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Geometric Analysis and PDEs pp 1–33Cite as

PDEs in Conformal Geometry

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Part of the book series: Lecture Notes in Mathematics ((LNMCIME,volume 1977))

Abstract

In these lectures I will discuss two kinds of problems from conformal geometry, with the goal of showing an important connection between them in four dimensions. The first problem is a fully nonlinear version of the Yamabe problem, known as the σk-Yamabe problem. This problem is, in general, not variational (or at least there is not a natural variational interpretation), and the underlying equation is second order but possibly not elliptic. Moreover, in contrast to the Yamabe problem, there is very little known (except for some examples and counterexamples) when the underlying manifold is negatively curved.

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Authors and Affiliations

  1. Department of Mathematics, University of Notre Dame, Research supported in part by NSF Grant DMS-0500538., 46556, Notre Dame, IN

    Matthew J. Gursky

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  1. Matthew J. Gursky
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Correspondence to Matthew J. Gursky .

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Editors and Affiliations

  1. Dept. Mathematics, Princeton University, Washington Rd., Princeton, 08544-1000, U.S.A.

    Sun-Yung Alice Chang

  2. Department of Mathematics, SISSA, Via Beirut 2-4, Trieste, 34014, Italy

    Antonio Ambrosetti

  3. Studi Avanzati (SISSA), Scuola Internazionale Superiore di, via Beirut 2-4, Trieste, 34014, Italy

    Andrea Malchiodi

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Gursky, M.J. (2009). PDEs in Conformal Geometry. In: Chang, SY., Ambrosetti, A., Malchiodi, A. (eds) Geometric Analysis and PDEs. Lecture Notes in Mathematics(), vol 1977. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-01674-5_1

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