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Verifying spiral minimal product structure through the Takahashi Theorem

Yongsheng Zhang

TL;DR

The paper addresses constructing spiral minimal products of $C$-totally real minimal immersions in spheres and verifying their minimality via the Takahashi theorem. It develops a two-input spiral construction based on a curve on $\mathbb{S}^3$ with components $\gamma_1,\gamma_2$, showing that under $C_1=-1$ the product preserves the $C$-totally real property and is amenable to iteration. The main result provides a detailed computation proving $\Delta_{g_\gamma} G_\gamma=-(k_1+k_2+1)G_\gamma$, implying the image lies on a sphere of radius $r$ with $r^2=\frac{k}{k_1+k_2+1}$ (where $k=1+k_1+k_2$), and yields a corollary describing all spiral minimal products with $\gamma_1\gamma_2\neq 0$. This framework connects to special Legendrian inputs and, via calibrations, to families of special Lagrangian cones and generalized Delaunay constructions.

Abstract

We verify the spiral minimal product structure through the Takahashi Theorem with full computational details which were omitted in [LZ].

Verifying spiral minimal product structure through the Takahashi Theorem

TL;DR

The paper addresses constructing spiral minimal products of -totally real minimal immersions in spheres and verifying their minimality via the Takahashi theorem. It develops a two-input spiral construction based on a curve on with components , showing that under the product preserves the -totally real property and is amenable to iteration. The main result provides a detailed computation proving , implying the image lies on a sphere of radius with (where ), and yields a corollary describing all spiral minimal products with . This framework connects to special Legendrian inputs and, via calibrations, to families of special Lagrangian cones and generalized Delaunay constructions.

Abstract

We verify the spiral minimal product structure through the Takahashi Theorem with full computational details which were omitted in [LZ].

Paper Structure

This paper contains 3 sections, 3 theorems, 35 equations.

Table of Contents

  1. Introduction
  2. Spiral minimal product
  3. Verifying spiral minimal product through the Takahashi Theorem

Key Result

Theorem 3.1

Let $(M, g)$ be a $k$-dimensional Riemannian manifold and $f:M\longrightarrow \Bbb{R}^{n+1}$ an isometric immersion. If there exists $\lambda\neq 0$ such that ${\boldmath {\Delta_{g}}}f=-\lambda f,$ then it follows that (1) $\lambda>0$; (2) $f(M)\subset S^n(r)$ where $r^2=\frac{k}{\lambda}$; (3) $f:

Theorems & Definitions (5)

  • Theorem 3.1: T
  • Theorem 3.2
  • proof
  • Corollary 3.3
  • Remark 3.4