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The structure of fully nonlinear equations and its applications to prescribed problems on complete conformal metrics

Rirong Yuan

Abstract

This paper investigates the structure of fully nonlinear equations and their applications to geometric problems. We solve some fully nonlinear version of the Loewner-Nirenberg and Yamabe problems. Notably, we introduce Morse theory techniques to construct admissible metrics under a weak condition on the underlying metric, which can be further relaxed in a broad setting. Furthermore, we provide some topological obstruction to demonstrate the optimality of our structural conditions.

The structure of fully nonlinear equations and its applications to prescribed problems on complete conformal metrics

Abstract

This paper investigates the structure of fully nonlinear equations and their applications to geometric problems. We solve some fully nonlinear version of the Loewner-Nirenberg and Yamabe problems. Notably, we introduce Morse theory techniques to construct admissible metrics under a weak condition on the underlying metric, which can be further relaxed in a broad setting. Furthermore, we provide some topological obstruction to demonstrate the optimality of our structural conditions.

Paper Structure

This paper contains 34 sections, 69 theorems, 298 equations.

Table of Contents

  1. Introduction
  2. The structure of nonlinear operators
  3. Prescribed problems on the conformal classes of complete metrics
  4. Conformal bending of closed manifolds
  5. A fully nonlinear version of Loewner-Nirenberg problem
  6. A complete noncompact version of fully nonlinear Yamabe problem
  7. Geometric optimal condition via topological obstruction
  8. Construction of admissible conformal metrics via Morse theory
  9. Construction of metrics on closed manifolds
  10. Construction of metrics on compact manifolds with boundary
  11. Construction of metrics on complete noncompat manifolds
  12. Description of the test cone
  13. Test cone
  14. Determine the maximal test cone
  15. On the structure of nonlinear operators
  16. ...and 19 more sections

Key Result

Theorem 1.2

Let $n\geq 2$, and let $\mathcal{S}$ be as in Definition def-testcone. Suppose, in addition to concave and elliptic-weak-3, that $\partial\Gamma^\sigma\subseteq\mathcal{S}$ for some $\sigma\in(\sup_{\partial\Gamma}f, \sup_\Gamma f).$ Then $\mathcal{C}^{\mathfrak{m}}_{\mathcal{S},f} =\bar{\Gamma}_{\

Theorems & Definitions (127)

  • Definition 1.1
  • Theorem 1.2
  • Remark 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Definition 1.6
  • Definition 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Theorem 1.10
  • ...and 117 more