytical techni-ques. The combination of those with geometricalstructures of the manifold may yield improvementsor even the development of new strategies toapproach thsese equations. The method we have been using studies a non-linear elliptic equationof the type $\Delta u +f(x)exp(2u)-\lambda=0$ ona Riemann Surface. The main results we have got-ten deal with uniqueness of solutions, but wealso proved existence or non-existence in somecases. The geometric traits are decisive to getthsese proofs, and it does not seem trivial toobtain them by using plain Analysis. The resultsof this research are in a paper [Go2] that isin the references of the project, and that has been already accepted for publication on "Diffe-rential Geometry and Its Applications". The con-tinuation of this work is the core of our project, and can be summarized in finding maximal para-meters for existence and/or uniqueness for theabove equation, using the cardinality of the divisor of the holomorphic section, whose squa-red norm is the very function $f(x)$. We alsostudy applications of these results to other geometric problems, like in the study of theisometric immersions of constant mean curvatureinto space-forms [GU]. (AU)
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