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Planar doubling nodal solutions to the Yamabe equation with maximal rank

Yuanli Li, Liming Sun

Abstract

This article constructs two families of nodal solutions to the Yamabe equation, each concentrating along two planar circles. One family is conformally equivalent to the one previously obtained by Medina--Musso. The second family is a twisted variant of the first; it is new and is derived from ansatzes that are not Kelvin invariant, in contrast to a standard assumption in earlier works. In addition, in dimension 3, these solutions attain maximal rank. By means of a continuous family of conformal transformations, we then analyze the interaction of the two circles, which display a crossing phenomenon reminiscent, in some sense, of leap-frogging behavior in vortex dynamics.

Planar doubling nodal solutions to the Yamabe equation with maximal rank

Abstract

This article constructs two families of nodal solutions to the Yamabe equation, each concentrating along two planar circles. One family is conformally equivalent to the one previously obtained by Medina--Musso. The second family is a twisted variant of the first; it is new and is derived from ansatzes that are not Kelvin invariant, in contrast to a standard assumption in earlier works. In addition, in dimension 3, these solutions attain maximal rank. By means of a continuous family of conformal transformations, we then analyze the interaction of the two circles, which display a crossing phenomenon reminiscent, in some sense, of leap-frogging behavior in vortex dynamics.

Paper Structure

This paper contains 6 sections, 7 theorems, 208 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Conformally equivalent counterparts of the solutions
  3. A First Approximation
  4. Gluing Scheme
  5. Symmetry Decomposition
  6. Balancing the parameters

Key Result

Theorem 2

For $k$ sufficiently large, there exists a finite-energy solution $u$ to the Yamabe equation, which is of the form with $o_k(1)\to 0$ uniformly as $k\to \infty$. The parameters $\mu_m,r_m,\lambda_m, R_m$ are determined through the following relations: there exists a fixed positive number $\eta<1$ and some $\eta<l_m,t_m<\eta^{-1}$ such that the $\mu_m$ and $d_m:=R_m-r_m$ and Moreover, we have the

Figures (2)

  • Figure 1: The planar twin solutions
  • Figure 2: Clips of the movement of two circles under the transform $\Phi_a$ for various $a$. The bubbles of bar circle are denoted by filled dots, while that of hat circle are denoted by hallow dots.

Theorems & Definitions (14)

  • Definition 1
  • Theorem 2
  • Remark 3
  • Theorem 4
  • proof
  • Remark 5
  • Proposition 6
  • Proposition 7
  • Remark 8
  • Proposition 9
  • ...and 4 more