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Barycenter technique for the higher order $Q$-curvature equation

Saikat Mazumdar, Cheikh Birahim Ndiaye

Abstract

Let $k\ge1$ be an integer, and $(M,g)$ be a smooth, closed Riemannian manifold of dimension $2k+1\le n\le 2k+3$, or $(M,g)$ be locally conformally flat of dimension $n\ge 2k+1$. Applying the Bahri-Coron barycenter method, we show the existence of a conformal metric with constant $Q$-curvature of order $2k$, or equivalently, the existence of a positive solution for the $2k$-th order $Q$-curvature equation involving the GJMS operator $P_{g}$. We only assume a natural positivity preserving condition on $P_{g}$ and do not suppose any condition on the sign of the {\emph{mass}} of $P_{g}$. In particular, we obtain existence without using a positive mass theorem.

Barycenter technique for the higher order $Q$-curvature equation

Abstract

Let be an integer, and be a smooth, closed Riemannian manifold of dimension , or be locally conformally flat of dimension . Applying the Bahri-Coron barycenter method, we show the existence of a conformal metric with constant -curvature of order , or equivalently, the existence of a positive solution for the -th order -curvature equation involving the GJMS operator . We only assume a natural positivity preserving condition on and do not suppose any condition on the sign of the {\emph{mass}} of . In particular, we obtain existence without using a positive mass theorem.
Paper Structure (3 sections, 2 theorems, 23 equations)

This paper contains 3 sections, 2 theorems, 23 equations.

Table of Contents

  1. Introduction
  2. Notations and preliminaries
  3. Bubbles and Error estimate

Key Result

Theorem 1.1

Let $k\ge1$ be an integer, and $$M,g$$ be a smooth, closed Riemannian manifold of dimension $2k+1\le n\le 2k+3$ or $(M,g)$ be locally conformally flat. Assume that the $2k$th order GJMS operator $P_{g}$ satisfies the positivity condition positivity. Then eq. eq:one has a smooth positive solution for

Theorems & Definitions (3)

  • Theorem 1.1
  • Remark 1.1
  • Proposition 2.1