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Fully nonlinear prescribed curvature problems on closed manifolds with negative curvature

Jiaogen Zhang

TL;DR

This work studies fully nonlinear prescribed curvature problems in the negative curvature regime on closed manifolds via the modified Schouten tensor $A_g^t$ and a broad class of concave elliptic operators. Recasting the problem as a scalar PDE $F(W[u])=\psi e^{2u}$, the authors establish robust a priori estimates ($C^0$, gradient, and $C^2$) under Condition $\mathscr{T}$, then solve by a continuity method to obtain existence and uniqueness of a conformal metric within the given conformal class. A key contribution is showing solvability for all $t<1$ under Condition T, with the result holding for all $O(n)$-invariant Gårding-Dirichlet operators, thereby generalizing classical results such as Gursky–Viaclovsky. The paper further applies the framework to the prescribed $\mathcal{M}_p$-curvature problem, providing a unified treatment that encompasses a broad family of curvature functionals between scalar and Ricci curvatures.

Abstract

In this manuscript, we investigate fully nonlinear prescribed curvature problems for the modified Schouten tensor on closed Riemannian manifolds with negative curvature. We prove that whenever the corresponding concave elliptic operator satisfies a structural Condition $T$, which encompasses all $O(n)$-invariant Gårding-Dirichlet operator, such prescribed curvature problems are always solvable.

Fully nonlinear prescribed curvature problems on closed manifolds with negative curvature

TL;DR

This work studies fully nonlinear prescribed curvature problems in the negative curvature regime on closed manifolds via the modified Schouten tensor and a broad class of concave elliptic operators. Recasting the problem as a scalar PDE , the authors establish robust a priori estimates (, gradient, and ) under Condition , then solve by a continuity method to obtain existence and uniqueness of a conformal metric within the given conformal class. A key contribution is showing solvability for all under Condition T, with the result holding for all -invariant Gårding-Dirichlet operators, thereby generalizing classical results such as Gursky–Viaclovsky. The paper further applies the framework to the prescribed -curvature problem, providing a unified treatment that encompasses a broad family of curvature functionals between scalar and Ricci curvatures.

Abstract

In this manuscript, we investigate fully nonlinear prescribed curvature problems for the modified Schouten tensor on closed Riemannian manifolds with negative curvature. We prove that whenever the corresponding concave elliptic operator satisfies a structural Condition , which encompasses all -invariant Gårding-Dirichlet operator, such prescribed curvature problems are always solvable.

Paper Structure

This paper contains 10 sections, 10 theorems, 87 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Condition T
  4. Ellipticity of the prescribed curvature problems
  5. A priori estimates
  6. Zeroth order estimate
  7. First order estimate
  8. Second order estimate
  9. Proof of Theorem \ref{['main']}
  10. Prescribed $\mathcal{M}_p$-curvature problem

Key Result

Theorem 1.1

Suppose the concave elliptic operator $(f,\Gamma)$ satisfies hypotheses $(f1)-(f4)$ and Condition T. Let $(M,g)$ be a closed Riemannian manifold with $\lambda(-g^{-1}A_{g}^{t})\in \Gamma$ for some $t<1$, and let $\psi$ be a smooth positive function on $M$. Then there exists a unique conformal metric

Theorems & Definitions (18)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Proposition 3.1
  • ...and 8 more