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A Condition in Mean Curvature Prescriptions for Conformal Metrics on the Ball

Alvaro Ortiz, Gonzalo Garcia

Abstract

This paper considers the prescribed zero scalar curvature and mean curvature problem on the n-dimensional Euclidean ball for $n \geq 3$. Given a rotationally symmetric function $H:\partial B^{n}\rightarrow R$, in this work, we will prove that if $H'(r)$ changes signs where $H>0$ and $H(r)$ also satisfies a flatness condition then there exists a metric $g$ conformal to the Euclidean metric, with zero scalar curvature in the ball and mean curvature $H$ on its boundary.

A Condition in Mean Curvature Prescriptions for Conformal Metrics on the Ball

Abstract

This paper considers the prescribed zero scalar curvature and mean curvature problem on the n-dimensional Euclidean ball for . Given a rotationally symmetric function , in this work, we will prove that if changes signs where and also satisfies a flatness condition then there exists a metric conformal to the Euclidean metric, with zero scalar curvature in the ball and mean curvature on its boundary.

Paper Structure

This paper contains 9 sections, 15 theorems, 110 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Sub-critical Case
  4. Estimates on $J_{p}$
  5. Proof of the Sub-critical Case.
  6. A priory Estimates
  7. $H$ Negative and away from zero
  8. $H$ Small and close to zero
  9. $H$ Positive and away from zero

Key Result

Theorem 1.1

Let $n\geq 3$ and let $H = H(r)$ be a smooth function on $\partial B^{n}$ symmetric along the $x_{n}$ axis. Assume that $H$ has at least two positive local maximums and satisfies a flatness condition near every critical point $\tau_{0}$ as follows. $H(r) = H(\tau_{0})+ a|r-r_{0}|^{\alpha}+k(|r-r_{0} where $1<p\leq \frac{n}{n-2}$ and $k'(s) = o(s^{\alpha-1})$, have a smooth positive solution.

Theorems & Definitions (22)

  • Theorem 1.1
  • Definition 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Proposition 3.1
  • proof
  • Lemma 3.2
  • Proposition 3.3
  • ...and 12 more