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Fully nonlinear Yamabe-type problems on non-compact manifolds

Jonah A. J. Duncan, Yi Wang

TL;DR

The paper studies conformal deformations on non-compact manifolds that solve fully nonlinear Yamabe-type equations governed by the eigenvalues of the Schouten tensor, both in negative and positive regimes. It develops an exhaustion-based framework, leveraging Dirichlet boundary-value problems and local gradient/Hessian bounds to obtain complete metrics with $\lambda(-g^{-1}A_g)\in\overline{\Gamma}$ or $\partial\Gamma$, and, in the positive case, uses Guan–Guo-type results and barrier arguments to produce solutions of $f(\lambda(g^{-1}A_g))=\psi$ under suitable admissibility and decay assumptions. The work furnishes explicit geometric examples with warped-product ends and asymptotically flat ends, including Schwarzschild-type metrics, and demonstrates how perturbations yield nontrivial conformal classes supporting boundary- or viscosity-solution metrics. Together, these results advance the understanding of fully nonlinear conformal curvature problems on non-compact manifolds and connect to σ_k-Yamabe theory and geometric notions of mass.

Abstract

We obtain existence results for a class of fully nonlinear Yamabe-type problems on non-compact manifolds, addressing both the so-called positive and negative cases. We also give explicit examples of manifolds with warped product ends and asymptotically flat ends satisfying the hypotheses of our theorems.

Fully nonlinear Yamabe-type problems on non-compact manifolds

TL;DR

The paper studies conformal deformations on non-compact manifolds that solve fully nonlinear Yamabe-type equations governed by the eigenvalues of the Schouten tensor, both in negative and positive regimes. It develops an exhaustion-based framework, leveraging Dirichlet boundary-value problems and local gradient/Hessian bounds to obtain complete metrics with or , and, in the positive case, uses Guan–Guo-type results and barrier arguments to produce solutions of under suitable admissibility and decay assumptions. The work furnishes explicit geometric examples with warped-product ends and asymptotically flat ends, including Schwarzschild-type metrics, and demonstrates how perturbations yield nontrivial conformal classes supporting boundary- or viscosity-solution metrics. Together, these results advance the understanding of fully nonlinear conformal curvature problems on non-compact manifolds and connect to σ_k-Yamabe theory and geometric notions of mass.

Abstract

We obtain existence results for a class of fully nonlinear Yamabe-type problems on non-compact manifolds, addressing both the so-called positive and negative cases. We also give explicit examples of manifolds with warped product ends and asymptotically flat ends satisfying the hypotheses of our theorems.
Paper Structure (7 sections, 13 theorems, 90 equations)

This paper contains 7 sections, 13 theorems, 90 equations.

Table of Contents

  1. Introduction
  2. Existence results in the negative case
  3. Proof of Theorem \ref{['H']}
  4. Further existence results for possibly incomplete metrics
  5. Examples in the negative case
  6. Existence results in the positive case
  7. Examples in the positive case

Key Result

Theorem 1.1

Suppose $(f,\Gamma)$ satisfies 21'--24' and $\mu_\Gamma^+>1$. Let $(M^n,g_0)$ be a smooth complete non-compact manifold such that, for some constant $c>0$ and some compact set $K_0\subset M$, it holds that $f(\lambda(-g_0^{-1}A_{g_0})) \geq c>0$ and $\lambda(-g_0^{-1}A_{g_0})\in\Gamma$ on $M\backsla

Theorems & Definitions (25)

  • Theorem 1.1
  • Remark 1
  • Remark 2
  • Remark 3
  • Corollary 1.2
  • Corollary 1.3
  • Remark 4
  • Theorem 1.4
  • Remark 5
  • Corollary 1.5
  • ...and 15 more