An elliptic geometric pde on a compact surface

• Grant number: 06/02023-4
• Support Opportunities: Regular Research Grants
• Duration: September 01, 2006 - 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)

Scientific publications

(References retrieved automatically from Web of Science and SciELO through information on FAPESP grants and their corresponding numbers as mentioned in the publications by the authors)

NASCIMENTO, ARNALDO S.; GONCALVES, ALEXANDRE C.. INSTABILITY OF ELLIPTIC EQUATIONS ON COMPACT RIEMANNIAN MANIFOLDS WITH NON-NEGATIVE RICCI CURVATURE. Electronic Journal of Differential Equations, MAY 8 2010. (06/02023-4)